9. S. Feferman - Por Qué Los Programas de Nuevos Axiomas Deben Ser Cuestionados

TheBul l et in of Symbol ic LogicVolume 6, Number 4, Dec. 2000

DOES M ATHEM ATICS NEED NEW AX IOM S?

SOLOM ON FEFERM AN , HARVEY M . FRIEDM AN , PENELOPE M ADDY , AND JOHN R. STEEL

Does mathematics need new axioms? was the second of three plenarypanel discussionsheld at the ASL annual meeting,ASL 2000, in Urbana-Champaign, in June, 2000. Each panelist in turn presentedbrief openingremarks, followed by a secondround for respondingto what the othershadsaid; the session concluded with a lively discussion from the oor. Thefour articles collected here representreworked and expanded versions ofthe rst two parts of those proceedings,presented in the same order asthe speakersappeared at the original panel discussion:Solomon Feferman(pp. 401413), PenelopeMaddy (pp. 413422), John Steel(pp. 422433),and Harvey Friedman (pp. 434446). The work of eachauthor is printedseparately, with separatereferences,but theportions consistingof commentson and repliesto othersareclearly marked.

PenelopeMadd y

WHY THE PROGRAMS FOR NEW AXIOMSNEED TO BE QUESTIONED

SOLOM ON FEFERM AN

The point of departure for this discussion is a somewhat controversialpaper that I published in the American M athematical M onthly under theti tle Does mathematics need new axioms? [4]. The paper itself was basedon a lecture that I gave in 1997 to a joint sessionof the American Mathe-matical Societyand the Mathematical Association of America, and it wasthus written for a general mathematical audience. Basically, it was in-tended as an assessment of G odels program for new axioms that he hadadvanced most prominently in his 1947 paper for the Monthly , entitledWhat is Cantors continuumproblem? [7]. My paper aimed to be an as-sessmentof that program in the light of research in mathematical logic inthe intervening years, beginning in the 1960s,but especially in more recentyears.

Received September10,2000;revisedOctober 10,2000.

c! 2000, Association for Symbolic Logic1079-8986/ 00/ 0604-0001/ $5.60

401

402 S. FEFERM AN, H . FRIEDM AN, P. M ADDY, AND J. R. STEEL

In my presentation here I shall be following [4] in its main points, thoughenlarging on someof them. Somepassagesare even taken almost verbatimfrom that paper where convenient, though of course all expository back-ground material that wasnecessarythere for a general audienceis omitted.1

For a logical audience I have written before about various aspectsof thequestionsdealt with here, most particularly in the article Godels programfor new axioms: Why, where, how and what? [2] and prior to that in Innityin mathematics. IsCantor necessary? (reprinted asChs. 2 and 12 in [3]).

My paper [4] openedas follows:

The question, Does mathematics neednew axioms?, isambigu-ous in practically every respect.

What do wemean by mathematics ? What do wemeanby need? What do we meanby axioms? You might even ask, What

do wemeanby does?

Amusingly, this was picked up for commentby The New Yorker in its issueof May 10,1999,in oneof its little end llers (op. cit., p. 50), as follows:

Ne w apparently speaksfor itself.

I had to admit, they had me there.Part of themultiple ambiguitiesthat weseein theleadingquestionherelies

in thevariouspointsof view from which it might beconsidered. Thecrudestdi!er encesare between the point of view of the working mathematiciannot in logic related elds (under which are counted, roughly, 99% of allmathematicians), thenthat of themathematical logician, and, nall y, that ofthephilosopherof mathematics. Even within eachof theseperspectivesthereare obviously divergent positions. M y own view is that the question is anessentially philosophical one: Of coursemathematicsneedsnewaxioms weknow that from G odels incompletenesstheoremsb ut then the questionsmust be: Which ones?and Why those?

Lets start by making somepreliminary distinctions as to the meaningofaxiom. TheOxford EnglishDictionary denes axiom asusedin logic andmathematicsby: A self-evident proposition requiring no formal demonstra-tion to prove its truth, but received and assentedto assoon asmentioned.I think its fair to say that something like this de nition is the rst thing wehave in mind when we speakof axioms for mathematics: this is the idealsenseof theword. I tssurprising how far themeaningof axiom hasbecomestretched from the ideal sensein practice, both by mathematicians and lo-gicians. Somehave even taken it to mean an arbitrary assumptionand sorefuseto takeseriously what statusaxiomsare to hold.

1The parts taken from [4] are reprinted here with the kind permission of the AmericanM athematical M onthly.

DOES M ATHEM ATICS NEED NEW AX IOM S? 403

When the working mathematician speaksof axioms, he or she usuallymeansthosefor someparticular part of mathematics suchasgroups, rings,vector spaces,topological spaces,H ilbert spaces,and so on. Thesekindsof axioms have nothing to do with self-evident propositions, nor are theyarbitrary starting points. They are simply de nitions of kinds of structureswhich have beenrecognized to recur in various mathematical situations. Itake it that thevalueof thesekindsof structural axiomsfor theorganizationof mathematical work isnow indisputable.

In contrast to the working mathematicians structural axioms, when thelogician speaksof axioms, heor shemeans, rst of all, lawsof valid reasoningthat aresupposedto apply to all partsof mathematics,and, secondly, axiomsfor such fundamentalconceptsasnumber, set and function that underlie allmathematical concepts;theseareproperly called foundationalaxioms.

The foundational axiomscorrespondto suchbasicpartsof our subjectthatthey hardly needany mention at all in daily practice, and many mathemati-cianscan carry on without calling on themeven once. Somemathematicianseven question whether mathematicsneedsany axiomsat all of this type: forthem, so to speak, mathematics is as mathematics does. According to thisview, mathematics isself-justifying,andany foundational issuesarelocalandresolved according to mathematical need, rather than global and resolvedaccording to possibly dubious logical or philosophical doctrines.

One reasonthe working mathematician can ignore the question of needof foundational axioms and I think that we [members of the panel] areall agreedon thisis that the mathematics of the 99% group I indicatedearlier can easily be formalized in ZFC and, in fact, in much weaker sys-tems. Indeed,research in recentyearsin predicativemathematics and in theReverseMathematicsprogram shows that the bulk of it can be formalized insubsystemsof analysishardly stronger than " 11-CA ,

2 and moreover thesci-enticall y applicable part canbe formalized in systemsconservativeover PAand even much weaker systems.3 So, foundationally, everyday mathematicsrests in principle on unexceptionable grounds.

Before going on to the perspectives of the mathematical logician andthe philosopher of mathematics on our leading question, lets return toG odels program for new axioms to settle undecidedarithmetical and set-theoretical problems. Of course, the part of G odels program concerningarithmetical problemsgoesback to his fundamental incompletenessresults,as rst indicated in ftn. 48aof his famous1931paper [6]. I t was there thatG odel assertedthe true reasonfor incompletenessto be that the formation

2For predicativemathematics, cf. [3], Chs. 13and 14; for ReverseM athematics, cf. Simpson[19].

3Cf. [3], Chs. 13 and 14 for theclaim about PA . Feng Yehasshown in hisdissertation [20]that substantial portions of scienti cally applicable functional analysis can be carried outconstructively in a conservativeextension of PRA .


of ever higher types can be continued into the transnite; he repeatedthis reason periodically since then, but did not formulate in print the exactnature of such further axioms. An explicit formulation of the program inpursuit primaril y of settling CH only appearedin the1947article on CantorsContinuum Problem (and its1964revision in the light of subsequentevents).It wasalso there that G odelmadethedistinction betweennew axiomsbasedon intrinsic reasons and those based on extrinsic reasons. Concerning theformer he pointed to axioms of M ahlo type, of which he said that theseaxioms show clearly, not only that the axiomatic system of set theory asknown today is incomplete, but also that it can be supplemented withoutarbitrarinessby new axiomswhich areonly thenatural continuation of thosesetup so far. ([7], p. 520) SinceG odel thought CH is falseand recognizedthat Mahlo-type axioms would be consistent with V = L , he proposedother reasonsfor choosingnew axioms;hopefully, thesewould be basedonhitherto unknown principles . . . which a more profound understanding ofthe conceptsunderlying logic and mathematicswould enableus to recognizeas implied by these concepts , and if not that, then one should look foraxioms which are so abundant in their veria ble consequences. . . thatquite irr espective of their intrinsic necessitythey would have to be assumedin the samesenseasany well-establishedphysical theory. ([7], p. 521)

My co-panelistsare better equipped than I to report on the subsequentprogress on G odels program in the caseof set theory.4 Briey , the researchin this direction has concentrated primaril y on higher axioms of innity ,also known as large cardinal axioms (LCAs). Theseare divided roughlybetweenthe so-called small largecardinals suchasthosein theMahlo hi-erarchiesof inaccessiblecardinals,and the large largecardinals,a divisionthat correspondsroughly to existenceaxiomsacceptedon intrinsic grounds(or consistentwith V = L ) and thoseacceptedon extrinsic grounds. Thedivision is not a sharp one but falls somewhere below the rst measurablecardinal.5 By all accounts from thespecialists, thehigh point in thedevelop-ment of large largecardinal theory is the technically very impressiveworkextending nice properties of Borel and analytic sets, such as Lebesguemeasurability, the Baire property, and the perfect subsetpropertyvia thedeterminatenessof associated innitary gamesto arbitrary setsin the pro-jectivehierarchy, all under theassumptionof the existenceof innitel y manyWoodin cardinals.6

But the striking thing, despite all such progress, is thatcontrary toG odels hopes the Continuum Hypothesis is still completely undecided,

4That was indeed stressed in thepresentations of M addy and Steel.5Cf. K anamori [11], p. 471.6This is due, cumulatively, to the work of many leading workers in higher set theory; cf.

M artin and Steel [15] and Steels presentation to this panel discussion, for the history ofcontributions.


in the sensethat it is independentof all remotely plausibleaxiomsof innity ,including all large large cardinal axioms which have beenconsidered sofar.7 In fact, it is consistentwith all thoseaxiomsif they are consistentthat the cardinal number of thecontinuum isanything it ought to be, i.e.,anything which isnot excluded by K onigsTheorem.8 That may leadone toraise doubts not only about G odels program but its very presumptions. Isthe Continuum Hypothesis a de nite problem as G odel and many currentset-theoristsbelieve?

Heresa kind of testof onesviewsof that: ashasbeenwidely publicized,aClay M athematicsInsti tute that hasrecently been established in Cambridge,Massachusettsis o!ering what it calls Millennium Prizesof $1,000,000eachfor thesolution of seven outstandingopen mathematical problems, includingP = NP, theRiemann Hypothesis, the Poincare conjecture, and so on. Butthe Continuum Problem is not on that list. Why not? I ts one of the fewin H ilberts list from one hundred years ago thats still open. Would youfeel condent in going to the scientic board of that institute and arguingthat the Continuum Problem has unaccountably been left o!, and that itssolution, too, should beworth a cool million?

M y own view as iswidely known is that theContinuum Hypothesis iswhat I havecalled an inherently vague statement, and that thecontinuumitself, or equivalently the power setof the natural numbers, is not a denitemathematical object. Rather, its a conception we have of the totality ofarbitrary subsetsof thesetof natural numbers,a conception that is clearenough for us to ascribemany evident properties to that supposedobject(suchas the impredicativecomprehensionaxiom scheme)but which cannotbe sharpenedin any way to determine or x that object itself. On my view,it follows that the conception of the whole of the cumulative hierarchy, i.e.,the transnitel y cumulatively iterated power set operation, is even more soinherently vague, and that one cannot in general speak of what is a fact ofthe matter under that conception. For example, I deny that it isa fact of thematter whetherall projectivesetsareLebesguemeasurableor have theBaireproperty, and so on.

What thenon this viewe xplains the common feelingthat set theory issucha coherent and robust subject, that our ordinary set-theoretical intu-itions are a reliable guide through it (as in any well acceptedpart of mathe-matics), and that thousandsof interesting and prima facieimportant resultsabout setswhich wehaveno reasonto doubt havealready beenestablished?

7Interestingly, the only detailed approach we know of to settle CH that G odel himselftried rst negatively and then positively wasnot via axioms for largecardinalsbut rathervia proposed axioms on scales of functions between alephs of nite index. Whatever themerits of those axioms qua axioms, his attempted proofs (c. 1970)using them proved to bedefective; cf. [8], pp. 405425.

8Cf. M artin [14]. Thesituation reported there in 1976 is unchanged to date.


Well, I think that only shows that in settheory as throughout mathematics,a littlebit goesa long way in other words, that only thecrudest featuresofour conception of the cumulative hierarchy are neededto build a coherentand elaborate body of results. M oreover, one can expect to make steadyprogress in expanding this body of results, but even so there will always liebeyond this a permanently grey area in which such problemsas that of thecontinuum fall.

While G odels program to nd new axioms to settle the Continuum Hy-pothesishas not beenand will likely never ber ealized, what about theorigins of his program in the incompletenessresults for consistent formalsystemsextendingnumber theory? Throughout his life G odel said wewouldneed new, ever-stronger set-theoretical axioms to settle open arithmeticalproblemsof even the simplest, purely universal form, problems that he fre-quently referred to asof Goldbach type. But the incompletenesstheoremby itself gives no evidencethat any openarithmetical problemsor equiva-lently, nite combinatorial problemsof mathematical interest will requirenew such axioms.

Were all familiar with the fact that the " 01 statementshown undecidableby the rst incompletenesstheorem for a given formal systemS (containingarithmetic) is cooked up by a diagonal construction, while the consistencystatement Con(S) shown independentby the second incompletenesstheo-rem is of denite metamathematical interest, but not of mathematical inter-est in the usual sense. A lso familiar is the work of Paris and Harringtonproving the independencefrom PA of a special nite version of RamseysTheorem, and, beyond that, the work of Harvey Friedman proving the in-dependenceof a nite version of K ruskals Theorem from a moderatelyimpredicative system and of an ExtendedK ruskal Theorem from the sys-tem of " 11-Comprehension.

9 Each of theseis a " 02 statement shown trueby ordinary mathematical means(i.e., in a way understandable to mathe-maticians without invoking any mention of what axioms they depend on,or of any metamathematical notions) and is established to be independentof the respective S by showing that it implies (or is even equivalent to) the1-consistencyof S, 1- Con(S).10

For a number of years, Friedman has been trying to go much farther,by producing mathematically perspicuousnite combinatorial statements!whose proof requires the existence of many M ahlo cardinals and even ofstrongeraxiomsof innity (lik e thosefor theso-calledsubtle cardinals), and

9Cf. Paris and Harrington [16] and, for results related to K ruskals Theorem, Simpson[19], p. 408.

101-Con(S) is the statement of " -consistency of S restricted to #01 sentences; in otherwords, it says that each such sentence provable in S is true.


hehascomeup with variouscandidatesfor such ! .11 From thepoint of viewof metamathematics, thiskind of result isof thesamecharacteras theearlierwork just mentioned;that is, for certain very strong systemsS of settheory,the ! producedis equivalent to (or isslightly stronger than) 1- Con(S). Buttheconclusion to bedrawn isnot nearly asclear as for theearlier work, sincethe truth of ! is now not a result of ordinary mathematical reasoning,butdependsessentially on accepting 1-Con(S). In my view, it is begging thequestion to claim this shows we needaxioms of largecardinals in order todemonstrate the truth of such ! , since this only shows that we need their1-consistency. However plausible we might nd the latter for one reasonoranother, it doesnt follow that we should accept thoseaxioms themselves as rst-classmathematical principles.

My point here is simply that there is a basicdi!er encebetweenacceptingsystemssuch as ZFC + LCA , where LCA is the applicable large cardinalaxioms,andaccepting1-Con(ZFC + LCA). As to thequestionof theneedoflargecardinal assumptionsto settlenite combinatorial problemsof thesortproducedby Friedman, there is thus, in my view, an equivocation betweenneedinga given axiom and needingits 1-consistency;it is only the latter thatis demonstrated by his work. But if one does not grant that there is a factof the matter whetherstatementsLCA of various large cardinal axiomsaretrue, is therea principled reasonfor accepting1-Con(ZFC + LCA) withoutacceptingZFC + LCA itself? Of course, if one does think that there is afact of the matter as to whether such statements LCA are true, then theequivocation is a non-issue. But then, what is it that leads one to recognizeLCA rather than its negation to be true?

Returning to the question of mathematical interest, there is not a shredof evidenceso far that we will needanything beyond ZFCor even muchweaker systemsto settle outstanding combinatorial problemsof interestto the working mathematician, such as thoseon the Millennium Prize list,nor is thereany evidencethat the kind of metamathematical work weveseenfrom Paris-Harrington to Friedman will bearany relevanceto the solutionsof theseproblems,if theyareever solvedat all.

Thus,asI saidat theoutset, I think weareleft to regard thequestion: Doesmathematicsneednew axioms?,asprimaril y aphilosophicalone. And if youagreewith meon that, then wehave thediscouraging conclusion that wecanexpectasmany answersto thequestionastherearevarietiesof thephilosophyof mathematics; among those that have been seriously supported in onequarter or another, we have the platonic-realist, structuralist, naturalist,

11In [4] I referred to Friedman [5] for his then most recent work in that direction. M orerecently, Friedman has been promoting rather di! erent statements derived from Booleanrelation theory; cf. my comments on FriedmansU rbana presentation below.


predicativist, constructivist, and formalist philosophies.12 In other words, ifthe problem is indeeda philosophical one, we can hardly expect an answerthat will commandanywhereneargeneralassent.

But as a mathematical logician, if not as a working mathematician orphilosopher of mathematics, I can end with a bit more positive conclusion.Even if mathematics doesnt convincingly neednew axioms, it may needforinstrumental and heuristic reasonsthe work that hasbeendoneand contin-ues to bedone in higher set theory. For example, in my own subject prooftheoryanalo guesof large cardinal notions have proved to be very impor-tant in theconstruction of recursiveordinal notation systemsfor theor dinalanalysis of various subsystemsof analysisand admissible set theory.13 Sofar, thesejust employ symbols that act in the notation systemslike smalllarge cardinals, and do not depend on the assumption that such cardinalsactually exist. The widespread appearance of analogue large cardinal no-tions (and, more generally, large set notions) also in admissible set theory,constructive set theory, constructive type theory and my own systemsofexplicit mathematics14 suggeststhat thereshouldbea generaltheoryof suchnotions which includesall theseas special cases. So far, theseanaloguescorrespond mostly to small largecardinals. A t any rate, without thecon-siderablework in higher set theory that led to suchnotions, theseother areasof mathematical logic might still beback where theywere in theearly 1960s.It remainsto beseenwhetherthebulk of that work, which ison lar ge largecardinals, can havesimilar applications, and if not why not.

COM M ENTSAND RESPONSES

Responseto M addy. M addy argues from a position that she calls thenaturalistic point of view asto the philosophy of mathematics.15 Accordingto this, mathematical practice, and set-theoretical practice in particular, isnot in need of philosophical justi cation. Justi cation . . . comes fromwithin . . . in . . . terms of what meansare most e!ecti ve for meeting therelevant mathematical ends. Philosophy follows afterwards, as an attemptto understandthepractice, not to justify or to criticiz e it. From that pointof view, theoriginal panelquestion is a bit o! target. Rather, it would bemoreappropriate to askwhetheror not someparticular axiom . . . would orwould not help thisparticular practice . . . meetoneor moreof its particular

12Perhaps one should even add the philosophy that is implicit in the view that categorytheory provides theproper foundation of mathematics. What to call it?

13Cf., e.g., Pohlers [17] and Rathjen [18].14For these, cf. Aczel and Richter [1], Gri! or and Rathjen [9], and Jager and Studer [10].15M addy haselaborated thisposition in [13]. That isincidentally a retreat from her attempt

in [12] to formulatea compromisebetween G odelian platonic realism and Quinean scienti crealism (one form of naturalism) that would justify current higher set-theoretical practice.


goals. Theexamplegiven,from contemporary set theory, istheassumptionof many Woodin cardinals.

The naturalistic point of view in philosophy, as usually described, is thattheentities to beadmitted arejust thosepositedby andstudied in thenaturalsciences,and that the methodsof justica tion and explanation aresomehowcontinuous with thoseof the natural sciences.16 One of the foremost expo-nents of the naturalist position in this sense is Quine and, according to hisview, only so muchof mathematics is justied asis indispensable to scienticpractice. Thus, M addys use of naturalism to describe her point of view isstrikingly contrary to that, sinceher aim, above all, is to account for and insomesensegive approbation to that part of current set-theoretical practicewhich acceptsvarious largecardinal axioms(that happen to be inconsistentwith V = L , among other things). This she doesby taking mathematics ingeneraland set theory in particular as a science to be studied in its ownright , independently of its relationships to the natural sciences.

While Maddy keepsinvoking mathematical practicein general in thescopeof her naturalism, shedoesnot reect on the many instancesin its historyin which the question of what entities are to be admitted to mathematicsand what methods are legitimate had to be faced, leading to substantialrevisions from whats OK to whats not OK and vice versa. In bindingitself to mathematical practice, this kind of naturalism is in dangerof beingunduly transitory. Even if one takes theproposed naturalistic point of viewand mathematical practice as exemplied in set theory for granted, there isa crucial questionas to what determinesthe ma thematical ends for whichthe most e!ective meansare to be sought. And having chosenthe ends,in what sensedoese!ecti venessjustify the means?Why is it to bepresumedthat the good propertiesof Borel and analytic setsshould generalize to allprojectivesets,given that theydont hold for all sets?

Maddy says her naturalist neednt concern herselfwith w hether the CHhasa determinate truth value in some Platonic world of sets or confrontthe question of whether or not it is inherently vague. Why then is itassumed that therehas to be a determinateanswer to whether all projectivesetshave the perfectsubsetproperty or the property of Baire, etc.? Is theresomethingessentially di!er ent about the character of theseset-theoreticalproblemsthat makes the latter determinate but not (necessarily) the former?At any rate, admitting the possibility of somekind of indeterminatenessforCH seemsto me to bea slippery slopefor thenaturalist.

As a nal point , M addy suggeststhat I m in favor of limiting mathematics,though the essenceof pure mathematics is its freedom. Surely she doesnot think that anything goesin mathematics. Old-style innitesimals, D iracdelta-functions,unrestrictedcomprehension? I f not, what justi eswhat istobeadmitted to mathematics? Once it isagreed that therehas to besomesort

16Cf. (R. Audi, ed.) TheCambridgeDictionary of Philosophy, 2nd ed. (1999), p. 596.


of justica tion, intrinsic or extrinsic, then one is in the game of potentiallylimiting mathematics in some way or other. I dont hew to any sort ofabsoluteprinciple in favor of limiting mathematics.

Response to Steel. In hisdiscussion of my contention that thecontinuumproblem is inherently vague, Steel says that if the language of 3rd orderarithmetic [in which it is couched]permits vague or ambiguoussentences,then it is important to trim or sharpenit so as to eliminate these. . . it maybe that, in the end, our solution to the Continuum Problem is bestseenasresolving someambiguity.

It is useful, in response, to elaborate my ideas about vaguenessmoregenerally. These can be illustrated, to begin with, in the context of veryfamiliar , set-theoretically low down mathematics. The conception of thestructure N of the natural numbers is not a vague one (at leastin my view);statementsabout N have a denite truth value, and the axioms of PA areamong those and (on reection) are evident for it. By comparison, thenotion of feasible (or feasibly computable) natural number is a vague one,and inherently so; there is no reasonableway to make it denite . Though wemight well admit certainstatementsabout feasiblenumbersasbeingevident,e.g., if n and m are feasible, so is n + m, we cannot speakof truth or falsityof statementsabout feasible numbersin general. Nevertheless,the notion offeasibly computable number is su$ciently suggestive to act asa heuristic fora reasonable mathematical theory. Similarly, the notion of random numberbetween 1 and 10 is vague, but the conception of it makes it evident thatthe probability of such a number being lessthan 6 is 1/ 2. It is from suchvague beginningsthat substantial, coherent, and even robust mathematicaltheoriescanbedevelopedwithout committing oneselfto a notion of truthasto the notions involved.

In the caseof set theory, it is at the next level (over N) that issues of ev-idence, vagueness,and truth arise. Once the conception of the structure ofarbitrary sets of natural numbers is presentedto us and we reect on it , theaxiomsof second-order arithmetic (anal ysis) becomeevident for it. Nev-ertheless,in my view, themeaningof arbitrary subsetof N isvague, and soI would strongly resist talking about truth or falsity of analytic statements.In opposition to my view it might be argued that the structure of the con-tinuum, whenconceived geometrically, is not vague, and hencethat analyticstatementshavea denite truth valuevia the interpretation of analysisin thereal numbers. Probably if apoll weretaken,few mathematicianswould agreewith methat the notion of arbitrary real number isvague, and so I would notwant to makean issueof it. But I believeI would garnersubstantially greatersupport of my consequent view that thenotion of arbitrary subset of the realnumbers (existing independently of any human de nitionsor constructions)is vague, since we no longer have the anchor of geometric intuition there.M oreover, I would argue that it is inherently vague, in the sense that there


is no reasonable way the notion can be sharpened without violating whatthe notion is supposedto be about. For example, the assumption that allsubsetsof the realsare in L or evenL (R) would besuch a sharpening, sincethat violatesthe ideaof arbitrariness. In the other direction, it is hard toseehow therecould beany non-circular sharpening of the form that thereasmany suchsetsaspossible. It is from suchconsiderations that I have beenled to theview that the statementCH is inherently vagueand that it ismean-inglessto speakof its truth value; the fact that no remotely plausibleaxiomsof higher set theory serve to settleCH only bolstersmy conviction. Fromthe quote above, Steel is apparently willing to countenancean ambiguity inthe notions involved in CH. I f, asheputs it, the bestthing then to do wouldbeto resolvetheambiguity, it would show CH to bevaguebut not inherentlyso; that is thenub of our disagreement.

Relatedly, Steelcharacterizes my views asbeing instrumentalistic, whichhe takes to be a dodge, but he oversimplies my position in that respect.One kind of instrumentalism that I have espoused, to the extent that I havedone so in one place or another, is very much a H ilbertian one (in therelativized sense): given a system S that one understandsand accepts, ifanother systemT is reducedto S, conservatively in the language of S, thenthat justies the useof T , even if one does not grant de nite meaning tothe language of T beyond that of S. As an example, the overwhelmingpart (if not all) of scientically applicable mathematics canbe formalized incertain higher order systemsT which are conservative over PA , and in factmuch of it is already conservative over PRA; that thereby justies the useand applicationsof such T (cf. [3], Chs. 13,14). Similar resultshold for thebulk of everyday mathematics (whetherpureor applied) conservatively (forcertain analytic statements)over constructively justied systems(cf. [3], pp.201!). This kind of instrumentalism is thusphilosophically satisfactory.

I have also argued (e.g., in [3], p. 73), that ones picture of thecumulativehierarchy is clear enough as a whole to justify condence in the use ofZFC (and like theories) for deriving number-theoretical results. This is apragmatic instrumentalism which is not philosophically satisfactory sincethere is thus far no philosophically satisfactory justica tion for ZFC, at leastnone in my view. But the result of the case studies cited above shows thatthough this kind of instrumentalism admits much more in principle thanthe preceding, there is no real di! erence in practice(i.e., with respectto themathematics of the 99% of all mathematicians.

Responseto Friedman. The core of Friedmans presentation consistsoftwo daring predictionsabout thee! ect of hisnew work on Boolean relationtheory, which it isclaimedwill eventually forcethemathematical communityto accept fully (perhaps after a period of controversy) new large cardinalaxioms. Which those are is not speci ed, and in particular it is not saidwhether thesewill just be small large cardinals (presumably palatable to


mathematicians with a certain amount of encouragement) or also lar gelargecardinals. I t isalso not predicted how long thiswill take. I f he is right,time will tell. If not . . . ? The criteria ag he proposes for the adoption ofnew axiomssetaveryhigh bar (Olympic sized), but in my view appropriatelyso.

Finally, Friedmanaddressesmy point that thereisan equivocation betweenneedinga largecardinal axiom andneedingthestatementof its1-consistency(over ZFC). He says that the choicesare essentially equivalent for the pur-poses of proving " 02 statements. Of course. He also says that it is morenatural to develop suchconsequencesof, say, Booleanrelation theory underthe assumptionof the axiom rather than thestatement of its1-consistency. Ialso agree to that. But I do not agreewith hisconclusion that thiswill showwe need largecardinal axioms. It is neither here nor there that he meansby ! needslarge cardinals to prove that an y reasonable formal systemthat proves! must interpret largecardinals in the senseof Tarski. If ! isequivalent to the statementof 1-consistencyof a largecardinal axiom LCAand PA ! T and T proves ! then of courseLCA is interpreted in T by theformalizedcompletenesstheorem. But that doesnt show that weneedeitherLCA or its consequence! in thenormal senseof theword.

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[2] S. Fef er man, Godels program for new axioms: Why, where, how and what?, in Godel96, (P. H ajek, editor), LectureNotesin Logic, vol. 6, 1996,pp. 322.

[3] , In the light of logic, Oxford University Press, New York, 1998.[4] , Does mathematics need new axioms?, American M athematical M onthly, vol.

106 (1999), pp. 99111.[5] H . Fr iedman, Finite functions and the necessary use of large cardinals, Annals of

Mathematics, vol. 148 (1998), pp. 803893.[6] K. G odel , Uber formal unentscheidbare Satze der Principia M athematica und ver-

wandter SystemeI , M onatshefte f ur M athematik und Physik, vol. 38 (1931), pp. 173198.Reprinted, with English translation, in Collected Works,, Vol. I ., Publications 19291936 (S.Feferman, et al., editors), Oxford University Press, New York, 1986, pp. 144195.

[7] , What is Cantors continuum problem?, American M athematical M onthly, vol.54 (1947), pp. 515525; errata 55, 151. Reprinted in Collected Works, Vol. I I ., Publications19381974 (S. Feferman, et al., editors), Oxford University Press, New York, 1990, pp. 176187.(1964revisedand expanded version, ibid., pp. 254270.)

[8] , Collected Works, Vol. I I I ., Unpublished essays and lectures (S. Feferman, etal., editors), Oxford University Press, New York, 1995.

[9] E. Gr i! or and M. Rathjen , The strength of some M artin-L of type theories, Archivefor M athematical Logic, vol. 33 (1994), pp. 347385.

[10] G. Jager and T. St uder , Extending the system T0 of explicit mathematics: the limitand M ahlo axioms, to appear.

[11] A . K anamor i, Thehigher in nite, Springer-Verlag, Berlin, 1994.


[12] P. M addy, Realism in mathematics, Clarendon Press,Oxford, 1990.[13] , Naturalism in mathematics, Clarendon Press,Oxford, 1997.[14] D. A . M ar t in, H ilberts rst problem: The continuum hypothesis, in M athematical

developmentsarising from H ilbert problems (F. Browder, editor), Proceedingsof Symposia inPureM athematics, vol. 28, American M athematical Society, Providence, 1976, pp. 8192.

[15] D. A . M ar t in and J. St eel , A proof of projectivedeterminacy, Journal of theAmericanMathematical Society, vol. 2 (1989), pp. 71125.

[16] J. Par is and L . Har r ingt on, A mathematical incompletenessin Peanoarithmetic,in Handbook of mathematical logic (J. Barwise, editor), Nort h-Holland, Amsterdam, 1977,pp. 11331142.

[17] W. Pohl er s, Subsystems of set theory and second order number theory, in Handbookof proof theory (S. R. Buss, editor), Elsevier, Amsterdam, 1998, pp. 209335.

[18] M. Rathjen , Recentadvancesin ordinalanalysis: " 12-CA andrelatedsystems, Bulletinof Symbolic Logic, vol. 1 (1995), pp. 468485.

[19] S. Simpson, Subsystemsof second order arithmetic, Springer-Verlag, Berlin, 1998.[20] F. Ye, Strict constructivism and the philosophy of mathematics, Ph. D. Dissertation,

Princeton University, 1999.

DEPARTM ENT OF M ATHEM ATICSSTANFORD UN IVERSITY

STANFORD, CAL IFORN IA 94305, USAE-mail : [email protected]

DOESM ATHEM ATICSNEED NEW AX IOM S?

PENELOPE MA DDY

AsFefermanhasmadeclear,17 wecant beginto addressour title questiondoesmathematics neednew axioms?without rst askingneed for whatpurpose? I f webegin with thepurposeof providing tools for current science,I think we must agree with him that ZFC is surely enough, and indeed thatweaker systemswould probably do (p. 109). Granted, therearesome linger-ing worries about whether sciencecould get by in practiceon theseweakersystems;in the context of discovery, the more free-wheeling impredicativeand innitary methodsmight be needed.But Feferman himself grants this(p. 109). So, we agree that mathematics doesnt neednew axioms for itspractical usesin current science, and that it probably needsfar fewer axiomsthan it already hasfor its strictly formal usesin current science.

Still, theresan odd bias implicit in theshapeof thisdiscussion, and in theformulation of our title question. Why ask doesmathematicsneed such-andsuch?When theywere rst developed,did mathematicsneed n-dimensionalspaces,non-Euclidean geometry, or abstract algebra? I t surely didnt needthem for the scienceof the time, though I ve chosenexamplesthat did later

17In his [2], from which his remarks in this panel discussion begin. Unreferenced pagenumbers in the text refer to theprinted paper.


becomeessentialto science. I think the moral of the story of the devel-opment of mathematics in the 19th and 20th centuries is that attempts tolimit mathematics are a bad idea, that the essenceof pure mathematics isits freedom, that mathematicians should be allowed to follow their math-ematical noseswherever they leadand that this would be true even if allwe cared about were the usefulnessof mathematics in science.18 Perhapswed do better to replacethe original questiondoes mathematicsneednewaxioms?with the more open-endedquestionw ould mathematics benetfrom new axioms?

Ill comeback to this in a moment, but for now lets skip directly to thebroadest version of our title question: what is needed for all of modernmathematics?Here, of course, weknow that new axiomsareneededto settlebasicquestionsof descriptive set theory, questions about the properties ofsimple setsof real numbers, questions raised nearly a century ago by theearly analystsLuzin and Suslin. Though thesenew axiomsdont yet appearon o$cial lists at the beginningsof our textbooks, theyare largely adopted,sometimes without comment, by those working in the eld. Thus it iscommon, in thesediscussions,to assumethe existenceof an array of largelargecardinals, most often a generousstoreof Woodin cardinals.

Now anyone who works in the generous spirit of modern mathematicstouchedon a moment ago, anyone moved by his faith that the more math-ematically interestingstructuresthe better, any suchpersonwould applaudthis development in set theory, but it seemsthat Fefermandoesnot. Thoughhe insists that his main concern is simply to determine what, fundamen-tally, is neededfor what? (p. 109), we can hardly ignore his disapproval ofe!orts to nd new axioms to settle theCH. Though theproblem of theCHis more complex than the earlier problemsin descriptive set theory and theoutcomeof e!orts to settle it is still doubtful, it is unclearwhy the e!ort tosettleCH should be fundamentally misguidedin a way that the (successful)e! orts to settl e the others were not. In fact, I suspect that Feferman is asuncomfortable with the largecardinals neededto settletheold questionsofdescriptive set theory ashe is with current e!orts to settle the CH. What Iwant to do here is investigate two interrelated reasonsthat seemto me to liebehind this distaste.

The rst of theseis his view that Platonismw hich he rejects as thor-oughly unsatisfactory (p. 110)pr ovides the fundamental justica tion forcontemporary set theory (pp. 109110).I take it Feferman has in mindthe useof Platonism to arguethat statementslikeCH, though independent,neverthelesshave objective, determinate truth values.19 Feferman emphati-cally rejects this point of view, insisting that the Continuum Hypothesisis

18For morediscussion of the roleof applied mathematics, see [5].19As in [3], p.260.


an inherently vague problem, and thus that no new axiom will settle[it] ina convincingly denite way (p. 109).

Now perhaps I dont nd Platonism asrepugnant asFeferman does,but Ido agreewith him on theoperativeclaim: that Platonism cannot justify thepractice of set theory, in particular, the practice of seekingnew axioms tosettletheCH. Where I disagreewith Feferman is in the implicit assumptionthat Platonism is theonly possible justi cation for thisset theoretic practice,and thus, that any failureof Platonism leaves thepracticeunjusti ed.

What I propose is that philosophical considerations like those surround-ing Platonism arelargely irr elevant to thevery realmethodological questionsof set theoretic practice: is CH a legitimate mathematical question,despiteits independence?;what reasonscould there be for accepting or rejectingany given candidate for a new axiom of set theory? What matters for thesequestions,I would argue ([4]), is a wide rangeof specically mathematicalconsiderations, considerations directly linked to the goals of the particularmathematical practice in question. To seehow this played out in one his-torical case, recall the debate over the Axiom of Choice. M uch heat wasgenerated in metaphysical battles betweenPlatonism and various versionsof Denabilism or Constructivisma debate unresolved to this dayb utin the end, as Greg Moor e demonstrates in his delightful casestudy ([8]),the axiom wasadopted becauseit isessentialto achieving the mathematicalgoals of a stunningly diverse array of practices. To take just one such goal,closeto home, if you want a well-behaved theoryof transnite numbers,youwill want your set theory to include theaxiom of choice. This isoneamongmany mathematically sound reasons for preferring that our fundamentaltheory of setsinclude the axiom.

The idea, then, is that set theoretic practice in particular, and mathemat-ical practice in general,are not in needof justica tion from philosophicalquarters. Justica tion, on this view, comesfrom within, couched in simpleterms of what meansare most e!ecti ve for meetingthe relevant mathemat-ical ends. Philosophy follows afterwards, as an attempt to understand thepractice, not to justify or to criticize it. From thisnaturalistic point of view,our original questiondoes mathematics need new axioms?is a bit o!target. I t would bemoreappropriate to askwhetheror not someparticularaxiomsa y one assertingthe existenceof many Woodin cardinalsw ouldor would not help this particular practicecontempor ary set theorymeetone or more of its particular goals. In this case, the answer is yes: theassumption of many Woodins provides a complete theory of the projectivesetsof reals, somethingset theoristshad sought for decades,and it doessoin a mannerconsistentwith various other set theoretic goals. I wont try tospell out the details here; I just want to indicate the type of justica tion Ihave in mind for this naturalistic approach to settheoretic method.


What, then, does naturalism suggest for the case of the CH? First, thatweneednt concernourselveswith whetheror not the CH hasa determinatetruth value in somePlatonic world of sets; weneednt confront thequestionof whether or not it is inherently vague, assuming this involvessomeextra-mathematical theory of themeaningof settheoretic vocabulary; weneedntevenarguethat theanswer to the CH issomehow pre-determined, that thereisa pre-existing right answerout therefor us to discover. Instead,weneedtoassessthe prospectsof nding a new axiom that is well-suited to the goalsofset theory and also settlesCH. As Feferman points out, we dont currentlyhave suchan axiom. Perhaps it will turn out that there is no such axiom,perhaps for some reason of principle, but especially in light of recent work,it seems to mepremature to declare thecasehopeless.

So, I suggestthat Fefermans rst reason for discomfort over the searchfor new set theoretic axioms is his mistaken belief that the only possiblejustica tion for this practicemust lie in philosophical Platonism. Wecan getat the secondreason if we ask ourselveswhy he should be so pessimisticsosoon.

To seethis,notice that themathematical reasonsfor adding oneor anotheraxiom to ZFC are often divided into two varieties: intrinsic and extrinsic.Extrinsic reasonsare easy to recognize; they involve the consequencesof agiven axiom candidate, its fruits, if you will. Intrinsic reasonsaredescribedin various waysby various writers, using terms like self-evident, intuiti ve,and part of the very conceptof set. The justica tions Ivegestured towardsso far haveall beenextrinsic.

Now Feferman says nothing directly against the notion of an extrinsicjustica tion, but I think its apparent that hes lessthan impressedby them.To begin with, hegivesconsiderableattention to theOED sde nition of anaxiom asa self-evident proposition (p. 100),which clearly excludesextrinsicjustica tions for axioms, and he praisesthe Peano axioms for coming ascloseasanything wehave to meetingthe ideal dictionary senseof theword(p. 101). Hecontinues

If [the axioms of ZFC] are to be considered axioms in the ideal,dictionary sense, they should be evident for some pre-axiomaticconceptthat wehave in mind. (p. 102)

One candidate for this pre-axiomatic concept is Freges nave notion of theextensionof a property, but this falls prey to the paradoxes. Theonly otheroption is Zermelos iterative conception, and this, according to Feferman,involves us in the Platonism he nds so unpalatable (pp. 102103). Noticethat the possibility of extrinsic supports for the axioms of ZFC isnt evenraised.

Now, obviously, if Feferman is unmoved by extrinsic justica tions, itsno surprise that he would disapprove the contemporary search for new settheoretic axioms, which relieson them so heavily. So we must ask for the


ground of this preference: whats wrong with extrinsic justica tions, andwhatsso good about intrinsic ones?

As a warm-up for this question notice that a Platonist might balk atextrinsic justi cations of the broad variety under consideration here. Thesetake the form: we want this particular mathematical theory to do such-and-such; using method so-and-so is an e!ective way of achieving such-and-such;therefore, its reasonable for us to usemethod so-and-so. A truePlatonist might well object that it isnt enough that our mathematical theorydo all the things we want it to do, it must also be true that is, true in theobjectiveworld of mathematical objectsthat thetheory purports to describe.My naturalist doesnt care about thesephilosophical niceties;what mattersare the intr a-mathematical goals and the e!ectivenessof various meansofachieving them.

So a Platonist might object to our extrinsic justi cations, but of course,Feferman is no Platonist; he puts no stock in mathematical objects. Philo-sophically, he takes thecentral question to be What is thenatureof math-ematical concepts?and theseare

conceptionsof certain kinds of ideal worlds, . . . . presented moreor lessdirectly to the imagination, from which basicprinciplesarederived by examination ([1], p. 124).

Now if theseconceptsare taken to be objectively real,we could imagine anobjection to extrinsic reasons that runs parallel to the Platonists: it isntenough that a theory be e!ective, it must also be true of those concepts!But here again, Feferman rejects the realist line. Still, he attaches greatimportance to teasingout what is implicit in the conceptsand principles([1], p. 122), so there doesseemto be someobjective content about whichwemight go wrong if we trusted to merely justica tions.

The question the naturalist can hardly help asking at this point is why itmatterssomuchwhat isandwhat isnt part of our existing concept?Imagineourselves, for example, around 1870, working with Felix K lein or SophusLie to isolate the notion of group. We would have various mathematicalgoalsin mind, and wewould mold theaxiomsof group theory to meetthosegoals as best we could. 20 What we would not do is stop and ask ourselvesif the existence of inverses is or is not part of our concept of group. Whyshould set theory beany di!er ent?Why should a contemporaryset theorist,appreciative of the power of largecardinals, stop to ask himself whetherornot they arecontained in theconceptof set?

Here Feferman has a clear answer. The axioms of group theory arestructural axioms; they aim to provide a framework and their value forthe organization of mathematical work is now indisputable, but they havenothing to do with self-evident propositions (p. 100). M ore could be said

20For more discussion of this case, and its analogies and disanalogies with set theory, see[5].


here, for example, contrasting the way one set of structural axiomsthedenition of groupis usedto bring out similarities betweenmathematicalstructuresthat otherwiseseemquite dissimilar, while another thedenitionof topological spaceis usedto isolateinvariant properties,andso on. Eachcasein which themathematical community successfully settleson the rightdenition of someconcept shows once again how a concept was moldedto meetcertain mathematical needs. Following Fefermans line of thought,wecan fully agree that intrinsic considerations are largely irrelevant in suchcases. But, Feferman continues, theaxioms of set theory aredi! erent; theyarent structural, but foundational, and he suggests,they are neededtojusti fy mathematical practice in the end (p. 100). He then proceeds withthe analysissketched above, in which ZFC comesup short in the matter ofintrinsic justi cations.

All this strongly suggeststhat Feferman regards intrinsic justica tions asessential and extrinsic justi cations as inappropriate in the case of founda-tional axioms like thoseof set theory. Now I fully agree that thesettheoreticaxioms are (at least partly) aimed at satisfying a foundational goal; as anaturalist I would argue that an appreciation of this foundational goal helpsus seewhy certain widespread settheoretic methodsare rational. The ironyis that I seethe e!ecti venessof an axiom candidate at helping set theoreticpractice reach its foundational goal as a sound extrinsic reason to adoptit as a new axiom! Given that Feferman seesthe foundational goal as re-quiring intrinsic justi cation, there must be some stark di! erence in ourunderstandings of the foundational goal itself.

In fact, I think that di!er encelies fairly close to the surface. M y ownunderstanding is that set theory seeks to providea uni ed arena in which settheoretic surrogates for all classicalmathematical objectscan be found andthe classicaltheoremsabout theseobjectscan beproved. This sort of foun-dation brings the various structuresof mathematics onto one stage, wherethey canbecontrastedand compared; it providesa uniform answer to ques-tions of mathematical existenceand proof. I trust there is no disagreementthat theseservicesare mathematically valuable. But set theoretic founda-tions in thissensedo not do two of the things that earlier thinkershad hopedfor: they do not reveal what mathematical entities really are, in somedeepmetaphysicalsense;and, more to thepoint for our presentpurposes,theydonot providean epistemic foundation; they do not show ushow to derive thevarious truths of mathematics by transparent steps from absolutely certaintruths. Thisgoal, it seemsto me, isonethat thedevelopmentof mathematicshasforcedus,however reluctantly, to abandon.

But suppose you have not surrendered this goal. Then it might makeperfect senseto insist that foundational axioms be conceptual truths ofsomesort or other, and thusself-evident and absolutely certain. I f you insist


on beginning from indubitable premises, asa trueepistemic foundationalistmust, you simply cant allow extrinsic justi cationsof any kind.

My suggestion,then, is that the fundamental di!er encebetweenFefer-man and my naturalist, the di!er encethat leadsto their disagreementoverthe legitimacy of extrinsic justica tions in set theory, and from there to adisagreementover the viability of the search for new set theoretic axioms,is simply this: Feferman cleaves to a foundational goal that the naturalistregardsasoutmoded. On this point , I leaveyou to your consciences.


Comments on Friedman and Steel. Hoping it might clarify the issuesatstake, Id like to takea moment to lay out what I take to be thedisagreementbetweenSteeland Friedman on our title question. It begins:

St eel : Of course mathematics needs new axioms. It needsthem to settlethe outstanding questions of descriptive set theory, and it needsthem iftheres to beany hopeof settling theCH.

Fr iedman: Given the goalsof contemporary set theory, this isa perfectlyrational answer.

Thenaturalist might wonder why this isnt theend of thestory, but Fried-man hasmore to say:

Fr iedman: But the goals of set theory are very di!er ent from those ofcoremathematics.

St eel : So what? The algebraists goals are di!er ent from thoseof thetopologist. Both di!er from thegoalsof thenumber theoristor thegeometer.Weshould expect that the set theorist would havehis own goals.

Fr iedman: In the casesyou cite, the various parties respectthe othersgoals,even if they dont share them. But thecoremathematician thinks theset theorists goals are outright wrong, not just di!er ent. In particular, thecore mathematician thinks the set theorist is too concernedwith generality,with pathologies,etc.

St eel : But surely these are irrational prejudices on the part of the coremathematician. Weshould try to educatehim.

At this point , I m not sure which of two tacksbestrepresentsFriedmansresponse. Perhaps it issomeamalgamof the two.

Fr iedman number one: No, thecoremathematiciansreasonsfor rejectingthe set theoristsgoalsareperfectly rational. They canonly becounteredbysomedevelopment likemy Boolean relation theory, which will show thecoremathematician that heneedsset theory to attain his own goals.


Fr iedman number t wo: Even if thecoremathematician is not rational inhis rejectionof set theoretic goals,even if his objectionsaremereprejudices,the set theoretic community needs to counter them with something likeBoolean relation theory. If it doesnt, then as a matter of sociological fact,set theory will go unsupported and eventually dieout.

Notice that on either interpretation, Friedman is not arguing that the settheorist is irrational to carry on as he does, given the goals of his practice.Rather, Friedman is arguing that the set theorist should modify his goals,either for rational or for practical reasons.

CommentsonSteel. Steelsuggeststhat philosophy may haveamoreactiverole to play than the naturalist allows, in particular, that philosophy maybeneededto spellout what countsasa solution to the Continuum Problem.Perhaps. But it seemsto me that Steelhas masterfully outlined, as only adiscerning expert can, the purely mathematical reasonswhy adding manyWoodins (and hencePD ) to ZFC countsasa solution to theold problemsofdescriptiveset theory, 21 and that hesgivenusa tantalizing vision of purelymathematical developments that might one day qualify as a solution to thecontinuum problem. I tshard to seewhat persuasive forcephilosophy couldadd to this appealing (and purely naturalistic) picture. I fear Steel is toomodest!

Comments on Feferman. Feferman has raisedseveral concernsabout mynaturalism,22 perhaps the most troublesomeof which are the suggestionsthat it is unduly transitory or relativistic or parochial.23 Of course itis true that our rational judgment of which mathematical principles andmethods are best will changeas we learn more: we have lessreason nowto embraceinnitesimals than we did before Cauchy and Weierstrass;wehavemore reason now to embrace largecardinals than wedid, for example,when Ulam rst dened measurables in 1930. I dont seewhy this is anymorealarming than the fact that scientistshavemore reasonnow to believein atoms than they did before Einstein and Perrin; we learn new things,acquirenew evidence, modify our theories,in both mathematicsandscience.So naturalistic justi cations will shift as our understanding increases,but I

21The words true and believe crop up occasionally in Steels discussion, but I cant seethat they do any serious philosophical work. In practice, he gives de ationary readingsof such terms, e.g., to believe that there are measurable cardinals is to seek to naturallyinterpret all mathematical theoriesof sets, to theextent that they havenatural interpretations,in extensions of ZFC + there is a measurable cardinal . (The discussion referenced infootnote 26 concerns another such de ationary reading.)

22Feferman is right to note that my naturalism di! ers in somestriking ways from Quines.Those di! erences, and my reasons for continuing to use the word naturalism, are detailedin [4]. (Those with further interest in this version of naturalism might also see [5], [6], and[7].

23The last two descriptions come from an earlier version of Fefermans remarks, whichwassubsequently shortened for purposes of publication.


dont think thismakes those justi cationsany more unduly transitory thanour scientic theories.

Naturalism is relativistic, in the sensethat the justica tion for a givenmathematical method is given in termsof the goalsof the practice in whichit is embedded. Certainly it is true, as Friedman has emphasized, thatdi!er ent groups of mathematicians will share di!er ent goals: a topologisthopesto isolate invariants; an algebraist hopesto uncoverhiddensimilaritiesbetweenwidely varying structures; a set theorist hopesto provide a broad,rigorous foundation for classicalmathematics. And it is, asFeferman notes,a delicatetaskto understandthe implicit goalsof a given practice.24 But thisdoesnt mean that anything goes(e.g., early innitesim als o!ended againstthe overarching mathematical goal of consistency)25 or that there isno suchthing as justifying a method in suchterms (e.g., the rejection of V=L mightbe justied in termsof the foundational goal of settheory).

Obviously, theconstructivist hasdi!er ent mathematical goals from thoseof the set theorist, goals in terms of which he adjudicatesbetweenvariousmethods for his practice (e.g., rejecting proof by reductio ad absurdum).Analysisof thesegoalsand justi cations fallssquarely within thenaturalistseld of study; naturalism isnot parochial to set theory, thoughset theory hasbeen my own focus. What fallsoutsidenaturalistic scruples isphilosophicalincursions into mathematics (e.g., rejection of classicalmethodsfor reasonsof a priori semantic theorizing, as in Dummett).

Finally, onepoint of clari cation: thenaturalistic methodologist takesnostand on whether or not either the CH or the questionsof descriptive settheory have determinate answersdespitetheir independence.26 The di!er -encebetweenthe two is simply that we have attr active axiom candidatesattr active when viewed in terms of the goals of the practice that settl e thequestionsof descriptionset theory, and wedont havesuchcandidatesin thecaseof CH. I f there is a slippery slopehere, asFeferman fears,the naturalistdoesnt climb it in the rst place.

REFERENCES

[1] Sol omon Fef er man, In the light of logic, Oxford University Press, New York, 1998.

24Each of the cases Feferman lists e.g., in nitesimals in the eighteenth century, Fourierexpansions, Delta functions is well worthy of extended naturalistic analysis. I have givenmost attention to the con ict between V = L and large cardinals, trying to argue rst foran analysis of (some of ) the goals of set theoretic practice and second that V = L is a badchoice, given thesegoals. (See [4], part I I I .)

25Part of an extended naturalistic analysis of this episode would most likely reveal howthegeneral goal of consistency wasnot enough to swamp thegoal of providing mathematicaltools for a wide range of scienti c applications. But consistency remained a goal of themathematical practice, and waseventually reached, without sacri cing theothers.

26For discussion,see[4], pp. 194-196,211-212.


[2] , Does mathematics need new axioms?, American M athematical M onthly, vol.106 (1999), pp. 99111.

[3] K ur t G odel , What is Cantors continuum problem?, reprinted in his Collected works,volume I I (S. Feferman et al., editors), Oxford University Press, New York, pp. 254270,1990.

[4] Penel opeM addy, Naturalism in mathematics, Oxford University Press, Oxford, 1997.[5] , Somenaturalistic re ectionsonsettheoretic method, to appear in Topoi.[6] , Naturalism and the a priori, to appear in New essays on the a priori (P.

Boghossian and C. Peakcocke, editors).[7] , Naturalism: friends and foes, to appear in Philosophical Perspectives 15,

M etaphysics2001 (J. Tomberlin, editor).[8] Gr egory M oor e, Zermelos axiom of choice, Springer-Verlag, New York, 1982.

DEPARTM ENT OF LOGIC AND PH ILOSOPHY OF SCIENCEUN IVERSITY OF CAL IFORN IA

IRVINE, CAL IFORN IA 92697-5100, USAE-mail : [email protected]

MA THEMA TICS NEEDS NEW AXIOMS

JOHN R. STEEL

1. Denitions. Let mebeginby clarifying the question.By new I shall simply mean: not a consequence of ZFC. This is of course

somewhat arbitrary. If onetakesa long view, thefull strength of ZFCappearsto be quite new; on the other hand, to a modern set theorist, large cardinalhypotheses going well beyond ZFC are rather old hat. Someof the weakerlarge cardinal hypotheseswere actually discovered and studied before ZFCitself was fully isolated.

By axiom I shall mean: assumption to be adopted by all, as part of abroadest point of view. The broadest point of view proviso is meant toexcludefrom attention thetemporary adoption of restrictiveassumptionsasa convenient device for avoiding irrelevant structure. V = L isoften assumedtemporarily for such reasons by set theorists who do not believe it, just asall functions areC ! issometimesassumed by di! erential geometerswhodo not believe it.

The old self-evidencerequirementon axioms is too subjective, and moreimportantl y, too limiting. In the future, what f orcesitself upon usastrueis more likely to be a theory asa whole, and theprocessis more likely to begradual. We may very well never reach the level of con dence in the newtheory that we have in, say, Peano Arithmetic. Nevertheless,new axiomsmay emerge, and berationally justied. Theself-evidencerequirementwouldblock this kind of progresstoward a stronger foundation.

What is important is just that our axiomsbetrue, and asstrongaspossible.Let meexpand on thestrength demand,asit is a fundamentalmotivation inthe search for new axioms.


It is a familiar but remarkable fact that all mathematical language can betranslated into the language of set theory, and all theoremsof or dinarymathematicscan be proved in ZFC. In extending ZFC, we are attempting tostrengthen this foundation. Surely strength isbetter than weakness!Profes-sor M addy haslabelledthe str onger is better rule of thumb maximize, anddiscussedit at somelengthin herrecentbook [4]. I would say that what weareattempting to maximizehere is the interpretativepower of our set theory. Inthis view, to believe that therearemeasurablecardinals is to seekto naturallyinterpret all mathematical theoriesof sets,to theextent that theyhavenaturalinterpretations, in extensionsof ZFC+ there isa measurablecardinal.

Maximizing interpretative power entails maximizing formal consistencystrength, but the converse is not true, as we want our interpretations topreservemeaning.

In this light we can seewhy most set theorists rejectV = L as restrictive:adopting it restricts the interpretative power of the language of set theory.The language of set theory as usedby the believer in V = L can certainlybe translated into the language of set theory as used by the believer inmeasurable cardinals, via the translation ! "# ! L . There is no translationin the other direction. While it is true that adopting V = L enablesone tosettlenew formal sentences, this is in fact a completely sterilemove, becauseonesettles! by giving it the sameinterpretation as ! L , which can besettledin anyones theory. I f by question we mean interpreted formal sentence,adopting V = L settlesnonewquestions. I t simply preventsusfrom asking asmany questions,sinceweare then forbidden to askabout the world outsideL .

Of course, the Axiom of Foundation also has the appearance of beingrestrictive. However, so far as we know, it is not in reality. We know of nomathematical structure outside the classof wellfounded sets. On the otherhand, weknow of plenty of interestingstructureoutsideL .

It is sometimesmaintained that one could obtain all the logical strengthof measurable cardinals in an extension of V = L by adopting ZFC + V =L + there isa transitivemodel of ZFC + therearemeasurablecardinals .This is an example of what I would call the instrumentalistdodge. For anytheory T and classof sentences%,the instrumentalist versionof T over %,or Inst(T, %), is the theory: A ll theoremsof T in %are true.27 Thus

Inst(T, " 01) $ Con(T ),

Inst(T, " 02) $ 1-Con(T ).

27I shall generally use theory to mean axiomatizable theory in the language of settheory extending ZFC . I t might beslightly morenatural to let Inst(T, %) beaxiomatized bytheschema (F ! ! ) " ! , for ! # %and niteF $ T , since the latter theory haspreciselythesame%-consequences asT .


Onecould obtain all the#12 consequencesof measurablesin

ZFC+ V = L + Inst(Therearemeasurables, #12).

(This is close to the theory ZFC + V = L + there is a transitive modelwith a measurable cardinal, although the latter is slightly stronger.) Onecould go even further, elim inating not just all thenon-constructiblesets,butalso all the innite sets,and neverthelessobtain all the " 01 consequencesofmeasurablesin

PA + Inst(Therearemeasurables, " 01).

There are endlessvariations here. The theory Inst(T, %), usedsimply asa device to avoid directly assertingT while retaining all its %-predictions,is a mathematical parallel of the physical theory: There are no electrons,but mid-size objects behave as if there were. That is, it is a parallel of thistheory as it might be used by a philosopher of today, not as it might havebeenusedby a physicist in 1900. Perhaps a cleanerparallel would be thetheory that the world popped into existence5 minutesago, looking exactlyas it would if there had been a past like the one we believe in. M is-usedin this way, Inst(T, %) is no more than an odd way of assertingT ; it onlybecomes more than that if one has a program for nding a tool for making%predictions which is better than T , and incompatible with T in the realmof non-%sentences. In evaluating a retreat from T to Inst(T, %), oneshouldaskwhether its proponent hasany suchcompeting tool in mind.28

Finally, letmemakea few remarksregarding mathematicsand need. First,it is true that most, if not all, of the mathematics applied in Physics,Chem-istry, and theother sciencesrequiresmuch lessthanZFC. On theother hand,G odels incompletenesstheorems guaranteethis will never be more than areport of the current state of a!airs, for Con(ZFC) doesleadto predictionsabout thephysical world (such as that no contradiction will beproved fromZFC in the next 20 years) which we do not know how to derive using onlyZFC. Second, it is true that most, if not all, of the most active areas in puremathematics require much less than ZFC. Once again, G odels theoremsguaranteethis will never be more than a description of the current state ofa!airs .

ProfessorFefermanhasstatedhis belief that the famousopenproblemsofnumber theory, such as Goldbachs conjectureor theRiemann Hypothesis,can be settled in ZFC. I agree that this is probably the case. However, weshould recognize that until the problemsareactually settled this will almostcertainly never be more than an educated guess.29 If our educated guesses

28I have included this brief discussion of instrumentalism because Professor Fefermansometimes makes such a move. H is class %of meaningful sentences does not seem to gomuch further than the languageof Peano A rithmetic. See for example [2], page73.

29There is the very remote possibility that one could show ZFC settles the questionswithout actually exhibiting therelevant ZFC-proofs. Goldbachsconjectureand theRiemann


as to what techniqueswill be useful in solving a given problem were highlyreliable, mathematics would be lessinterestingthan it is.

Of course, the permanent possibility that new axioms will be needed isnot the same as the reality of such a need. In this connection, we do knowthat many natural, well known problems in the more theoretical parts ofpure mathematics, such as the Continuum Problem or Suslins Problem,demonstrably require new axioms for their solution. The most concrete ofthesearetheproblemsabout projectivesetsof realsfrom classicaldescriptiveset theory. Thesearose in the early 1900s, in the work of Analysts likeLebesgueand Luzin who were interestedin the foundations of their subject.In what follows I shall sketch some reasonsfor believing that we do havethe new axiomsweneedto settleall the classicalquestionsabout projectivesets. I t is worth noting that Lebesgueand Luzin would probably have beenjust assurprised to nd that large cardinal axioms are useful in the theoryof projective setsas modern mathematicians would be to nd that they areuseful in number theory.

Does mathematics really need this theory of projective sets we get fromlargecardinal axioms?Does it needa decisionon the Continuum Problem?Let me just say that I prefer a di!er ent formulation of our question,namely:

Is the search for, and study of, new axioms worthwhile? Shouldpeoplebeworking in this direction?

It seemsto me that this getsmoredirectly at the practical question,and putsit in a morepositiveway. I t doesnt matter so much whetherwecanmakedowithout this line of research; the more important question iswhetherwearebetter o! with it. And it doesnt matter so much whether wecall thesubjectwhich isbetter o! (ordinary, core, normal) mathematics, or something else.It is this reformulatedquestionwhich I shall now discuss.

2. Someresults in Godels program. One of the chief inspirations of theline of research we are discussing is G odels paper [1], and so it is oftencalledGodelsprogram. How should weevaluate thisprogram? Aswith anyline of research, we should look at what has beendone, and whats left todo. Unfortunately, we are discussing at least 35 years of continuous workby a reasonably large number of people. I shall settle, then, for a briefenumeration of someof themain themes.

2.1. Wehave found natural new axioms, the largecardinal axioms. Theseare strengtheningsof the axiom of innity of ZFC. They are expressions ofan intrinsically plausible informal re ection principle.

2.2. Theseaxioms have proved crucial to organizing and understandingthe family of possible extensions of ZFC. Of course, there is nothing like a

Hypothesis are " 01 statements, so one cannot prove them independent of ZFC without alsoproving them.


systematic classication of all the possible extensionsof ZFC, but there ismoreorder here than onemight suspect:

(1) M any natural extensionsT of ZFC have beenshown to be consistentrelative to some large cardinal hypothesis H , via the method of forcing.This method is so powerful that, at the moment, we know of no interestingT extending ZFC which seemsunlikely to be provably consistent relativeto some large cardinal hypothesisvia forcing. Thus, at least if we allow Boolean-valued interpretations, the extensions of ZFC via largecardinalhypothesesseemto beconal in thepart of the interpretability order on allnatural extensionsof ZFC which weknow about.

(2) Often, it hasbeenshown that theconsistencyof the largecardinal hy-pothesisH must beassumed, in that Con(T ) impliesCon(H ). This involvesconstructing a canonical inner model for H . These canonical inner mod-elsadmit a systematic, detailed, ne structure theory much like Jensenstheory of L . Such a thorough and detailed description of what a universesatisfying H might look like also provides good evidence that H is indeedconsistent.30

Here are someexamplesof relative consistencyresultsobtained by thesemethods. The rst four areequiconsistencies.31

Con(ZF + A ll setsLebesguemeasurable)

% Con(ZFC+ There is an inaccessible),

Con(ZFC+ There is a total extensionof Lebesguemeasure

% Con(ZFC+ There is a measurable),

Con(ZFC+ GCH rst fails at &" )

% Con(ZFC+ There is a measurable # of order #+ + ),

Con(ZFC+ A ll gamesin L (R) aredetermined)

% Con(ZFC+ Thereare in nitely many Woodin cardinals),

Con(ZFC+ There is a supercompact cardinal)

# Con(ZFC+ Proper forcing axiom)

# n < " Con(ZFC+ Therearen Woodin cardinals).

Of course, all the implications displayed are provable in Peano A rithmetic.Concerning the last set of results, it is generally believed that the ProperForcing Axiom (PFA) is equiconsistentwith the existenceof supercompactcardinals. Wedo not havea consistencystrength lower bound on PFA betterthan the one displayed because our tool for producing such lower bounds,

30The introduction to [7] contains an essay on the inner model program.31These results are due to Baumgartner, Gitik, M itchell, Schimmerling, Shelah, Solovay,

Steel, Todorcevic, and Woodin, building on earlier work of Dodd, Jensen, Kunen, M agidor,M artin, Prikry, Silver, and others.


the construction of canonical inner models for large cardinal hypotheses,does not yet produce models with more than small numbers of Woodincardinals. Extending inner model theory so that it can produce inner modelswith supercompact cardinals isoneof the most important openproblemsinG odelsprogram.

2.3. Thepattern aboveextends to many moreexamples. I t seems that ev-erynatural extensionof ZFC isequiconsistentwith anextensionaxiomatizedby something like largecardinal axioms! If S and T are extensions of ZFCby largecardinal axioms,then it isgenerally easyto comparetheconsistencystrengths of S and T ; moreover, the consistencystrengthsof large cardinalextensions of ZFC fall into a wellordered hierarchy. Thus it seemsthat theconsistencystrengthsof all natural extensionsof ZFC are wellordered, andthe largecardinal hierarchy providesa sort of yardstick which enablesus tocompare theseconsistencystrengths.32

The ordering of consistencystrengthscorrespondsto the inclusion orderon the setsof " 01 (or in fact arithmetical, or even #

12) consequencesof the

theoriesin question. That is, for any class%of sentencesand any theory T ,let

(%)T = { ! ( %| T ) ! } .

The linearity of the ordering of natural consistencystrengthsmeansthat fornatural S and T extending ZFC,

(" 01)S ! ("01)T or ("

01)T ! ("

01)S.

The fact that our relativeconsistencyproofsactually produce reasonable in-terpretations, and in particular, wellfoundedmodels,meansthat for naturalS and T

(" 01)S ! ("01)T % (#

12)S ! (#

12)T .

Thus, at the level of #12 sentences, weknow of only one road upward, andlarge cardinal hypothesesare its central markers. Does this road lead to #12truth? It is certainly our best guessat the moment; moreover, there is nohint of a competing guess.

2.4. The phenomenondescribedin the last item extends to all sentencesin the language of second order arithmetic, provided we restrict ourselvesto consistencystrengthswhich are su$ciently great. M ore precisely, for Sand T natural theoriesof consistencystrength at leastthat of Ther e are nWoodin cardinals with a measurableabove them all , wehave

(" 01)S ! ("01)T % (#

1n+ 2)S ! (#

1n+ 2)T .

32We know of no way to compare the consistency strengths of PFA and the existence ofa total extension of Lebesgue measure except to relate each to the large cardinal hierarchy.Thesame is true for most other comparisons: the largecardinal hierarchy isessential.


Closely related to this is the phenomenonthat any natural theory of consis-tency strength at least that of PD33 actually implies PD. For example, theProper Forcing Axiom impliesPD. So doestheexistenceof a homogeneoussaturated ideal on " 1. (Neither of thesepropositions has anything to dowith PD on its surface.)

Thus, at the level of sentencesin the languageof secondorder arithmetic,weknow of only one road upward. Largecardinal hypothesesare its centralmarkers; moreover, it goes through PD. We are led to this road in manydi!er ent ways. A long it lies our bestguessat the truth for sentencesin thelanguage of second order arithmetic, and we have no hint of a reasonablecompetingguess.

2.5. The relationship between large cardinal hypothesesand PD is ex-plained by the following theorem.34

Theor em2.1 (M artin, Steel,Woodin) . The followingareequivalent:

(1) PD,(2) For all n < " , every #1n consequence of ZFC + there are n Woodin

cardinals is true.

According to this theorem, PD is just the instrumentalists trace ofWoodin cardinals in the languageof secondorder arithm etic.

2.6. The theory in the languageof secondorder arithmetic basedon largecardinal axioms contains answers to all the questionsabout projective setsfrom classicaldescriptive set theory. The theory of projective setsone getsthis way extends in a natural way the theory of low-level projective setsdeveloped by the classicaldescriptive set theoristsusing only ZFC; indeed,in retrospect, much of the classicaltheory can be seenas basedon opendeterminacy, which is provable in ZFC. Virtually nothing about sets in theprojectivehierarchy beyond the rst few levelscan bedecidedin ZFC alone,but largecardinal hypotheses,via PD, yield a deepand powerful extensionof the classical theory to the full projective hierarchy. (Kechris, Martin,Moschovakis, andSolovay areamong theprincipal architectsof this theory.)By placing the classicaltheory in this broader context, we have understoodit better.

In this realm of evidence through consequences, I would like to mention aclassof examplespointed out by D. A. Martin in [5]. Namely, sometheoremsare rst proved under a large cardinal hypothesis,and then later the proofis re ned or a new proof given in such a way as to use only ZFC. Boreldeterminacy is a prime example of this: originally it was proved assumingthere are measurable cardinals, and then later a more complicated proof

33PD is theassertion that all projective gamesaredetermined.34The breakthrough results of Foreman, M agidor, and Shelah [3], and then Shelah and

Woodin [8], wereacrucial step toward thistheorem. Seetheintroduction to [6] for ahistoricalessay.


which usesonly ZFC was found.35 Martin points out other instancesof thisphenomenon, for example some cases of his cone theorem for the Turingdegreeswhich can beprovedby directconstructions. Another niceexampledown lower isBorel Wadgedeterminacy. This isan immediate consequenceof Borel determinacy, but by work of Louveau and Saint Raymond, it ismuchweakerin fact, provable in secondorder arithmetic. There are interestingpotential instancesof this phenomenon in which we have a proof from alarge cardinal hypothesis,and at leastsomepeoplesuspectthere is a proofin ZFC, but that proof isyet to bediscovered. The point here is that theuseof largecardinals to provide proofs of statementswhich can then beprovedtrue more laboriously by elementarymeansconstitutes evidence for largecardinals.36

It is sometimesclaimed that there is an alternate, equally good, theory ofprojectivesetswhich weget from V = L. Thereare two replieshere.

First , the central idea of descriptive set theory is that denable sets ofreals are free from the pathologies one gets from a wellorder of the reals.Since V = L implies there is a &12 wellorder of the reals,under V = L thiscentral ideacollapseslow in theprojective hierarchy, and after that there is,in an important sense, no descriptive set theory. One has insteadinnitarycombinatoricson &1. This iscertainly not thesort of theory that looksusefulto Analysts.

Mor e importantl y, even if therewerea wonderful, useful theory of projec-tive sets based on V = L, adopting largecardinal axioms would in no wayeliminateor devalue it. For the believer in largecardinals, this theory wouldmake perfect senseas a wonderful, useful part of the rst-or der theory ofL . On the other hand, the believer in V = L cannot give the appropriatesenseto thetheory of projectivesetsweget from PD. (Hemay resort to someversion of the instrumentalist dodge, but that amounts to uttering thewords V = L while acting as if you believe something else.) Adopting V = Lbrings with it a loss in this situation. Adopting measurable cardinals givesus 0$, and with that, a much clearer view of L than we get if we are onlyallowed to look at it from inside .

2.7. Large cardinal axioms seem to decideall natural questions in thelanguage of secondorder arithmetic. There is metamathematical evidenceof this completenessin the fact that no sentencein the languageof second

35By M artin. Thisproof uses the small cardinal hypothesis that thepower set operationcan be iterated %1 times. H . Friedman showed that such a hypothesis isneeded for theproof.Borel determinacy is a natural statement in the language of second order arithmetic whichcannot beproved without appealing to arbitrary setsof reals, setsof setsof reals, etc.

36G odels result on speeding up proofs shows that some form of this phenomenon occurswhenever one increases consistency strength, but it does not guarantee that any mathemati-cally interesting theorems haveshorter proofs in thestronger system.


order arithmetic can be shown independent of existence of arbitrarily largeWoodin cardinals by forcing:37

Theor em2.2 (Woodin). Supposethere is an iterable innermodelsatisfying there are " Woodincardinals; thenif M and N are set-generic extensionsof V , wehave

L (R)M $ L (R)N .

There isonly one theory with this kind of generic completeness:Theor em2.3 (Woodin). Suppose that whenever M and N are set generic

extensions of V , we have L (R)M $ L (R)N ; then there is an iterable innermodel satisfying thereare " Woodincardinals.

2.8. If we regard Theorem 2.2 as a key indicator of the completenessoflarge cardinal axioms for sentencesin the language of second order arith-metic, then it is natural to ask whether this kind of completenessextendsto the language of third order arithmetic. There we immediately encounterCH, which is a #21 statement. None of our current large cardinal axiomsdecideCH, and consequently they do not yield a theory which isgenericallyabsoluteat the #21 level:

Theor em2.4 (Levy, Solovay).(1) Noneof thecurrent largecardinal axiomsdecidesCH.(2) Let A beoneof the current large cardinal axioms, andsupposeV |= A.

Then thereareset generic extensionsM and N of V which satisfy A, butarenot #21-equivalent.

Nevertheless,theremay besomenatural extrapolation of our current largecardinal axioms,somenatural markers of still higher consistencystrengths,which yield a complete theory in the languageof third order arithmetic,with anaccompanying genericabsolutenesstheorem. A second,betterdelin-eated alternative is an extensionof our current largecardinal axiomswhichdoesnot increaseconsistencystrength, with an accompanying conditionalgenericabsolutenesstheorem for sentencesin the language of third orderarithmetic. Indeed,Woodin hasshown that simply adding CH givesussucha theory at the#21 level:

Theor em2.5 (Woodin). Suppose V |= Ther e are arbitrarily large mea-surable Woodin cardinals. Let M and N be set-generic extensions of VsatisfyingCH; then M and N are#21-equivalent.

It is open whether if V |= There are arbitraril y large supercompactcardinals , and M and N areset-genericextensionsof V satisfying * , thenM and N are #22-equivalent. I f so, this would indicate that in the presence

37The hypothesis of Theorem 2.2 follows from the existence of arbitrarily large Woodincardinals. Woodin and the author noticed independently that the weaker hypothesis usedhere su$ ces. I have stated 2.2 this way in order to point out its converse. The converse isclosely related to work of theauthor.


of large cardinal hypotheses,* yieldsa complete theory at the#22 level, justasCH doesat the#21 level.

2.9. There are many interesting open problemsbearing on G odels pro-gram. Hereare two which seemparticularly important to me.(1) Develop a theory of canonical inner models satisfying There is a su-

percompact cardinal.(2) Find conditional genericabsolutenesstheoremsat the #2n level, for all

n < " .


Reply to Feferman. Feferman stateshis belief that CH doesnot expressade nite proposition , that it is inherently vague . The trouble is supposedto be that the set of all reals is not a denite mathematical object. Butif that is the trouble, it would seemthat ther e is a set of all real numbersdoesnot expressa denite proposition; indeed,mathematics would seemtobe shot through with vague statements. Worse than that, with inherentlyvaguestatements,so that there is no hopeof rectifying the situation, except,I guess,by chucking the whole mess. Taken seriously, this analysis leadsus into a retreat to some much weaker constructivist language, a retreatwhich would tossout good mathematics in order to save inherently vaguephilosophy. Feferman himself doesnot seemto take that route; instead,headopts an instrumentalist stance, according to which higher set theory hasno denite meaning,but somehow hasvaluenevertheless.38

Fefermans instrumentalismcomesto the fore in the following passage:My point here is simply that there is a basic di!er encebetweenacceptingsystemssuchasZFC + LCA, whereLCA is theapplica-ble largecardinal axioms,and accepting1-Con(ZFC + LCA). Asto the questionof theneedof largecardinal assumptionsto settlenite combinatorial problemsof the sort producedby Friedman,there is thus, in my view, an equivocation betweenneedinga givenaxiom and needing its 1-consistency;it is only the latter that isdemonstrated by his work.

To this I would reply: there certainly could be a basic di!er ence betweenacceptingLCA and accepting1-Con(LCA), but except for the fact that hewitholds the honoric rst-class mathematical principle in the caseofLCA, Feferman hasnt told us what the practical, behavioral content of thedi! erence would be in his case. He proposes no tools for generating " 02truths which arenot subsumedundersomenatural interpretation by axiomsof the form LCA , and indeed,as I have emphasized, there is presently no

38I refer here to what Feferman calls his pragmatic instrumentalism. H ilbert-style reduc-tions to constructivist systemsarenot a dodge, they area retreat which tossesout too muchgood mathematics.


hint of such a tool. The suggestionthat we might retreat from LCA toInst(LCA , " 02), which isat leastimplicit in this passage, isan empty one, asfar as I can tell. No onesmathematical behavior would change.

Feferman goeson to say:But if

9. S. Feferman - Por Qué Los Programas de Nuevos Axiomas Deben Ser Cuestionados

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