Dean Kurokawa Slides - philosophy.nd.eduWhat is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Context and assumptions I Georg Kreisel: “On informal rigour and

On the methodology of

informal rigour:

Set theory, semantics, and

intuitionism

MWPMW

2 December 2020Notre Dame

Walter DeanUniversity of Warwick

Hidenori KurokawaKanazawa University

What is ‘informal rigour’? Creating Subject La Predicativite Envoi

Context and assumptions

I Georg Kreisel: “On informal rigour and completeness proofs” (1967b)

I Problems in the Philosophy of Mathematics (Lakatos 1967, ed.)I Speakers: Bernays, Easley, Kreisel, Kalmar, Korner,

Mostowski, A. Robinson, Szabo.I Attendees: Bar-Hillel

ú, Carnap, Dummett, Hacking, Harsanyi, Heyting

ú,

Hintikka, Kyburg, Luce, Jeffrey, Kleeneú

, Kuhn, Myhill†, Popper, Quine,

Suppes, Salmon, Tarski, Williams.

I Assumptions for this talk:

I Most of your are familiar with “Informal rigour” (1967b) wrt:1) Kreisel’s ‘squeezing’ argument about first-order validity.

2) Kreisel’s argument that CH has a definite truth value.

I Fewer are familiar with Kreisel’s Creating Subject argument aboutintuitionistic analysis and the status Markov’s principle.

I There is no general received understanding of what Kreisel meant by‘informal rigour’ or how it applies to these cases.

2/25


Outline and goals

I Goals for this talk:

1) What is ‘informal rigour’? – a precise answer.2) Two of Kreisel’s core examples wrt to 20th c. phil of maths.

i) the Creating Subject argument

ii) the “La Predicativite” argument

3) Convince you that the method is still relevant.i) Additional applications – e.g. squeezing non-classical validity?

ii) An informally rigorous argument that CH is true or false?

iii) Conceputal realism? (Kreisel vs Carnap vs Godel vs . . .)

I Other teasers:

I Kreisel was the original ‘K’ in the BHK interpretation.I Tait’s (1981) analysis of finitism – i.e. PRA – is a response to

Kreisel’s (1960b, 1965) earlier analysis – i.e. PA.I In his work on predicative definability (1960a/61/62a) and lawless

sequences (1958b/68) Kreisel anticipated the method of forcing.

3/25


What is informal rigour: Short answerTwo slogans:

I Kreisel: Informal rigour = ‘philosophical proof’.

I Our gloss: Informal rigour = conceptual analysis mediated by

mathematical theorems.

1967a: “Mathematical logic: what has it done for the philosophy of mathematics?”

Successes of mathematical logic Time and again it has turned out that traditionalnotions in philosophy have an essentially unambiguous formulation when one thinksabout them . . . [A]lso, when so formulated by essential use of mathematical logic, theyhave non trivial consequences for the analysis of mathematical experience . . . Amongthem are the well known cases

(i) the notion of mechanical process, its stability in the sense that apparentlydi�erent formulations lead to the same results . . .

(ii) the notion of aggregate which is analysed by means of the hierarchy (theory) oftypes, and, of course

(iii) the notions of logical validity and logical inference which are analysed . . . bymeans of (first order) predicate logic . . .

[T]he results are important as object lessons: once one has seen the simpleconsiderations [wrt (iii)] concerning Godel’s completeness theorem one cannotdoubt the possibility of philosophical proof or, as one might put it, of informal rigour.

(1967a, p. 202)4/25


What is informal rigour: Beginning of longer answer

Claim: All Ø 7 instances of (mid-)Kreisel’s application of informal rigour

can be subsumed under a single scheme and sub-scheme:

IRS: Reflection on common (or intuitive) and novel concepts is

combined with a mathematical theory to resolve an open question.

SS: A common concept analyzed by showing that it is ‘squeezed

between’ two precise concepts.

The ‘old fashioned’ idea is that one obtains rules and definitions by analyzing

intuitive notions and putting down their properties . . . Informal rigour wants

(i) to make this analysis as precise as possible . . . in particular to eliminate

doubtful properties of the intuitive notions when drawing conclusions . . . and

(ii) to extend this analysis [by] not to leav[ing] undecided questions which

can be decided by full use of evident properties of these intuitive notions.

Below the principal emphasis is on intuitive notions which do not occur in ordinary

mathematical practice (so-called new primitive notions) . . . (1967b, p. 138)

5/25


Formal versus informal rigour

Extensionaldefiniteness

Predicativedefinability

Kreisel 1965, 1967 a,b,c

Foundational standpoints

FOValidity

CreatingSubject

Standardvs

nonstandardmodels

ContinuumHypothesis

Predicativism(1958c/60a/60b/62a)

Finitism(1951/60b/65/70a)

Intuitionism(1958a/62b/68/70b)

Predicativeproof

Finitistproof

Intuitionisticvalidity

Choice sequence

Mechanical procedure

Formal system

Formal rigour

Formalism

Informal rigour

Markov’sPrinciple

ZF2PA2

FOL HPC PA1 ZF1HYP

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�11-CA

FIM

PRA

Cumulativetype

structure

Naturalnumber

structure

Pre-Kreisel… -1949

Early-Kreisel1950 -1962

Mid-Kreisel1963 -1971

Late-Kreisel1972-…

SOL

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Realism

Positivism Explication

FREGE

GÖDEL

CARNAP

DEDEKINDZERMELO

HILBERT &BERNAYS

KLEENE

TURING

COHENROBINSONBOURBAKI BROUWER

HEYTINGKLEENE

POINCARÉ

RUSSELL

Finitistfunction

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KeyInformal RigourConcepts StructuresStatements THEORIESPositionsFIGURESMethods

Kreisel 1987a,b, 1989


The Informal Rigour SchemeI Three types of concepts:

I Common (or ‘intuitive’): C1, C2, . . . (e.g mechanical, valid)I Precise: fi1, fi2, . . . (e.g. recursive, true in all models)I Novel (or ‘new’): N1, N2, . . . (e.g. truth, the CS has evidence for)

I Let Ï be an open questions about common or precise concepts.1) Reflection on concepts and practice yields:

i) �1 expresses relations between common concepts.ii) �2 bridging principles btw common and precise concepts.iii) �3 bridging principles btw novel, common, precise concepts.

2) TP a mathematical theory in the precise language – e.g. ZF.3) TK = TP + �1 + �2 + �3.4) Philosophical Theorem: TK „ Ï or TK „ ¬Ï.

I Example: Kreisel’s CH argument.I Def (Ï) is the common property of mathematical definiteness.I Kreisel argues ’Ï(Def (Ï) … Z

2 |=s2 Ï or Z

2 |=s2 ¬Ï).

I Zermelo’s Categoricity Thm allows us to conclude Def (CH).7/25


Brouwer on the Creating SubjectBrouwer suggested that the notion of a Creating Subject [CS] – or

idealized mathematician – can be used within intuitionisitic mathematics.

Let [A] be a mathematical assertion that cannot be tested, i.e. for which no

method is known to prove either its absurdity or the absurdity of its absurdity.

Then the creating subject can, in connection with the assertion A, create an

infinitely proceeding sequence of rational numbers a1a2a3, . . . (1948, p. 478)

I A cannot be tested i� no method is known for proving ¬A or ¬¬A.I There are untested statements – e.g. Goldbach’s conjecture. (Kreisel

called this an empirical fact.)I Brouwer made use of untested statements together with the CS in his

weak counterexamples – i.e. implausibility arguments which illustratewhy certain principles should not be constructively provable. E.g.

(LEM) A ‚ ¬A(Apart) ’x’y(x ”= y æ x#y)

(MP) ’x(A(x) ‚ ¬A(x)) æ (¬¬÷xA(x) æ ÷xA(x))

(GMP) ’–(¬¬÷x–(x) = 0 æ ÷x–(x) = 0)

8/25


Brouwer on non-identity and apartnessWithin intuitionistic mathematics . . .

I Reals are given by sequences of rationals ÈrnÍ s.t. |rn ≠ rn+1| < 2≠n .

I A distinction is drawn between1) Non-identity: x ”= y – i.e. equality of x, y is constructively absurd.2) Apartness: x#y – i.e. there is a positive di�erence between x and y.

I Classically x ”= y and x#y are equivalent.I Can we show that the following is not constructively provable?

(Apart) ’x’y(x ”= y æ x#y)

Weak counterexample: Let A be untested and define

rn =

Y]

[

0 the CS does not have evidence for A or ¬A at stage n2

≠mthe CS obtained evidence for A at stage m Æ n

≠2≠m

the CS obtained evidence for ¬A at stage m Æ n

We can show constructively 1) r ”= 0 and 2) not r#0.9/25


Kreisel’s contributions to intuitionism

Brouwer (1927/48)[Creating Subject

arguments]

Heyting (e.g.1930)[towards intuitionistic analysis]

Heyting (1930/56)[proof/BHK interpretation,

HPC]

Beth (1956)[Beth models, completeness]

Kreisel & Gödel (1958/62)

[internal semantic,incompleteness of HPC]

Kreisel (1958)[lawless sequences

Open Data]

Dyson & Kreisel (1961)[König’s Lemma]

Kripke (1965)[Kripke models]

Cohen (1963)[forcing]

Kreisel (1963/65)[BC-N, BC-C]

Kleene & Vesley (1965)

[Bar induction, FIM]

Kreisel (1962b/65)[Theory of Constructions]

Kreisel (1967b)[Informal Rigour, Creating Subject] Kripke scheme

(1965?)

Myhill (1967)[Inconsistency of BC-C

with CS/KS]

Kleene (1945)[realizability]

Troelstra (1977b)[Choice Sequences]

Gödel (1958)[Dialetica]

Kreisel (1959)[Modified realizability]

Scott (1970)[Constructive Validity]

Kreisel-HowardCorrespondence

[formulas as types]

???

TYPEDUNTYPED

Kreisel (1970a)[Church’s Thesis]


Kreisel and the Creating Subject

I In the mid-1960s, MP, GMP, Apart, etc. were controversial.I Kreisel (1967b) called Brouwer’s CS argument “empirical”.I Wanted to replace it so as “not to leave undecided questions which

can be decided by full use of evident properties of . . . intuitive notions”.I Proposed to do this by formalizing the novel concept

the CS has evidence of A at stage nI Kreisel wished to turn Brouwer’s argument into a formal refutation

of (Apart) by reasoning about this relation.I Kreisel’s strong counterexample is constructed as follows:

1) Propose and motivate axioms about the Creating subject (CS).2) Extend intuitionistic analysis (FIM0) to the language of CS (FIM

+0 ).

3) Show that FIM+0 + CS „ ¬GMP where

I GMP is ’–(¬¬÷x–(x) = 0 æ ÷x–(x) = 0)

I FIM0 „ ¬GMP æ ¬Apart

11/25


Axiomatizing the Creating SubjectI Kreisel’s take on the Creating Subject (1967b, p. 179):

I “The essential point: . . . proofs arranged in an Ê-order (each

proof of course being a mental, not necessarily finite, object on

the intuitionistic conception).”

I Notation: n = 0, 1, 2, . . . for stages and

⇤nA = the Creating Subject has an evidence of A at the stage n.I Kreisel’s axiomatization:

(CS1) ⇤nA ‚ ¬⇤nADecidability of ⇤nA – i.e. “we can recognize a proof when we see one.”

(CS2) A æ ¬¬÷n(⇤nA)

“[T]he only grounds we could have for asserting that a proposition

would never be proved are that we already know it to be absurd – and

not e.g. that people are too stupid.”

(CS3) ÷n(⇤nA) æ AReflection principle expressing the soundess of the constructive proof.

I Let CS = CS1 + CS2 + CS3.

12/25


An axiomatization of intuitionistic analysisI The language of FIM0 (LFIM0) is that of 2nd-order arithmetic with

I x, y, z, . . . as numerical variables

I names s, t, . . . for primitive recursive functions

I variables –, —, “, . . . for choice sequences of type N æ NI —(x) =df È—(0), —(1), . . . , —(x ≠ 1)Í.I – œ K0 =df ’x’y(–(x) ”= 0 æ –(x ı y) = –(x)) · ’—÷x(–(—(x)) ”= 0)

(K0 is the class of continuous functionals)

I The axioms of FIM0 (a subsystems of Kleene & Vesley, 1965)

I Primitive Recursive Analysis:(PrAn1) The axioms of first-order Heyting Arithmetic, inclusive of

I induction in the full language of FIM0I identity axiom ’x’y(x = y æ –(x) = —(y))

I defining equations of all primitive recursive terms

(PrAn2) The comprehension scheme÷–’x(–(x) = t(x))

where t(x) is any term of LFIM0 which does not contain – free.I BC-N: Brouwer’s function-number continuity principle

’–÷xA(–, x) æ ÷“ œ K0’–A(–, “(–))

13/25


Extending the language with ⇤n

I Kreisel’s argument requires that we define a choice sequence

– : N æ N in terms in terms of a formula involving ⇤n.

I L+FIM0 includes statements ⇤nA with n free.

I A(x) œ �+0 if built up from �

00-formulas and ⇤nB.

I FIM+0 is obtained from FIM0 by

I Adding terms and axioms stating ‰A(x) is the characteristic function

of A(x) œ �+0 .

I Extending function comprehension to �+0 -formulas.

I Lemma: �+0 -formulas are decidable in FIM+

0 .

I Proof: By CS1.I Let TK =df FIM+

0 + CS – i.e. this is the Kreiselian theory in

which the refutation of GMP (and hence Apart) is carried out.

14/25


Kreisel’s arugumentOur reconstruction of Kreisel’s (1967b) result:

Theorem FIM+0 + CS „ ¬’–(¬¬÷x–(x) = 0 æ ÷x–(x) = 0)

i) Over FIM0 + CS2 + GMP implies

’—÷m[(÷x < m)(—(x) ”= 0) ‚ ⇤m’x(—(x) = 0)]ii) Given an arbitrary — : N æ N we use �+

0 -comprehension to define

–(m) = 0 ¡ ((÷x < m)(—(x) ”= 0) ‚ ⇤m’x(—(x) = 0))iii) Via CS3 (Reflection) and BC-N we can show that this implies

(GLEM) ’—(’x(—(x) = 0) ‚ ¬’x(—(x) = 0))

Corollaries Let TK = FIM+0 + CS. i) TK „ ¬Apart;

ii) TK is non-conservative over FIM0.

I Open question: Are Apart and GMP constructively justifiable?I Kreisel: We can give an informally rigorous answer as follows:

i) Analyze the novel concept the CS has evidence for A.

ii) Prove a “philosophical theorem” – i.e. TK „ ¬Apart.iii) The use of ⇤n is “essential” since GMP is applied to “an

empirically defined sequence –”.I Our take: Kreisel’s CS argument is paradigmatic of informal rigour.

I But it’s also largely forgotten because of the the Kripke Schema and

Myhill’s inconsistency with BC-C.

15/25


The context of “La Predicative” (1960a)

Kreisel (1955)The class [of hyperarithmetical predicates] provides a precise and satisfactory definition of the notion of predicative sets, based on the concept of constructive ordinal.

Kreisel (1960a), “La Prédicativé”Without affirming the identification of predicative definitions with the class [HYP] of hyper-arithmetic definitions, I will describe some results which bear on this identification: they demonstrate for [HYP] properties which are evident for the intuitive notion of predicativity.

Kleene (1955)“Hierarchies of number-

theoretic predicates”[HYP] Poincare (1910)

Russell (1908)

Kreisel (1959)[Cantor-Bendixson]

Spector (1955)

Kleene (1938)

Grzegorczyk, Mostowski,

Ryll-Nardzewski (1958)

Kreisel (1961)“Set theoretic problems suggested by the notion of potential infinity”

[extensional definiteness]

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[�ck1 ]

Addison (1959)

Kreisel (1958b)[lawless

sequences,open data]

Kreisel (1962a)“The axioms of choice and the hyperarithmetical functions”

???

Feferman (1963/65)[arithmetical forcing]

Kreisel (1965)

Cohen (1963)

???

Gödel (1938)


The common concept: predicative definabilityPoincare and Russell introduced the notion of predicativity with regard to a

very general philosophical problem: which sentences of everyday language or, more

generally, which symbolic expressions can we regard as definitions? . . . Poincare

was very impressed with the following property of paradoxical definitions: . . . the

defined object constitutes a particular value of a variable which appears in the

expression of the property. (“La Predicative” 1960a, p. 371)

I X is predicative in the common sense i� it is definable in manner whichdoes not involve quantification over the totality to which it belongs.

I N is predicative.I Then the arithmetical sets are predicative

Arith = {X ™ N : N |= ’x(x œ X ¡ Ï(x)), Ï(x) œ L1a}

I Sets defined in terms of second-order quantification over P(N) arenot predicative (at least prima facie).

I Do the intuitively predicative sets Pred properly extend Arith?

17/25


The Squeezing Scheme

I Let C (x) be a common concept.

I Do we have ’x(C (x) ¡ fi(x)) for some precise concept fi(x)?I Does C (x) admit an extensionally adequate mathematical analysis?

I A squeezing argument is a schema for demonstrating this:

1) ’x(fin(x) æ C (x)) œ �2 – i.e. fin(x) is a su�cient condition2) ’x(C (x) æ fiw(x)) œ �2 – i.e. fiw(x) is a necessary condition3) TP „ ’x(fiw(x) æ fin(x)) – i.e. a mathematical theorem

4) TK „ ’x(C (x) ¡ fiw(x)) – i.e. a philosophical theoremI Example: Kreisel’s validity argument.

I C (x) is the common concept of first-order validity Val(x)

I fin(Ï) =df „1 Ï – i.e. first-order derivability D(x)

I fiw(Ï) =df |=1 Ï – i.e. truth in all models V (x)

I WKL0 „ ’x(V (x) æ D(x)) – i.e. Godel’s Competeness Thm.I Let TK = WKL0 + 1) + 2).I Then TK „ ’x(Val(x) ¡ V (x)) is the result of the squeezing.

18/25


The narrow concept: hyperarithmeticality[T]he fundamental idea of predicativity: in a predicative definition only quantifiers

relating to already constructed sets are used. (1960a, p. 377)

I I.e. X is predicative in the narrow sense i� it is definable using

quantifiers over sets which have already been constructed.

I Kleene’s (1955) ramified analytical hierarchy :

I RA0 = Arith

I RA–+1 = the class of sets definable using restricted quantifiers ofthe form ’X œ RA–, ÷X œ RA–

I RA⁄ =t

–<⁄RA–

I HYP =df RAÊck1

I Kleene showed HYP = {X : X ÆT H– for – Æ Êck1 }.

I Recall: HYP ) Arith.I Kreisel: [This definition] puts in a precise form the intuitive idea

expressed by “already”. (1960a, p. 377)

I Thesis 1: ’X(X œ HYP æ X œ Pred).

19/25


The wide concept: extensional definitenessThe following theorems concern an another idea of Poincare (1910, p. 47)

concerning the definition of predicativity: a definition of D is said to be

predicative if an enlargement of the class of the sets considered does notchange the set defined by D. (Kreisel 1960a, p. 378)

I call a classification predicative if it is not changed by the introduction ofnew elements . . . What is here meant by the word ‘predicative’ is best illustrated byan example. If I am to deposit a set of objects into a number of boxes two things canoccur: either the objects already deposited are conclusively in their places, or, when Ideposit a new object, I must always take the others out again (or at any rate some ofthem). In the first case I call the classification predicative, in the second not.

(Poincare 1910, p. 47)

Basic idea: X ™ N is predicative in the wide sense if there is Ï(x) œ L2a s.t.

X = {n œ N : M |= Ï(n)} = {n œ N : N |= Ï(n)}for every Ê-submodel M of N satisfying appropriate axioms – i.e.

I Dom1(M) = NI Dom2(M) ™ P(N)

I M |= “Predicativism” 20/25


Extensional definiteness and �11-definability

Defnition: X ™ N is �11-definable just in case there are L2

a-formulasÂ1(x, X) and Â2(x, X) not containing second-order quantifiers s.t.

X = {n œ N : N |= ’XÂ1(n, X)} = {n œ N : N |= ÷XÂ2(n, X)}

Proposition (Kreisel 1961): If an Ê-model M |= �11-CA0 and X is

�11-definable – say with via the �

11-definition ’XÂ1(x, X) – then

X = {n œ N : M |= ’XÂ1(n, X)} = {n œ N : N |= ’XÂ1(n, X)}

– i.e. ’XÂ1(x, X) is absolute (or “extensionally definite”).

Thesis 2: ’X(X œ Pred æ X œ �11)

Kreisel: At first glance this notion is broader than the fundamental idea of

predicativity; because it allows the . . . use of quantifiers relating to an

indeterminate class of sets while the other idea does not. (1960a, p. 378)

21/25


Squeezing predicative definabilityTheorem (Kleene, 1955) X is �1

1-definable if any only if X œ HYP.

So we can now argue:1) ’X(X œ HYP æ X œ Pred) Thesis 1 œ �2

2) ’X(X œ Pred æ X œ �11) Thesis 2 œ �2

3) ’X(X œ �11 æ X œ HYP) ACA0 = TP „ Kleene’s Theorem

4) ’X(X œ Pred ¡ X œ HYP) TK „ philosophical theorem

I Our take: This analysis of predicativity is paradigmatic of informal rigour.I Kreisel: “This theorem clearly expresses Poincare’s idea.” (1960a, p. 380)

I I.e. it expresses a “definitional completeness” of HYP:

If X œ �11 = HYP, then there exists an – < Êck

1 and an –-ramified Ï(x)

s.t. X = {n : N |= Ï(n)} œ RA–.I Compare the role of the Completeness Theorem in the validity argument:

If |=1 Ï, then there exists a derivation D showing „1 Ï.22/25


Summary and further topicsI We have argued

1) ‘Informal rigour’ corresponds to a definite method (IRS/SS).2) Kreisel used this method to evaluate principles and positions central

to 20th century debates in philosophy of maths.2Õ) Kreisel’s arguments illustrate the (non-trivial) interplay between

mathematical theorems and philosophical conclusions.I We also have promised you an argument for

3) Informal rigour is still relevant today.I Further examples and questions:

i) Other applications of informal rigour – e.g. lawless/randomsequence, feasibly computable function, fair electoral method?

iÕ) Squeezing arguments for non-classical validity notions?ii) What was Kreisel’s role in the discovery/reception of forcing?

- App. of informal rigour absolute, generic, continuum?

- Can informal rigour be used to decide CH?

-Frege

Hilbert=

? Kreisel

Cohen, Robinson, Mostowski=

? Woodin, Koellner

Feferman, Hamkins

iii) What was Kreisel’s background view of (math.) concepts?

23/25


Full paper

“On the methodology of informal rigour: Set theory, semantics, andintuitionism”, forthcoming in Intuitionism, Computation, and Proof:Selected themes from the research of G. Kreisel, M. Antonutti-Marfori and M. Petrolo (editors), Springer.

I Thanks to Mic Detlefsen (et al.) for discussion.

I Comments very welcome.

24/25

https://warwick.ac.uk/fac/soc/philosophy/people/dean/dean-kurokawa-2020-v-0.94.pdf

https://warwick.ac.uk/fac/soc/philosophy/people/dean/dean-kurokawa-2020-v-0.94.pdf

https://www.ihpst.cnrs.fr/en/activites/conferences/colloque-intuitionism-computation-and-proof-selected-themes-research-g-kreisel

https://www.ihpst.cnrs.fr/en/activites/conferences/colloque-intuitionism-computation-and-proof-selected-themes-research-g-kreisel


Overview of Kreisel’s applications of informal rigourTable 1

Concept or question Section Illustrative original sources Illustrative Kreisel references

Labeled informalrigour?

Notes

first-order validity 4.1 Bolzano 1834, Frege 1879,Gödel 1929, Tarski & Vaught 1956

1950, 1965, 1967a, 1967b, 1967c

Yes See Dean (2020) for a reconstruction of Kreisel's (1950) engagement with the arithmetized completeness theorem.

second-order validity 4.3 Henkin 1950 1967b, 1967c Yes Kreisel (1967b,c) mentions but does not develop this possibility. See Kennedy and Väänänen (2017) for discussion.

Creating Subject, Apart? GMP?

4.2 Brouwer 1948, Kleene & Vesley 1965

1967b Yes See van Atten (2018) for a reconstruction.

Is the CH a definite statement?

4.3 Zermelo 1930b, Gödel 1947/64 1965, 1967a, 1967b, 1967c, 1969, 1971

Yes Requires a subargument for the correctness of Kreisel's analysis of mathematical definiteness.

set (of things) A.1 Zermelo 1930a, Gödel 1947/64 1965, 1967a, 1967b, 1967c

Yes Kreisel contrasts the concept set (of things) with class and property. His most detailed argument that reflection on this concept as embodied in the cumulative hierarchy leads to ZF appears in 1965 §1 and resembles Scott's (1974) "levels" theory.

Standard vs non-standard models: which comes first?

A.2 Kreisel 1950, Scott 1961 1967b Yes Requires a subargument for the correctness of Kreisel's analysis of structure S1 is more fundamental than S2.

mechanical process 3.1 Turing 1936 1967a, 1987a Yes In 1967a, Kreisel accepted Turing's analysis. In (e.g.) 1987a this is less clear.

finitist function / proof A.3.1 Hilbert & Bernays 1934 1951, 1958, 1965, 1970a

No See Dean (2015) for a reconstruction and critique.

predicative definability/provability, extensional definiteness

A.3.2 Poincaré 1910 1960a, 1960b, 1961 No See Hallett (2011) for a reconstruction of Poincaré's anticipation of absoluteness.

predicative provability A.3.2 Turing 1939, Wang 1954 1958, 1970a No See Feferman (2005) and Dean & Walsh (2017) reconstructions.

intuitionistic validity A.3.3 Heyting 1930, 1956 1958, 1962b, 1970b No See Dean & Kurokawa (2016) on Kreisel's (1962b) analysis via his Theory of Constructions.

absolutely free (or lawless) sequence

A.3.3 Brouwer 1942 1958b, 1965, 1968 No See Troelstra (1977b) for a reconstruction.

�1

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Documents

Dean Kurokawa Slides - philosophy.nd.eduWhat is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Context and assumptions I Georg Kreisel: “On informal rigour and