On the methodology of informal rigour: Set theory, semantics, and intuitionism
MWPMW 2December2020 Notre Dame
Walter Dean University of Warwick Hidenori Kurokawa Kanazawa University What is ‘informal rigour’? Creating Subject La Pr´edicativit´e 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-Hillelú, Carnap, Dummett, Hacking, Harsanyi´ , Heytingú, 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 about intuitionistic 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 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi 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 Pr´edicativit´e” 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 G¨odel vs ...) I Other teasers:
I Kreisel was the original ‘K’intheBHK 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’? Creating Subject La Pr´edicativit´e Envoi What is informal rigour: Short answer Two 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 traditional notions in philosophy have an essentially unambiguous formulation when one thinks about them ... [A]lso, when so formulated by essential use of mathematical logic, they have non trivial consequences for the analysis of mathematical experience ... Among them are the well known cases (i) the notion of mechanical process, its stability in the sense that apparently di erent formulations lead to the same results ... (ii) the notion of aggregate which is analysed by means of the hierarchy (theory) of types, and, of course (iii) the notions of logical validity and logical inference which are analysed ... by means of (first order) predicate logic ... [T]he results are important as object lessons: once one has seen the simple considerations [wrt (iii)] concerning Godel’s¨ completeness theorem one cannot doubt the possibility of philosophical proof or, as one might put it, of informal rigour. (1967a, p. 202)
4/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi What is informal rigour: Beginning of longer answer
Claim:All 7 instances of (mid-)Kreisel’s application of informal rigour Ø can be subsumed under a single scheme and sub-scheme: IRS: Reflection on common (or intuitive)andnovel 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
What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi
Formal versus informal rigour Kreisel 1987a,b, 1989 Late-
Kreisel
1972-… Foundational standpoints
Predicativism Finitism Intuitionism
Mid- Kreisel 1965, 1967 a,b,c (1958c/60a/60b/62a) (1951/60b/65/70a) (1958a/62b/68/70b) Kreisel
1963 -
1971 Creating Predicative Choice Standard FO Continuum Informal Subject vs definability sequence rigour Validity Hypothesis Finitist nonstandard Extensional Markov’s function models definiteness Intuitionistic Principle validity Finitist
Explication Predicative proof Early- Positivism proof
Kreisel
BOURBAKI … -1949 BROUWER POINCARÉ
Formal HILBERT & PRA Realism rigour BERNAYS 2 GÖDEL ZF PA2 Key Formal Informal Rigour system ZERMELO Concepts DEDEKIND RUSSELL Structures TURING Statements Natural Cumulative THEORIES Mechanical FREGE number type Positions procedure FIGURES structure structure Methods What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi The Informal Rigour Scheme I 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: T Ï or T Ï. K „ K „¬ I Example: Kreisel’s CH argument. I Def (Ï) is the common property of mathematical definiteness. 2 s 2 s I Kreisel argues Ï(Def (Ï) Z = Ï or Z = Ï). ’ … | 2 | 2 ¬ I Zermelo’s Categoricity Thm allows us to conclude Def (CH).
7/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Brouwer on the Creating Subject Brouwer suggested that the notion of a Creating Subject [CS] – or idealized mathematician –canbeusedwithin intuitionisitic mathematics. Let [A]beamathematicalassertionthatcannot be tested,i.e.forwhichno 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,createan
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 illustrate why certain principles should not be constructively provable. E.g. (LEM) A A ‚¬ (Apart) x y(x = y x#y) ’ ’ ” æ (MP) x(A(x) A(x)) ( xA(x) xA(x)) ’ ‚¬ æ ¬¬÷ æ÷ (GMP) –( x–(x)=0 x–(x)=0) ’ ¬¬÷ æ÷ 8/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Brouwer on non-identity and apartness Within intuitionistic mathematics ... n I Reals are given by sequences of rationals rn s.t. rn rn+1 < 2≠ . È Í | ≠ | I A distinction is drawn between 1) Non-identity: x = y –i.e.equalityofx, 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 x#y) ’ ’ ” æ Weak counterexample:LetA be untested and define
0 the CS does not have evidence for A or A at stage n m ¬ rn = 2≠ the CS obtained evidence for A at stage m n Y m Æ 2≠ the CS obtained evidence for A at stage m n ]≠ ¬ Æ We can show[ constructively 1) r =0and 2) not r#0. ” 9/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Kreisel’s contributions to intuitionism
Troelstra (1977b) Kreisel-Howard [Choice Sequences] Correspondence Kreisel (1970a) Myhill (1967) [formulas as types] [Church’s Thesis] [Inconsistency of BC-C with CS/KS] Kreisel (1967b) Scott (1970) [Informal Rigour, [Constructive Validity] Creating Subject] Kripke scheme Kripke (1965) Kreisel (1962b/65) Cohen (1963) (1965?) [Kripke models] [forcing] [Theory of Constructions]
??? Kreisel Kleene & Vesley Kreisel & Gödel (1963/65) (1965) Dyson & Kreisel (1958) [BC-N, BC-C] [Bar induction, FIM] Kreisel (1959) Kreisel (1961) [lawless sequences (1958/62) [Modified realizability] [König’s Lemma] Open Data] [internal semantic, incompleteness of HPC] Gödel (1958) Beth (1956) [Dialetica] [Beth models, completeness] Kleene (1945) [realizability]
Heyting (1930/56) Heyting (e.g.1930) [proof/BHK interpretation, [towards intuitionistic analysis] HPC] UNTYPED TYPED
Brouwer (1927/48) [Creating Subject arguments] What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Kreisel and the Creating Subject
I In the mid-1960s, MP, GMP, Apart,etc.werecontroversial.
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 n
I 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)tothelanguageofCS (FIM0 ). 3) Show that FIM+ +CS GMP where 0 „¬ I GMP is –( x–(x)=0 x–(x)=0) ’ ¬¬÷ æ÷ I FIM GMP Apart 0 „¬ æ¬
11/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Axiomatizing the Creating Subject
I 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) A A ⇤n ‚¬⇤n Decidability of ⇤nA –i.e. “we can recognize a proof when we see one.” (CS2) A n( A) 欬÷ ⇤n “[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( A) A ÷ ⇤n æ Reflection principle expressing the soundess of the constructive proof.
I Let CS = CS1 +CS2 +CS3.
12/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi An axiomatization of intuitionistic analysis
I The language of FIM ( ) is that of 2nd-order arithmetic with 0 LFIM0 I x, y, z,... as numerical variables I names s, t,... for primitive recursive functions I variables –,—,“,... for choice sequences of type N N æ I —¯(x)= —(0),—(1),...,—(x 1) . df È ≠ Í I – K = x y(–(x) =0 –(x ı y)=–(x)) — x(–(—¯(x)) =0) œ 0 df ’ ’ ” æ ·’ ÷ ” (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 FIM0 I identity axiom x y(x = y –(x)=—(y)) ’ ’ æ I defining equations of all primitive recursive terms (PrAn2) The comprehension scheme – x(–(x)=t(x)) ÷ ’ where t(x) is any term of which does not contain – free. LFIM0 I BC-N: Brouwer’s function-number continuity principle – xA(–, x) “ K –A(–,“(–)) ’ ÷ æ÷ œ 0’ 13/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi
Extending the language with ⇤n
I Kreisel’s argument requires that we define a choice sequence – : N N in terms in terms of a formula involving ⇤n. æ+ I A n FIM0 includes statements ⇤n with free. L + 0 I A(x) 0 if built up from 0-formulas and ⇤nB. +œ I FIM0 is obtained from FIM0 by I Adding terms and axioms stating ‰A(x) is the characteristic function + of A(x) 0 . œ + I Extending function comprehension to 0 -formulas. + + I Lemma: 0 -formulas are decidable in FIM0 . I Proof: By CS1. + I Let TK =df FIM0 +CS–i.e.thisistheKreiselian theory in which the refutation of GMP (and hence Apart) is carried out.
14/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Kreisel’s arugument Our reconstruction of Kreisel’s (1967b) result: + Theorem FIM0 +CS –( x–(x)=0 x–(x) = 0) i) Over FIM0 +CS2 + GMP„¬implies’ ¬¬÷ æ÷ — m[( x < m)(—(x) =0) x(—(x)=0)] ’ ÷ ÷ ” ‚ ⇤m ’ ii) Given an arbitrary — : we use +-comprehension to define N æ N 0 –(m)=0 (( x < m)(—(x) =0) x(—(x)=0)) ¡ ÷ ” ‚ ⇤m ’ iii) Via CS3 (Reflection) and BC-N we can show that this implies (GLEM) —( x(—(x)=0) x(—(x)=0)) ’ ’ ‚¬’ Corollaries Let T =FIM+ +CS.i)T Apart; K 0 K „¬ ii) TK is non-conservative over FIM0.
I Open question:AreApart 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.T Apart. K „¬ 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 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi The context of “La Pr´edicativ´e” (1960a)
Kreisel (1965) ??? Feferman (1963/65) [arithmetical forcing]
Cohen (1963) Kreisel (1962a) “The axioms of choice and the hyperarithmetical functions” ???
Kreisel (1961) Kreisel (1960a), “La Prédicativé” “Set theoretic problems suggested Without affirming the identification of predicative definitions with the by the notion of potential infinity” class [HYP] of hyper-arithmetic definitions, I will describe some results Kreisel (1958b) [extensional definiteness] which bear on this identification: they demonstrate for [HYP] properties [lawless which are evident for the intuitive notion of predicativity. sequences, open data] Kreisel (1959) [Cantor-Bendixson] Kreisel (1955) The class [of hyperarithmetical predicates] Spector (1955) provides a precise and satisfactory definition of the notion of predicative sets, based on the Addison (1959) Grzegorczyk, Mostowski, concept of constructive ordinal. Ryll-Nardzewski (1958)
Kleene (1955) “Hierarchies of number- theoretic predicates”
Kleene
Poincare´ 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 Pr´edicativ´e”1960a, p. 371)
I X is predicative in the common sense i it is definable in manner which does not involve quantification over the totality to which it belongs.
I N is predicative. I Then the arithmetical sets are predicative
1 Arith = X N : = x(x X Ï(x)),Ï(x) a { ™ N| ’ œ ¡ œL }
I Sets defined in terms of second-order quantification over (N) are P not predicative (at least prima facie).
I Do the intuitively predicative sets Pred properly extend Arith?
17/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi 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) C(x)) 2 –i.e.fin(x) is a su cient condition ’ æ œ 2) x(C(x) fiw(x)) 2 –i.e.fiw(x) is a necessary condition ’ æ œ 3) T x(fiw(x) fin(x)) –i.e.amathematical theorem P „’ æ
4) T x(C(x) fiw(x)) –i.e.aphilosophical theorem K „’ ¡ I Example: Kreisel’s validity argument.
I C(x) is the common concept of first-order validity Val (x) I fin(Ï)= 1 Ï – i.e. first-order derivability D(x) df „ I fiw(Ï)= =1 Ï –i.e.truthinallmodelsV (x) df | I WKL0 x(V (x) D(x)) –i.e.G¨odel’s Competeness Thm. „’ æ I Let TK =WKL0 +1)+2). I Then T x(Val (x) V (x)) is the result of the squeezing. K „’ ¡
18/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi The narrow concept: hyperarithmeticality
[T]he fundamental idea of predicativity:inapredicativedefinitiononlyquantifiers 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 of the form X RA–, X RA– ’ œ ÷ œ I RA⁄ = –<⁄ RA– I HYP =df RAÊck 1 t ck I Kleene showed HYP = X : X H– for – Ê . { ÆT Æ 1 } 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 HYP X Pred). ’ œ æ œ
19/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi The wide concept: extensional definiteness The following theorems concern an another idea of Poincar´e(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 not change the set defined by D.(Kreisel1960a,p.378)
Icallaclassificationpredicativeifitisnotchangedbytheintroductionof new elements ... What is here meant by the word ‘predicative’ is best illustrated by an example. If I am to deposit a set of objects into a number of boxes two things can occur: either the objects already deposited are conclusively in their places, or, when I deposit a new object, I must always take the others out again (or at any rate some of them). In the first case I call the classification predicative, in the second not.
(Poincar´e1910, p. 47)
2 Basic idea: X N is predicative in the wide sense if there is Ï(x) a s.t. ™ œL X = n N : = Ï(n) = n N : = Ï(n) { œ M| } { œ N| } for every Ê-submodel of satisfying appropriate axioms – i.e. M N I Dom1( )=N M I Dom2( ) (N) M ™P I = “Predicativism” 20/25 M| What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi 1 Extensional definiteness and 1-definability
1 2 Defnition: X N is 1-definable just in case there are a-formulas ™ L Â1(x, X) and Â2(x, X) not containing second-order quantifiers s.t.
X = n N : = XÂ1(n, X) = n N : = XÂ2(n, X) { œ N| ’ } { œ N| ÷ } 1 Proposition (Kreisel 1961): If an Ê-model = 1-CA0 and X is 1 1 M| -definable – say with via the -definition XÂ1(x, X) –then 1 1 ’
X = n N : = XÂ1(n, X) = n N : = XÂ1(n, X) { œ M| ’ } { œ N| ’ }
–i.e. XÂ1(x, X) is absolute (or “extensionally definite”). ’ Thesis 2: X(X Pred X 1) ’ œ æ œ 1 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 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Squeezing predicative definability Theorem (Kleene, 1955) X is 1-definable if any only if X HYP. 1 œ So we can now argue: 1) X(X HYP X Pred) Thesis 1 ’ œ æ œ œ 2 2) X(X Pred X 1) Thesis 2 ’ œ æ œ 1 œ 2 3) X(X 1 X HYP) ACA =T Kleene’s Theorem ’ œ 1 æ œ 0 P „
4) X(X Pred X HYP) T philosophical theorem ’ œ ¡ œ K „
I Our take: This analysis of predicativity is paradigmatic of informal rigour.
I Kreisel: “This theorem clearly expresses Poincar´e’s idea.” (1960a, p. 380) I I.e. it expresses a “definitional completeness” of HYP: If X 1 =HYP, then there exists an –<Êck and an –-ramified Ï(x) œ 1 1 s.t. X = n : = Ï(n) RA–. { N| }œ I Compare the role of the Completeness Theorem in the validity argument:
If =1 Ï, then there exists a derivation showing 1 Ï. | D „ 22/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Summary and further topics I 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/random sequence, 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 ? Kreisel ? Woodin, Koellner - Hilbert = Cohen, Robinson, Mostowski = Feferman, Hamkins iii) What was Kreisel’s background view of (math.) concepts?
23/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Full paper
“On the methodology of informal rigour: Set theory, semantics, and intuitionism”, 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 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Overview of Kreisel’s applications of informal rigour Table 1
Concept or question Section Illustrative original sources Illustrative Kreisel Labeled Notes references informal rigour?
first-order validity 4.1 Bolzano 1834, Frege 1879, 1950, 1965, 1967a, Yes See Dean (2020) for a reconstruction of Kreisel's (1950) Gödel 1929, Tarski & Vaught 1956 1967b, 1967c 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, 4.2 Brouwer 1948, Kleene & Vesley 1967b Yes See van Atten (2018) for a reconstruction. Apart? GMP? 1965
Is the CH a definite 4.3 Zermelo 1930b, Gödel 1947/64 1965, 1967a, 1967b, Yes Requires a subargument for the correctness of Kreisel's statement? 1967c, 1969, 1971 analysis of mathematical definiteness.
set (of things) A.1 Zermelo 1930a, Gödel 1947/64 1965, 1967a, 1967b, Yes Kreisel contrasts the concept set (of things) with class 1967c 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- A.2 Kreisel 1950, Scott 1961 1967b Yes Requires a subargument for the correctness of Kreisel's
standard models: analysis of structure S1 is more fundamental than S2. which comes first?
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, No See Dean (2015) for a reconstruction and critique. 1970a
predicative A.3.2 Poincaré 1910 1960a, 1960b, 1961 No See Hallett (2011) for a reconstruction of Poincaré's definability/provability, anticipation of absoluteness. extensional definiteness
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 A.3.3 Brouwer 1942 1958b, 1965, 1968 No See Troelstra (1977b) for a reconstruction. lawless) sequence
1 References
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