Indefiniteness, definiteness and semi-intuitionistic theories of sets

Transcription

1 Indefiniteness, definiteness and semi-intuitionistic theories of sets Michael Rathjen Department of Pure Mathematics University of Leeds Brouwer Symposium Amsterdam December 9th, 2016

2 The continuum hypothesis, CH First version: Every infinite A R is either of size N or of the same size as R.

3 The continuum hypothesis, CH First version: Every infinite A R is either of size N or of the same size as R. Second version: The cardinality of R is ℵ 1 (or shorter: 2 ℵ 0 = ℵ 1 ).

4 Brouwer

5 Brouwer Die möglichen Mächtigkeiten Von anderen unendlichen Mächtigkeiten, als die abzählbare, die abzählbar unfertige, und die continuierliche, kann gar keine Rede sein.

6 Brouwer Die möglichen Mächtigkeiten Von anderen unendlichen Mächtigkeiten, als die abzählbare, die abzählbar unfertige, und die continuierliche, kann gar keine Rede sein. 1914: Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf.

7 Brouwer Die möglichen Mächtigkeiten Von anderen unendlichen Mächtigkeiten, als die abzählbare, die abzählbar unfertige, und die continuierliche, kann gar keine Rede sein. 1914: Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf. Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. (1919)

8 Brouwer Die möglichen Mächtigkeiten Von anderen unendlichen Mächtigkeiten, als die abzählbare, die abzählbar unfertige, und die continuierliche, kann gar keine Rede sein. 1914: Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf. Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. (1919) Gielen, de Swart, Veldman: The continuum hypothesis in intuitionism (1981)

9 Hilbert

10 Hilbert Hilbert (1900): List of 23 problems. CH is number one.

11 Hilbert Hilbert (1900): List of 23 problems. CH is number one. Hilbert (1925): Über das Unendliche. In it, Hilbert sketched a proof of CH. Instead of R he considers the set N N of all functions from N to N.

12 Hilbert Hilbert (1900): List of 23 problems. CH is number one. Hilbert (1925): Über das Unendliche. In it, Hilbert sketched a proof of CH. Instead of R he considers the set N N of all functions from N to N.,,Wenn wir die Menge dieser Funktionen im Sinne des Kontinuumproblems ordnen wollen, so bedarf es dazu der Bezugnahme auf die Erzeugung der einzelnen Funktionen.

13 Hilbert Hilbert (1900): List of 23 problems. CH is number one. Hilbert (1925): Über das Unendliche. In it, Hilbert sketched a proof of CH. Instead of R he considers the set N N of all functions from N to N.,,Wenn wir die Menge dieser Funktionen im Sinne des Kontinuumproblems ordnen wollen, so bedarf es dazu der Bezugnahme auf die Erzeugung der einzelnen Funktionen. If we want to order the set of these functions in the way required by the problem of the continuum, we must consider how an individual function is generated.

14 137 years and still going strong(?): Cantor s continuum problem The dream solution template à la Hamkins for determining truth in V. E.g. CH.

15 137 years and still going strong(?): Cantor s continuum problem The dream solution template à la Hamkins for determining truth in V. E.g. CH. Step 1. Produce a set-theoretic assertion Φ expressing a natural and intuitively true set-theoretic principle.

16 137 years and still going strong(?): Cantor s continuum problem The dream solution template à la Hamkins for determining truth in V. E.g. CH. Step 1. Produce a set-theoretic assertion Φ expressing a natural and intuitively true set-theoretic principle. Step 2. Prove that Φ determines CH. That is, prove Φ CH or prove that Φ CH

17 Three responses

18 Three responses A universe view: ( Woodin) CH has a truth value in the universe V.

19 Three responses A universe view: ( Woodin) CH has a truth value in the universe V. A multiverse view: ( Hamkins) There are many universes that set theorists study. They all exist. CH has different truth values in different universes.

20 Three responses A universe view: ( Woodin) CH has a truth value in the universe V. A multiverse view: ( Hamkins) There are many universes that set theorists study. They all exist. CH has different truth values in different universes. ( Feferman) CH is neither a definite mathematical problem nor a definite logical problem.

21 Gödel s intrinsic program: Reflection

22 Gödel s intrinsic program: Reflection Gödel 1964 says that: axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation set of. (260)

23 Gödel s intrinsic program: Reflection Gödel 1964 says that: axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation set of. (260) He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals

24 Gödel s intrinsic program: Reflection Gödel 1964 says that: axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation set of. (260) He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals and maintains that [t]hese axioms show clearly, not only that the axiomatic system of set theory as used today is incomplete, but also that it can be supplemented without arbitrariness by new axioms which only unfold the content of the concept of set as explained above ( ).

25 Gödel s intrinsic program: Reflection Gödel 1964 says that: axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation set of. (260) He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals and maintains that [t]hese axioms show clearly, not only that the axiomatic system of set theory as used today is incomplete, but also that it can be supplemented without arbitrariness by new axioms which only unfold the content of the concept of set as explained above ( ). Gödel later refers to such axioms as having an intrinsic necessary status.

26 What Hope for Gödel s Program to settle CH?

27 What Hope for Gödel s Program to settle CH? Theorem ( Cohen; Levy and Solovay 1967): CH is consistent with and independent of all small and large ) LCAs that have been considered to date, provided they are consistent with ZF.

28 What Hope for Gödel s Program to settle CH? Theorem ( Cohen; Levy and Solovay 1967): CH is consistent with and independent of all small and large ) LCAs that have been considered to date, provided they are consistent with ZF. Proof. By Cohen s method of forcing.

29 Gödel s Extrinsic Program (1947) There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole discipline...that quite irrespective of their intrinsic necessity they would have to be assumed in the same sense as any well-established physical theory.

30 Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition.

31 Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition. Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work.

32 Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition. Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work. I think the evidence now favors CH.

33 Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition. Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work. I think the evidence now favors CH. The picture that is emerging now [...] is as follows. The solution to the inner model problem for one supercompact cardinal yields the ultimate enlargement of L. This enlargement of L is compatible with all stronger large cardinal axioms and strong forms of covering hold relative to this inner model.

34 Feferman on CH

35 Feferman on CH Exploring the frontiers of incompleteness. Peter Koellner s Templeton project.

36 Feferman on CH Exploring the frontiers of incompleteness. Peter Koellner s Templeton project. Solomon Feferman: Is the continuum hypothesis a definite mathematical problem?

37 Feferman s analysis

38 Feferman s analysis Two Informal Notions of Definiteness

39 Feferman s analysis Two Informal Notions of Definiteness 1 Notion of a definite totality/ domain of things.

40 Feferman s analysis Two Informal Notions of Definiteness 1 Notion of a definite totality/ domain of things. 2 Notion of a definite proposition/ definite predicate.

41 Feferman s analysis Two Informal Notions of Definiteness 1 Notion of a definite totality/ domain of things. 2 Notion of a definite proposition/ definite predicate. Criteria for these can be given in logical terms

42 The Criteria

43 The Criteria A predicate P is definite over a domain D iff the Principle of Bivalence holds for it, i.e. x D [P( x ) P( x )].

44 The Criteria A predicate P is definite over a domain D iff the Principle of Bivalence holds for it, i.e. x D [P( x ) P( x )]. A totality D is definite iff quantification over D is a definite logical operation, i.e., whenever R( x, y) is definite over D, so are y D R(y, x ) and y D R(y, x ).

45 Feferman s analysis

46 Feferman s analysis Proposed logical framework for what s definite and what s not: What s definite is the domain of classical logic, what s not is that of intuitionistic logic.

47 Feferman s analysis Proposed logical framework for what s definite and what s not: What s definite is the domain of classical logic, what s not is that of intuitionistic logic. and = are definite. Sets are viewed as definite domains. Hence Classical logic for bounded ( 0 ) formulas. Intuitionistic logic for unbounded quantification.

48 Some Examples

49 Some Examples According to the (ultra) finitists, the natural numbers form an unfinished or indefinite totality, and quantification over the natural numbers is indefinite, while bounded quantification is definite.

50 Some Examples According to the (ultra) finitists, the natural numbers form an unfinished or indefinite totality, and quantification over the natural numbers is indefinite, while bounded quantification is definite. According to the predicativists, the natural numbers form a definite totality, but not the supposed collection of arbitrary sets of natural numbers.

51 More Examples

52 More Examples According to some classical Descriptive Set Theorists, the set R of real numbers is a definite totality but not the supposed totality of arbitrary subsets of R.

53 More Examples According to some classical Descriptive Set Theorists, the set R of real numbers is a definite totality but not the supposed totality of arbitrary subsets of R. Set theory identifies definite totalities with sets. Then V is not a definite totality by Russell s Paradox. ( Predicative Set Theory)

54 Indefinitely extensible concepts

55 Indefinitely extensible concepts Ich setze voraus, dass man wisse, was der Umfang eines Begriffes sei. I assume that it is known what the extension of a concept is. Frege: Die Grundlagen der Arithmetik (Breslau 1884) 68.

56 Indefinitely extensible concepts Ich setze voraus, dass man wisse, was der Umfang eines Begriffes sei. I assume that it is known what the extension of a concept is. Frege: Die Grundlagen der Arithmetik (Breslau 1884) 68. In Frege: Philosophy of Mathematics, Dummett s diagnosis of the failure of Frege s logicist project focusses on the adoption of classical quantification.

57 Indefinitely extensible concepts Ich setze voraus, dass man wisse, was der Umfang eines Begriffes sei. I assume that it is known what the extension of a concept is. Frege: Die Grundlagen der Arithmetik (Breslau 1884) 68. In Frege: Philosophy of Mathematics, Dummett s diagnosis of the failure of Frege s logicist project focusses on the adoption of classical quantification. Dummett argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept.

58 Dummett: What is Mathematics About?, 1994

59 Dummett: What is Mathematics About?, 1994 Cantor saw far more deeply into the matter than Frege did: he was aware, long before, that one cannot simply assume every concept to have an extension with a determinate cardinality. (p. 26)

60 Dummett: What is Mathematics About?, 1994 Cantor saw far more deeply into the matter than Frege did: he was aware, long before, that one cannot simply assume every concept to have an extension with a determinate cardinality. (p. 26) The fact revealed by the set-theoretic paradoxes was the existence of indefinitely extensible concepts - a fact of which Frege did not dream and even Cantor had only an obscure perception.

61 There can be no objection to quantifying over all objects falling under some indefinitely extensible concept, say over everything we should, given an intelligible description of it, recognize as an ordinal number, provided that we do not think of the statements formed by means of such quantification as having determinate truth-conditions; we can understand them only as making claims of the kind already sketched. They will not then satisfy the laws of classical logic, but only the weaker laws of intuitionistic logic.

62 Actualism versus Potentialism

63 Actualism versus Potentialism This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle.

64 Actualism versus Potentialism This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle. One way of formally regimenting this informal distinction is by employing intuitionistic logic for domains for which one is a potentialist and reserving classic logic for domains for which one is an actualist.

65 Actualism versus Potentialism This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle. One way of formally regimenting this informal distinction is by employing intuitionistic logic for domains for which one is a potentialist and reserving classic logic for domains for which one is an actualist. This is the approach Tait takes in his work on reflection principles. Feferman work on semi-intuitionistic systems of set theory can also be recast in those terms.

66 Toward Axiomatic Formulations

67 Toward Axiomatic Formulations Restrict quantifiers in the formulas that are supposed to represent definite properties, e.g. in Comprehension or Separation axioms.

68 Toward Axiomatic Formulations Restrict quantifiers in the formulas that are supposed to represent definite properties, e.g. in Comprehension or Separation axioms. Quantification over indefinite domains may still be regarded as meaningful, in order to state generic properties and closure properties of the realm of sets, but is governed by intuitionistic logic.

69 The theory SCS

70 The theory SCS Feferman: On the strength of some semi-constructive theories (2012)

71 The theory SCS Feferman: On the strength of some semi-constructive theories (2012) Basic axioms

72 The theory SCS Feferman: On the strength of some semi-constructive theories (2012) Basic axioms Extensionality

73 The theory SCS Feferman: On the strength of some semi-constructive theories (2012) Basic axioms Extensionality Pairing and Union

74 The theory SCS Feferman: On the strength of some semi-constructive theories (2012) Basic axioms Extensionality Pairing and Union Infinity

75 The theory SCS Feferman: On the strength of some semi-constructive theories (2012) Basic axioms Extensionality Pairing and Union Infinity Set Induction for any formula ϕ(x). x [ y x ϕ(y) ϕ(x)] x ϕ(x)

76 SCS continued

77 SCS continued LEM 0 is the schema ϕ ϕ for ϕ 0.

78 LEM 0 is the schema for ϕ 0. SCS continued ϕ ϕ BOS is the schema (for all formulas ϕ(x)): If x a [ϕ(x) ϕ(x)] then x a ϕ(x) x a ϕ(x).

79 LEM 0 is the schema for ϕ 0. SCS continued ϕ ϕ BOS is the schema (for all formulas ϕ(x)): If x a [ϕ(x) ϕ(x)] then x a ϕ(x) x a ϕ(x). AC full is the schema (for all formulas ϕ(x, y)): x a y ϕ(x, y) f [dom(f ) = a x a ϕ(x, f (x))]

80 LEM 0 is the schema for ϕ 0. SCS continued ϕ ϕ BOS is the schema (for all formulas ϕ(x)): If x a [ϕ(x) ϕ(x)] then x a ϕ(x) x a ϕ(x). AC full is the schema (for all formulas ϕ(x, y)): x a y ϕ(x, y) f [dom(f ) = a x a ϕ(x, f (x))] MP is the schema x θ(x) x θ(x) for θ(x) 0.

81 Why these axioms?

82 Why these axioms? Older roots: Pozsgay (1971, 1972); Tharp (1971); Wolf (1974).

83 Why these axioms? Older roots: Pozsgay (1971, 1972); Tharp (1971); Wolf (1974). Myhill s Constructive Set Theory (1974).

84 Why these axioms? Older roots: Pozsgay (1971, 1972); Tharp (1971); Wolf (1974). Myhill s Constructive Set Theory (1974). Feferman subjected these theories to functional interpretation (goes back a long way).

85 Feferman s Conjecture: CH is not a definite mathematical problem

86 Feferman s Conjecture: CH is not a definite mathematical problem Let SCS + be the theory SCS + R is a set.

87 Feferman s Conjecture: CH is not a definite mathematical problem Let SCS + be the theory SCS + R is a set. The formal version of the conjecture is that SCS + CH CH

88 Feferman s Conjecture: CH is not a definite mathematical problem Let SCS + be the theory SCS + R is a set. The formal version of the conjecture is that SCS + CH CH The theory SCS has too many axioms.

89 Feferman s Conjecture: CH is not a definite mathematical problem Let SCS + be the theory SCS + R is a set. The formal version of the conjecture is that SCS + CH CH The theory SCS has too many axioms. SCS + is quite strong. It proves every theorem of (classical) second order arithmetic. In strength it resides strictly between second order arithmetic and Zermelo set theory.

90 The conjecture is true

91 The conjecture is true R: Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman. JSL Techniques: Relativized constructible hierarchy L[A], forcing, realizability.

92 The conjecture is true Remark: R: Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman. JSL Techniques: Relativized constructible hierarchy L[A], forcing, realizability. The machinery can also be applied to furnish other indefinite statements.

93 Theorem: (Koellner, Woodin) Let DWOR be the statement that that there is a well-ordering of R in L(R). Assume Con(ZFC + There are ω-many Woodin cardinals ). Then SCS + DWOR DWOR.

94 Theorem: (Koellner, Woodin) Let NCR be the statement There is a non-constructible real. Then SCS NCR NCR.

95 Feferman s second conjecture

96 Feferman s second conjecture One of the basic considerations leading to the SCS systems in general is that a statement ϕ is recognized to be definite relative to such just in case LEM holds for it, in other words just in case SCS proves ϕ ϕ, and otherwise it is indefinite. Similarly, a formula ϕ(x) is recognized to be definite relative to SCS just in case it proves x [ϕ(x) ϕ(x)]. In the case of SCS, I conjecture that every such formula is equivalent to a 1 formula and hence is absolute for end-extensions. Furthermore, in the case of SCS + Pow(ω), the corresponding conjecture is that every such formula is 1 in the power set of ω.

97 1 Definition: For a predicate given by a formula A(x) with at most x free we say that A(x) is 1 with respect to a theory T SCS if there exist a Σ 1 -formula B(x) and a Π 1 -formula C(x) such that T x [A(x) B(x) C(x)]. A(x) is said to be definite or to satisfy LEM with respect to T if T x [A(x) A(x)]. In constructive mathematics, the latter property is also know as decidability (with respect to T ).

98 Global Choice

99 Global Choice The axiom of global choice, AC global, is expressed in an extension of the language by a new binary relation symbol R, via the following axioms pertaining to R:

100 Global Choice The axiom of global choice, AC global, is expressed in an extension of the language by a new binary relation symbol R, via the following axioms pertaining to R: (i) x y z[r(x, y) R(x, z) y = z] (1) (ii) x[x y x R(x, y)]. (2)

101 Global Choice The axiom of global choice, AC global, is expressed in an extension of the language by a new binary relation symbol R, via the following axioms pertaining to R: (i) x y z[r(x, y) R(x, z) y = z] (1) (ii) x[x y x R(x, y)]. (2) Moreover, the axiom schemes of 0 -Separation and 0 -Collection are now formulated for 0 -formulae of the extended language.

102 Theorem: Suppose ( ) SCS + AC global x [ ya( x, y) zb( x, z)] where A and B are 0. Then SCS + AC global x [ ya( x, y) ya( x, y)].

103 Counterexample One might think there is a more general result to the effect that a theory with decidable atomic formulae and Collection has the property that 1 predicates provably satisfy excluded third.

104 Counterexample One might think there is a more general result to the effect that a theory with decidable atomic formulae and Collection has the property that 1 predicates provably satisfy excluded third. Proposition. There is an extension T of HA such that there is a formula A that is 1 relative to T but T does not prove A A.

105 Theorem: Suppose SCS x [A(x) A(x)]. Then there exist a Σ 1 -formula B(x) and a Π 1 -formula C(x) in the language of SCS + AC global, L (R), such that SCS + AC global x [A(x) B(x) C(x)].

106 The previous result can also be slightly strengthened. Theorem: Suppose SCS + AC global x [A(x) A(x)]. Then there exist a Σ 1 -formula B(x) and a Π 1 -formula C(x) in the language of SCS + AC global, L (R), such that SCS + AC global x [A(x) B(x) C(x)].

107

108 One would like to get rid of global choice in the interpreting theory, i.e., weaken global choice to just choice.

109 One would like to get rid of global choice in the interpreting theory, i.e., weaken global choice to just choice. One idea is to use class forcing (as in the case of ZFC) to find an interpretation of SCS + AC global in SCS that preserves the formulae of L. In the case of ZFC the class of forcing conditions consists of local choice functions h, i.e., functions such that for all x in the domain of h, h(x) x if x and h(x) = if x =.

110 One would like to get rid of global choice in the interpreting theory, i.e., weaken global choice to just choice. One idea is to use class forcing (as in the case of ZFC) to find an interpretation of SCS + AC global in SCS that preserves the formulae of L. In the case of ZFC the class of forcing conditions consists of local choice functions h, i.e., functions such that for all x in the domain of h, h(x) x if x and h(x) = if x =. While forcing can be defined in SCS, the problem we encountered is that SCS seems to be too weak to be able to show that every theorem of SCS is forced, the particular culprit being Collection.

111 SCS + = SCS + R is a set Remark: Versions of the previous Theorem for stronger theories can also be proved, for instance for the theory SCS + = SCS + R is a set.

112 Theorem Suppose A is a Π 2 statement of L (R) that holds in L(b, ), where b is a transitive set and is well-ordering on b. If SCS + AC global A A, then L κ (b, ) = A holds for all admissible sets L κ (b, ) with κ > ω.

113 Corollary: Recall that a statement D is indefinite relative to SCS + AC global if SCS + AC global D D.

114 Corollary: Recall that a statement D is indefinite relative to SCS + AC global if SCS + AC global D D. One can now, e.g., take any of the statements that are equivalent to ATR 0 over ACA 0 from Simpson s book and conclude that they are indeterminate relative to SCS + AC global as they are of Π 2 form, hold in L but fail to hold in L ω ck. This was also observed by Koellner and 1 Woodin. Here are two examples.

115 Corollary: Recall that a statement D is indefinite relative to SCS + AC global if SCS + AC global D D. One can now, e.g., take any of the statements that are equivalent to ATR 0 over ACA 0 from Simpson s book and conclude that they are indeterminate relative to SCS + AC global as they are of Π 2 form, hold in L but fail to hold in L ω ck. This was also observed by Koellner and 1 Woodin. Here are two examples. 1 Comparability of well-orderings on subsets of ω.

116 Corollary: Recall that a statement D is indefinite relative to SCS + AC global if SCS + AC global D D. One can now, e.g., take any of the statements that are equivalent to ATR 0 over ACA 0 from Simpson s book and conclude that they are indeterminate relative to SCS + AC global as they are of Π 2 form, hold in L but fail to hold in L ω ck. This was also observed by Koellner and 1 Woodin. Here are two examples. 1 Comparability of well-orderings on subsets of ω determinacy.

117 Corollary: Recall that a statement D is indefinite relative to SCS + AC global if SCS + AC global D D. One can now, e.g., take any of the statements that are equivalent to ATR 0 over ACA 0 from Simpson s book and conclude that they are indeterminate relative to SCS + AC global as they are of Π 2 form, hold in L but fail to hold in L ω ck. This was also observed by Koellner and 1 Woodin. Here are two examples. 1 Comparability of well-orderings on subsets of ω determinacy. One can also take the statement Every real is constructible since there are admissible sets of the form L κ (x) where x is a constructible real but x fails to be constructible in L κ (x).

118 Remark. The machinery can also be applied to SCS + AC global + R is a set to give indefinite statements relative to that theory.

119 Remark. The machinery can also be applied to SCS + AC global + R is a set to give indefinite statements relative to that theory. Remark. Likewise, the machinery can be applied to SCS + AC global + Pow to give indefinite statements.

120 dank u wel