Introduction Abstraction Principles and Grounding Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Albert Visser Treatment of Arithmetic? OFR, Philosophy, Faculty of Humanities, Utrecht University Groundedness Workshop August 23, 2013, Oslo 1
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 2
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 2
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 2
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 2
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 2
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 3
What is this Talk About Introduction 1. Frege-style abstraction principles are basically asymmetric. Frege’s Program The identities between abstracts are grounded in the Grundgesetze equivalences. Similarly for comprehension principles. Predicative Frege Arithmetic 2. We review the basic Fregean framework and suggest that it is A Truly Predicative incomplete . One way of completing it, is, I hope, to add the Treatment of Arithmetic? motivational or explicit idea of conceptual ordering . 3. I explain why Predicative V and Predicative Hume’s Principle are not truly predicative. 4. I sketch what I think a truly predicative development should look like. These ideas have to be tested. Can I derive the totality of successor for the Beth-Burgess numbers? 5. Is Predicative Frege Arithmetic a reflective equilibrium? 4
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 5
Basic Setting We work in many-sorted logic. We have a sort of objects, a sort of Introduction extensions, a sort of concepts / classes. To simplify a bit we will Frege’s Program ignore the sort of objects. Grundgesetze Predicative Frege We have the usual formula classes Π 1 n , Σ 1 n and ∆ 1 n . ∆ 1 0 = Π 1 0 = Σ 1 Arithmetic 0 consists of all formulas without concept quantifiers. ∆ 1 A Truly Predicative 0 is also Treatment of Arithmetic? called the class of predicative formulas. Warning: we will not generally have pairing (of sufficiently low complexity), so e.g. Π 1 1 is of the form ∀ X 0 . . . ∀ X n − 1 φ , where φ is in ∆ 1 0 . We can also have sorts of binary concepts, etc. As long as we have ∆ 1 1 -comprehension and Law V this makes no difference since we will have pairing on the ground domain, but in general it will make a difference. 6
Abstraction Introduction Suppose E is an equivalence relation on D then we have a Frege’s Program function @ E and a domain A E such that: Grundgesetze ◮ @ E is surjective from D to A E . Predicative Frege Arithmetic ◮ @ E d = @ E e iff d E e . A Truly Predicative Treatment of Arithmetic? The basic idea is that underlying the equivalence there is a grounding relation: the lhs is grounded in the rhs. Compare this to Tarski biconditionals. Similarly for a concept introduced by comprehension. The entities falling under it and the parameters should be conceptually pre-existent. 7
Overview Introduction Introduction Frege’s Program Grundgesetze Predicative Frege Frege’s Program Arithmetic A Truly Predicative Treatment of Arithmetic? Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic? 8
The System Introduction The system GG (Γ) is defined as follows: Frege’s Program ◮ Γ -comprehension. Grundgesetze ◮ Extensionality of concepts. Predicative Frege Arithmetic ◮ Law V: ∂ X = ∂ Y iff X = Y . A Truly Predicative Treatment of ◮ Surjectivity of ∂ to the extensions: ∀ x ∃ X ∂ X = x Arithmetic? By the Russell Paradox GG (∆ 1 1 ) is inconsistent. If we omit surjectivity then GG (∆ 1 1 ) is consistent (Ferreira-Wehmeier, Walsh), but Π 1 1 -comprehension is inconsistent. GG − (∆ 1 1 ) proves that there are non-extensions. 9
Strength of GG (∆ 1 0 ) GG (∆ 1 0 ) is mutually interpretable with Q (Ganea, Visser). (This Introduction Frege’s Program result is very robust for variations of detail.) Grundgesetze Predicative Frege Repeating the construction gives us a hierarchy that corresponds Arithmetic to iterating consistency statements over Q (Visser). It also follows A Truly Predicative Treatment of a hierarchy of functions defined by Alex Wilkie (Visser, Arithmetic? unpublished). I ∆ 0 + supexp is the unattainable upperbound for the hierarchy. Even if, as we will suggest, the philosophical credentials of ∆ 1 0 -comprehension are suspect, it is a very meaningful expansion of a theory from a metamathematical point of view because of the connection with consistency. 10
Models of GG (∆ 1 0 ) part 1 We assume that for each n we have n -ary concepts. Let M be any first-order structure. We define: Introduction Frege’s Program M = ( M , Def ( M ) , Def ( M × M ) , . . . ) (1) Grundgesetze Predicative Frege where Def ( M n ) ⊆ P ( M n ) consists of the X ⊆ M n definable with Arithmetic parameters in M . A Truly Predicative Treatment of Arithmetic? If M is infinite, Def ( M ) and M have the same cardinality. Hence, choose ∂ : Def ( M ) → M to be any injection. Then the following structure is a model of GG (∆ 1 0 ) : M = ( M , Def ( M ) , Def ( M × M ) , . . . , ∂ ) (2) Injectivity implies Basic Law V. The verification of predicative comprehension is on next slide. 11
Models of GG (∆ 1 0 ) part 2 Consider predicative comprehension for A . Introduction ⊢ ∃ X n ∀ � x ∈ X n ↔ A ( � y , � x ( � x ,� Frege’s Program Z ) ) . (3) Grundgesetze Predicative Frege A class variable Z has three kinds of occurrences in A : Arithmetic A Truly Predicative Treatment of (a) in a formula of the form Z = U or U = Z , Arithmetic? (b) in a formula of the form t = ∂ Z or ∂ Z = t or ( . . . , ∂ Z , . . . ) ∈ U , (c) in a formula of the form � t ∈ Z . Eliminate subformulas (a) by replacing them by ∀ � u ( � u ∈ Z ↔ � u ∈ U ) . In subformulas (b) we replace ∂ P by the value p of ∂ P in the model. We replace in subformulas (c) � t ∈ P by t , � d ′ , � B ( � P ′ ) , where B is the first-order definition of P . We take the final formula as the definition of X n . 12
Simple Frege Models Introduction A simple Frege model is a structure of the following form Frege’s Program Grundgesetze M = ( ω, Def ( ω ) , Def ( ω × ω ) , . . . , ∂ ) Predicative Frege (4) Arithmetic A Truly Predicative where on ω we have the language of equality. Treatment of Arithmetic? The definable subsets of ω are finite or cofinite. These models are analogues of the ZF universe of pure sets. In these models we do not have the Julius Caesar problem since every object is an extension. 13
Many Non-Isomorphic Simple Frege Structures There are many non-isomorphic simple Frege models. Introduction Frege’s Program Let ( X i ) i ∈ ω be an enumeration without repetitions of the definable Grundgesetze classes of ω except the singletons. We define ∂ 0 ( X i ) := 2 i and Predicative Frege Arithmetic ∂ 0 ( { k } ) := 2 k + 1. So according to ∂ 0 no object codes its own A Truly Predicative singleton. Treatment of Arithmetic? Let ( Y i ) i ∈ ω be an enumeration without repetitions of the definable classes of ω except the singletons of odd numbers. We define ∂ 1 ( Y i ) := 2 i if X i and ∂ 1 ( { 2 k + 1 } ) := 2 k + 1. So according to ∂ 1 there are infinitely many elements that code their own singleton. One may wonder whether a good foundational approach should not uniquely determine the structure of the extensions. 14
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