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Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Abstraction Principles and Grounding

Albert Visser

OFR, Philosophy, Faculty of Humanities, Utrecht University

Groundedness Workshop August 23, 2013, Oslo

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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What is this Talk About

• 1. Frege-style abstraction principles are basically asymmetric.

The identities between abstracts are grounded in the

• equivalences. Similarly for comprehension principles.
• 2. We review the basic Fregean framework and suggest that it is
• incomplete. One way of completing it, is, I hope, to add the

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?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Basic Setting

We work in many-sorted logic. We have a sort of objects, a sort of extensions, a sort of concepts / classes. To simplify a bit we will ignore the sort of objects. We have the usual formula classes Π1

n, Σ1 n and ∆1

• n. ∆1

0 = Π1 0 = Σ1

consists of all formulas without concept quantifiers. ∆1

0 is also

called the class of predicative formulas. Warning: we will not generally have pairing (of sufficiently low complexity), so e.g. Π1

1 is of the form ∀X0 . . . ∀Xn−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.

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Abstraction

Suppose E is an equivalence relation on D then we have a function @E and a domain AE such that:

◮ @E is surjective from D to AE. ◮ @Ed = @Ee iff d E e.

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.

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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The System

The system GG(Γ) is defined as follows:

◮ Γ-comprehension. ◮ Extensionality of concepts. ◮ Law V: ∂X = ∂Y iff X = Y. ◮ Surjectivity of ∂ to the extensions: ∀x ∃X ∂X = x

By the Russell Paradox GG(∆1

1) is inconsistent.

If we omit surjectivity then GG(∆1

1) is consistent

(Ferreira-Wehmeier, Walsh), but Π1

1-comprehension is

• inconsistent. GG−(∆1

1) proves that there are non-extensions.

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Strength of GG(∆1

0) GG(∆1

0) is mutually interpretable with Q (Ganea, Visser). (This

result is very robust for variations of detail.) Repeating the construction gives us a hierarchy that corresponds to iterating consistency statements over Q (Visser). It also follows a hierarchy of functions defined by Alex Wilkie (Visser, 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

• f a theory from a metamathematical point of view because of the

connection with consistency.

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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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: M = (M, Def(M), Def(M × M), . . .) (1) where Def(Mn) ⊆ P(Mn) consists of the X ⊆ Mn definable with parameters in M. 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.

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Models of GG(∆1

0) part 2 Consider predicative comprehension for A. ⊢ ∃X n ∀ x ( x ∈ X n ↔ A( x, y, Z) ). (3) A class variable Z has three kinds of occurrences in A: (a) in a formula of the form Z = U or U = Z, (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 B( t, d′, P′), where B is the first-order definition of P. We take the final formula as the definition of X n.

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Simple Frege Models

A simple Frege model is a structure of the following form M = (ω, Def(ω), Def(ω × ω), . . . , ∂) (4) where on ω we have the language of equality. 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.

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Many Non-Isomorphic Simple Frege Structures

There are many non-isomorphic simple Frege models. Let (Xi)i∈ω be an enumeration without repetitions of the definable classes of ω except the singletons. We define ∂0(Xi) := 2i and ∂0({k}) := 2k + 1. So according to ∂0 no object codes its own singleton. Let (Yi)i∈ω be an enumeration without repetitions of the definable classes of ω except the singletons of odd numbers. We define ∂1(Yi) := 2i if Xi and ∂1({2k + 1}) := 2k + 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.

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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The Ackermann Model

We define the model A (the Ackermann model) as follows:

◮ ack(k) is the finite set coded by k, where we have i ∈ ack(k)

iff the i-th binary digit of the binary expansion of k (counting from the right) is 1. Note that ack is a bijection between numbers and finite sets. (The function ack is known as Ackermann coding.)

◮ ∂∗(X) := 2 · ack−1(X) if X is finite and

∂∗(X) := 2 · ack−1(ω \ X) + 1 if X is cofinite. Note e.g. that ∈∗ on the hereditarily finite extensions will be well-founded under this Frege function. We can embed A uniquely into any simple Frege model (for the right kind of morphism). A is not bisimulation minimal (e.g. 1 and 3 are bisimilar). In fact no model of GG(∆1

0) is bisimulation minimal.

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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The System

HP2(Γ) is defined like GG(Γ) with Law V replaced by:

◮ Hume’s Principle: #X = #Y iff there is a bijection F : X → Y

(if X and Y are equinumerous). One can show that Predicative V and Predicative Frege Arithmetic are mutually interpretable but as far as I know there is no quick and easy argument for any of the two directions.

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The Linnebo model

The Linnebo model L has as basic domain ω which is defined by: 0, 1, 2, . . . . . . , ∞ − 2, ∞ − 1, ∞. We just have identity on ω. The classes are the (parametrically) definable classes in the language of identity over ω. In the unary case these are the finite and cofinite subsets of ω. We define #X := n if X has n elements (for n a natural number) and #X := ∞ − m if ω \ X has m elements, for m a natural number. It seems that HP is doing better than V w.r.t. the uniqueness of

• models. Some further study is needed of the question: How

categorical (schmategorical) is HP?

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Some Definitions

◮ 0 := #∅. ◮ ∞ := #V,

where V = {x | x = x}.

◮ p ≤ q :↔ ∃X, Y ( #X = p ∧ #Y = q ∧ X ⊆ Y). ◮ p < q :↔ ∃X, Y ( #X = p ∧ #Y = q ∧ X ⊂ Y). ◮ S(p, q) :↔ ∃X, x (x ∈ X ∧ #X = p ∧ #X ∪ {x} = q). ◮ A cardinal p is Dedekind finite iff p < p. ◮ bebu(n) :↔ n = #{p | p < n} ∧ n < n. ◮ her(X) :↔ 0 ∈ X ∧ ∀p, q ((S(p, q) ∧ p ∈ X) → q ∈ X). ◮ freg(n) :↔ ∀X (her(X) → n ∈ X).

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Yes We Can and No We Cannot

We can verify many of the usual properties of arithmetic in HP2(∆1

0). Yes, S is an injective partial function and 0 is not in its

• range. Yes, < is a partial ordering with minimum 0 and maximum

∞. (It is surprising that Cantor-Schröder-Bernstein works in this context.) We can give alternative definitions of the natural numbers like bebu that are provably closed under successor (Beth, Burgess). One such definition even satisfies the induction axiom. No, we cannot prove that ∞ has a successor. So we cannot exclude that ∞ is Dedekind finite. No, we cannot prove that the virtual class freg of Frege’s natural numbers is closed under

• successor. In fact in L the natural numbers à la Frege are all the
• cardinals. In L bebu gives us the standard natural numbers.

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Overview

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

Introduction Frege’s Program Grundgesetze Predicative Frege Arithmetic A Truly Predicative Treatment of Arithmetic?

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Why Predicative V and Predicative Hume’s Principle are not Predicative

If we look at the models we see that GG(∆1

0) is highly

non-categorical. We can get some kind of unicity of models but some extra story is needed to explain why e.g. the Ackermann model would be the right one. Also the basic philosophical idea that a set is determined by its graph is overboard. The main problem is that we have unbounded quantification over extensions or numbers. The ∆1

0-comprehension principle does

conform with the idea that the totality of concepts might be

• pen-ended and growing, but it assumes that the extensions /

numbers are pre-given. In a defining formula of X we can have a quantification of the numbers / extensions, where presumably #X depends on X for its existence.

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Basics 1

We will just look at Arithmetic. We have intensional identity ≡ and extensional identity = on both concepts and numbers. We think of objects and concepts as both existing as tokens or intensions of objects and as the true objects modulo various notions of sameness. The tokens are not necessarily closed under =. We have a conceptual ordering ≺ and . This is intensional. So = is not a congruence for it. We have some basic insights:

◮ if x ∈ X, then x ≺ X ◮ X ≺ #X

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Basics 2

There are two options: have ≺, in the language or else use it to motivate the axioms. A formula is ∆1

0 iff all its quantifiers are

• bounded. (The bounding things may be terms like X ∪ Y.) If we do

not have ≺, in the language then we have no concept quantifiers and all object quantifiers are ∈-bounded. We write X = Y for: there is a bijection between X and Y. This bijection, say F is a bijection w.r.t. =. E.g. we have: if x F y and x′ F y′, then x = x′ iff y = y′. Notions of identity for concepts:

◮ X ≡ Y ◮ X =0 Y iff ∀z (z ∈ X ↔ z ∈ Y) ◮ X =1 Y iff ∃E (E : X

= Y ∧ ∀x ∈ X ∀y ∈ Y (x E y ↔ x = y)).

◮ X =2 Y iff ∀x ∈ X ∃y ∈ Y x = y ∧ ∀y ∈ Y ∃x ∈ X y = x ◮ X

= Y Probably X =2 Y and X = Y are incomparable. =1 and =2 are not ∆1

0.

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Possible Axioms

The system below is the minimal system I could think of. It does not employ the conceptual ordering but is motivated by it. hpp(1) ∅ exists, X × Y exists, dom(R) and cod(R) exist hpp(2) ⊢ ∀X ∃E ⊆ X 2 ∀x, x′ ∈ X (x = x′ ↔ x E x′) hpp(3) ⊢ ∃Y ∀x (x ∈ Y ↔ (x ∈ X ∧ φ(x))) similarly for relations hpp(4) ⊢ ∃X ∀x′ (x′ ∈ X ↔ x′ ≡ x) hpp(5) ⊢ ∃E ∀x ∈ X ∀y ∈ Y (x E y ↔ x = y) → ∃Z ∀z (z ∈ Z ↔ (z ∈ X ∨ z ∈ Y)) A preliminary investigation suggests that (an appropriate version

• f) bebu is provably closed under successor.

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Ruminations

The above system is as minimal as I could make it and conservative in the sense that we did not make the conceptual

• rdering explicit. Perhaps we need lower bounds and perhaps we

also need upper bounds for the levels of objects and concepts. A doubt arises: if we have the idea of abstract time anyway, why not go for a Kantian-Brouwerian treatment of number based on the intuition of time? Because the last simply does not deliver the notion of cardinal? Another doubt: why not make the tokens / intensions members of the club of countibilia? Since X ≺ #X, we have #X ∈ X and so infinity would be for free. (Note that we could have ∃x (#X = x ∧ x ∈ X, so the insight is strictly a token insight.

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Thank You