Abstract ModelTheory and SetTheoretic Multiverses Antonio Vincenzi - - PowerPoint PPT Presentation

Abstract Model–Theory and Set–Theoretic Multiverses

Antonio Vincenzi

A New Deal for Abstract Model Theory??

Altonaer Stiftung für philosophische Grundlagenforschung

Infinitary logics generalized quantifiers COMP Higer order logics INT ROB

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Model–Theoretic Logics

Episodic Relationships?

Assuming that the set–theoretic background of each logic is a set–theoretic universe M= (M,∈,…)

• The sensibility of Compactness, Löwenheim–Skolem,… to M,

namely the fact that, for some logic L, these properties can

◯ hold if M satisfies , GCH,… contains some large cardinal… ◯ fail otherwise

• Set–theoretic constraints on relationships among model–

theoretic properties, like Robinson = Compactness + Interpolation

• The relationships between logics

L = vocabulary + grammar + interpretation

and inner models C(L) of M

Set–Theoretic Logics

INTERPENETRATION OF INSTRUMENTS AND CONCEPTS

Construction

Concept r y c x P R X Y

fixed point prev next

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graft basic symbols new symbols formal tree basic rules new rules MEANING

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• a model–theoretic logic L generates class of constructions C(L) of M
• a set–theoretic logic K is a class of constructions K of M

If the operations used in K are inductive then L and K are equivalent

Basics on Set–Theoretic Logics

Conceptually

• The notion of formal language is not conceptually necessary to

define the notion of logic.

• The formal language of a logic can be naturally modified by its

set–theoretic background

• It is a technical trick that simplify some logic operation

(analogous to the table for the connectives)

Technically

Väänänen Problem: the second order logic is the only logic that

• Satisfies one–sorted interpolation
• Does not satisfy many–sorter interpolation
• Is different from its delta closure

Multiverses Logics

Since each set–theoretic logic K is relative to a set–theoretic universe M that contains K assume that

• a model–theoretic property (MTP) of ( K,M) is a property

◯ defined using the satisfaction relation of K,

◯ which possibly depends on some properties of M .

• a multiverse logic ( K,M) is a collection of ( K,M) such that

( K,M) satisfies MTP ⇔ ( K,M) satisfies MTP for each M

⇔ ‘( K,M)⎪ =MTP’ is M–absolute

• in particular M = Mod( ZFC), M = Mod( KP),…

WITTGENSTEIN–CHOMSKY LANGUAGES FRAMEWORKS

Surface Framework

Deep Framework Form Meaning Syntax Semantic

• An MTP is surface if satisfies the following equivalent properties

◯ the failure of MTP can be repaired acting only on K ◯ the failure of MTP is M–absolute ◯ its relative form is only MTP(( K +,M),( K,M)) with K< K*

• An MTP is deep if satisfies the following equivalent properties

◯ the failure of MTP can be repaired acting only on M ◯ the failure of MTP is M–relative ◯ its relative form is only MTP(( K,M),( K,M*)) with M*< M

Compactification

VOPENKA–LIKE CONPACTIFICATION THEOREM. M↾κ = the strengthening of M in which each M contains κ If [λ]–COMP(K,M) fails, then there is a Vopenka cardinal κ such that [λ]–COMP(K,M↾κ) holds. NON–STANDARD COMPACTIFICATION THEOREM. *M = non standard M in which each M is substituted by *M If λ be a regular cardinal and (K,M) be a set–theoretic logic with dependence number≤λ.

• If [λ]–COMP(K,M) fails and AM(K,M) holds then there is a *M

such that ‘cofinality λ’ is *M–absolute and [λ]–COMP(K,*M) holds.

• If both [λ]–COMP(K,M) and AM(K,M) fail then there is a *M such

that ‘cofinality λ’ is *M–absolute and [λ]–COMP((K,M),(K,*M)) holds.

1 1 2 2 3 3 4 4 0 0

Inexpressibility

HEISENBERG–LIKE UNCERTAINTY

A class of detectors of a fixed dimension cannot detect precisely

• bjects of their dimension

Consistency Inconsistency

INEXPRESSIBILITY THEOREM. For each bounded and relativizable logic (K,M) and each infinite cardinal λ the following are equivalent:

• (K,M) is [λ]–Compact.
• (λ,<) is not Σ1(K,M)–characterizable.
• The class of initial segments of (λ,<) is not Σ1(K,M)–characterizable.
• (λ,<) cannot be Σ1(K,M)–pinned down.

In particular, if λ is regular, then the above properties are equivalent to

• (λ,<) is not cofinally Σ1(K,M)–characterizable.

Alternatively, if λ is singular, then the same properties are equivalent to

• (λ,<) is not cardinallike Σ1(K,M)–characterizable.

EXPRESSIVITY an object is ( K,M)–expressible

⇕

( K,M) individuates univocally a class of objects containing this object

λ–LS λ–LST λ inexpressible [λ]–COMP

• PROJECT. Use the dichotomy

λ regular — λ cofinally characterizable λ singular — λ cardinallike characterizable to generalize the above results to each infinite cardinal

Abstract Deep Properties

Y

Consistency

X

Inconsistency Compactness

X Y

[λ,κ]

H(λ) H(κ)

Barwise

A admissible

Σ1(A) Abstract

H H

PROJECT: Find the properties of H and H that allow prove Abstract Inexpressibility, Compactification… results

Perspectives

SURFACE MTP Interpolation, Definability,… Inexpressibility TRASVERSE MTP Robinson… DEEP MTP Compactness, Löwenheim–Skolem,…