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A note on Δ₁ induction and Σ₁ collection

Neil Thapen (2005)

Fundamenta Mathematicae

Slaman recently proved that Σₙ collection is provable from Δₙ induction plus exponentiation, partially answering a question of Paris. We give a new version of this proof for the case n = 1, which only requires the following very weak form of exponentiation: " x y exists for some y sufficiently large that x is smaller than some primitive recursive function of y".

An application of a reflection principle

Zofia Adamowicz, Leszek Aleksander Kołodziejczyk, Paweł Zbierski (2003)

Fundamenta Mathematicae

We define a recursive theory which axiomatizes a class of models of IΔ₀ + Ω ₃ + ¬ exp all of which share two features: firstly, the set of Δ₀ definable elements of the model is majorized by the set of elements definable by Δ₀ formulae of fixed complexity; secondly, Σ₁ truth about the model is recursively reducible to the set of true Σ₁ formulae of fixed complexity.

An irrational problem

Franklin D. Tall (2002)

Fundamenta Mathematicae

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define X M to be X ∩ M with topology generated by U M : U M . Suppose X M is homeomorphic to the irrationals; must X = X M ? We have partial results. We also answer a question of Gruenhage by showing that if X M is homeomorphic to the “Long Cantor Set”, then X = X M .

Automorphisms of models of bounded arithmetic

Ali Enayat (2006)

Fundamenta Mathematicae

We establish the following model-theoretic characterization of the fragment IΔ₀ + Exp + BΣ₁ of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment IΔ₀ of Peano arithmetic with induction limited to Δ₀-formulae). Theorem A. The following two conditions are equivalent for a countable model of the language of arithmetic: (a) satisfies IΔ₀ + BΣ₁ + Exp; (b) = I f i x ( j ) for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here I f i x ( j ) is the...

Axioms which imply GCH

Jan Mycielski (2003)

Fundamenta Mathematicae

We propose some new set-theoretic axioms which imply the generalized continuum hypothesis, and we discuss some of their consequences.

Counting models of set theory

Ali Enayat (2002)

Fundamenta Mathematicae

Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and κ ω , , 2 . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then μ ( T , α ) = 2 . (3) If α < ω₁ and T ⊢ V ≠ OD, then μ ( T , α ) 0 , 2 . (4)...

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