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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".

A Word on the Joint Embedding Property

Milan Grulović (1995)

Matematički Vesnik

Addendum to “A note on the MacDowell–Specker theorem”

P. Clote (1998)

Fundamenta Mathematicae

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 $U \cap M : U \in \cap M$ . 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 $X = X_{M}$ .

An Ordering of the set of Sentences of Peano Arithmetic

Aleksandar Ignjatović (1985)

Publications de l'Institut Mathématique

Approximating the standard model of analysis

H. Enderton, Harvey Friedman (1971)

Fundamenta Mathematicae

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.

Compactness and homogeneity of saturated structures. II.

Josef Mlček (1983)

Commentationes Mathematicae Universitatis Carolinae

Completion closed algebras and models of Peano arithmetic

Petr Hájek (1981)

Commentationes Mathematicae Universitatis Carolinae

Constructing Kripke models of certain fragments of Heyting's arithmetic.

Wehmeier, Kai F. (1998)

Publications de l'Institut Mathématique. Nouvelle Série

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 $κ \in ω \cup ℵ ₀, ℵ ₁, 2^{ℵ ₀}$ . There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $μ (T, α) = 2^{ℵ ₀}$ . (3) If α < ω₁ and T ⊢ V ≠ OD, then $μ (T, α) \in 0, 2^{ℵ ₀}$ . (4)...

Counting $Δ_{0}$ sets

J. Paris, A. Wilkie (1987)

Fundamenta Mathematicae

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