### $\nabla $-model and distributivity in Boolean algebras

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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".

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.

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 $X={X}_{M}$.

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) $={I}_{fix}\left(j\right)$ for some nontrivial automorphism j of an end extension of that satisfies IΔ₀. Here ${I}_{fix}\left(j\right)$ is the...

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

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 $\kappa \in \omega \cup \aleph \u2080,\aleph \u2081,{2}^{\aleph \u2080}$. There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $\mu (T,\alpha )={2}^{\aleph \u2080}$. (3) If α < ω₁ and T ⊢ V ≠ OD, then $\mu (T,\alpha )\in 0,{2}^{\aleph \u2080}$. (4)...