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For a fixed q ℕ and a given Σ1 definition φ(d,x), where d is a parameter, we construct a model M of 1 Δ0 + ¬ exp and a non standard d M such that in M either φ has no witness smaller than d or phgr; is equivalent to a formula ϕ(d,x) having no more than q alternations of blocks of quantifiers.
After introducing the large set notion of Mahloness, this paper shows that constructive set theory with an axiom asserting the existence of a Mahlo set has a realizability interpretation in an extension of Martin-Löf type theory developed by A. Setzer.
We prove that on many inaccessible cardinals there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal λ is λ × ω-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra of cardinality
. Lastly, we give further...
Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a...
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