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Complexity of the axioms of the alternative set theory

Antonín Sochor — 1993

Commentationes Mathematicae Universitatis Carolinae

If is a complete theory stronger than Fin such that axiom of extensionality for classes + + ( ∃ X ) Φ i is consistent for 1 ≤ i ≤ k (each alone), where Φ i are normal formulae then we show + ( ∃ X ) Φ 1 + ⋯ + ( ∃ X ) Φ k + scheme of choice is consistent. As a consequence we get: there is no proper Δ 1 -formula in + scheme of choice. Moreover the complexity of the axioms of is studied, e.gẇe show axiom of extensionality is Π 1 -formula, but not Σ 1 -formula and furthermore prolongation axiom, axioms of choice and cardinalities are Π 2 -formulae,...

Revealed automorphisms

Jiří Sgall;  Antonín Sochor — 1991

Commentationes Mathematicae Universitatis Carolinae

We study automorphisms in the alternative set theory. We prove that fully revealed automorphisms are not closed under composition. We also construct some special automorphisms. We generalize the notion of revealment and Sd-class.

Forcing in the alternative set theory. II

Jiří Sgall;  Antonín Sochor — 1991

Commentationes Mathematicae Universitatis Carolinae

By the technique of forcing, some new independence results are proved for the alternative set theory (AST) and similar weak theories: The scheme of choice is independent both of AST and of second order arithmetic, axiom of constructibility is independent of AST plus schemes of choice.

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