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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".

Arithmetization of the field of reals with exponentiation extended abstract

Sedki Boughattas, Jean-Pierre Ressayre (2008)

RAIRO - Theoretical Informatics and Applications


 (1) Shepherdson proved that a discrete unitary commutative semi-ring A+ satisfies IE0 (induction scheme restricted to quantifier free formulas) iff A is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings. Let T range over axiom systems for ordered fields with exponentiation; for three values of T we provide a theory T in the language of rings plus exponentiation such that the ...

Complexity of the axioms of the alternative set theory

Antonín Sochor (1993)

Commentationes Mathematicae Universitatis Carolinae

If T is a complete theory stronger than ZF Fin such that axiom of extensionality for classes + T + ( X ) Φ i is consistent for 1 i k (each alone), where Φ i are normal formulae then we show AST + ( X ) Φ 1 + + ( X ) Φ k + scheme of choice is consistent. As a consequence we get: there is no proper Δ 1 -formula in AST + scheme of choice. Moreover the complexity of the axioms of AST 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,...

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