A Hamiltonian property of connected sets in the alternative set theory.
A nonstandard treatment with quantities
A note on the Mac Dowell-Specker theorem
A note on Δ₁ induction and Σ₁ collection
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: " exists for some y sufficiently large that x is smaller than some primitive recursive function of y".
A sequence of theories for arithmetic whose union is complete
Addendum to “A note on the MacDowell–Specker theorem”
Addition of initial segments. I.
Addition of initial segments. II.
An elimination of infinitely small quantities and infinitely large numbers (within the framework of AST)
An elimination of the predicate “to be a standard member” in nonstandard models of arithmetic
An Ordering of the set of Sentences of Peano Arithmetic
Arithmetic of cuts and cuts of classes
Arithmetically independent integers and values of rational functions.
Arithmetization of the field of reals with exponentiation extended abstract
(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 in the language of rings plus exponentiation such that the ...
Borel classes in AST. Measurability, cuts and equivalence
Bounds and definablity over fields.
Closure properties of countable non-standard integers
Completion closed algebras and models of Peano arithmetic
Complexity of the axioms of the alternative set theory
If T is a complete theory stronger than ZF such that axiom of extensionality for classes + T + is consistent for 1 (each alone), where are normal formulae then we show AST + + scheme of choice is consistent. As a consequence we get: there is no proper -formula in AST + scheme of choice. Moreover the complexity of the axioms of AST is studied, e.gẇe show axiom of extensionality is -formula, but not -formula and furthermore prolongation axiom, axioms of choice and cardinalities are -formulae,...
Conservative extensions of models of arithmetic.