An independence result concerning the axiom of choice

Independence, Randomness and the Axiom of Choice

A Conservation Result concerning Bounded Theofues and the Collection Axiom

We present two proofs, one proof-theoretic and one model-theoretic, showing that 0 adding the BE1-collection axioms to any bounded first-order theory R of arithmetic 0 yields an extension which is VC1-conservative over R. Preliminaries. A theory of arithmetic R contains the non-logic symbols 0, S, +, = , and 6 . R may contain further non-logical symbols; in particular, S2 is a theory of arithme...

A Conservation Result concerning Bounded Theories and the Collection Axiom

We present two proofs, one proof-theoretic and one model-theoretic, showing that adding the ß ^-collection axioms to any bounded first-order theory R of arithmetic yields an extension which is V ̂ -conservative over R. Preliminaries. A theory of arithmetic R contains the nonlogic symbols 0, S, +, ■, and < . R may contain further nonlogical symbols; in particular S2 is a theory of arithmetic [1]....

The Axiom of Choice

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...

The Axiom of Choice

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set X...