The axiom of choice, a benign matter for the non-logician, puzzles mathematicians. Today, it manifests itself in a strange way: it takes, depending on the axiom’s variants, either two or infinity of colors to resolve a coloring problem. Just as the parallels postulate seemed obvious, the axiom of choice has often been considered true and beyond discussion. The inventor of set theory, Georg Cant...

The existence of a Paretian and finitely anonymous ordering in the set of infinite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of ...

Dzik (1981) gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems. The problem therefore arises of giving a direct proof, not ...

It is know in topos theory that axiom of choice implies that the topos is Boolean. In this paper we want to prove and generalize this result in the context of allegories. In particular, we will show that partial identities do have complements in distributive allegories with relational sums and total splittings assuming the axiom of choice. Furthermore, we will discuss possible modifications of ...

Most work on infinitary algebras in the literature makes ample use of the axiom of choice AC (see, e.g., [10],[12],[14]). While the theory of arbitrary infinitary algebras can hardly be developed in a satisfactory manner without AC, in the case of Peano algebras (= word algebras, absolutely free algebras) many or even most of the basic results can be proved in Zermelo-Fraenkel set theory ZF wit...

We study computability on sequence spaces, as they are used in functional analysis. It is known that non-separable normed spaces cannot be admissibly represented on Turing machines. We prove that under the Axiom of Choice non-separable normed spaces cannot even be admissibly represented with respect to any compatible topology (a compatible topology is one which makes all bounded linear function...

In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (1) The axiom of dependent choice. (2) Products of compact Hausdorff spaces are Baire. (3) Products of pseudocompact spaces are Baire. (4) Products of countably compact, regular spaces are Baire. (5) Products of regular-closed spaces are Baire. (6) Products of Čech-complete...

For any set S let ∣ ∣ seq1-1(S) ∣ ∣ denote the cardinality of the set of all finite one-to-one sequences that can be formed from S, and for positive integers a let ∣ ∣a ∣ ∣ denote the cardinality of all functions from S to a. Using a result from combinatorial number theory, Halbeisen and Shelah have shown that even in the absence of the axiom of choice, for infinite sets S one always has ∣ ∣ se...