A Note on the Axiom of Countability.

The note discusses some considerations which speak to the plausibility of the axiom that all sets are countable. It then shows that there are contradictory but nontrivial theories of ZF set theory plus this axiom. In this note, I will make a few comments on a principle concerning sets which I will call the Axiom of Countability. Like the Axiom of Choice, this comes in a weaker and a stronger form (local and global). The weaker form is a principle which says that every set is countable: WAC ∀z∃f (f is a function with domain ω ∧ ∀x ∈ z∃n ∈ ω f (n) = x). (The variables range over pure sets—including natural numbers. ω is the set of all natural numbers.) The stronger form is that the totality of all sets is countable: SAC ∃f (f is a function with domain ω ∧ ∀x∃n ∈ ω f (n) = x). The stronger form implies the weaker. Any set, a, is a sub-totality of the totality of all sets. Hence, if the latter is countable, so is a. So I focus mainly on this. Let us start by thinking about the so called Skolem Paradox. Take an axiomatization of set theory, say first-order classical ZF. This proves that some sets, and a fortiori the totality of all sets, are uncountable. Standard model theory assures us that there are models of this theory (in which ‘∈’ really is the membership relation) where the domain of the model is countable. There is a function which enumerates the members of the domain. It is just one which has failed to get into the domain of the interpretation. Why should we not suppose, then, that the universe of sets Talk at the conference “Philosophy, Mathematics, Linguistics: Aspects of Interaction 2012” (PhML-2012), Euler International Mathematical Institute, May 22–25, 2012.

Authors

• Graham Priest

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