FOM: Re: Axiom of infinity

[Joe Shipman]

Another example, equivalent to the one below, would be the conjunction of
the axioms for a projective plane with Pappus's axiom and the negation of
Desargues's axiom.
-- JS

Joe Shipman wrote:

> Simpson asks for a simple, short axiom of infinity that doesn't
> interpret one of the three mentioned by Baldwin (discrete order, dense
> order, pairing function).
>
> How about the conjunction of the axioms for a division ring augmented by
> the negation of the commutativity axiom for multiplication?  By
> Wedderburn's theorem, a finite division ring is a field, so the only
> models of this sentence would be infinite.  This is a hard theorem to
> prove, so there can't be an easy proof that the sentence interprets one
> of the three infinity axioms Baldwin names.
>
> -- Joe Shipman