FOM: Re: The logical, the set-theoretical, and the mathematical

[Roger Bishop Jones]

Responding to: Joe Shipman Monday, September 11, 2000 10:54 PM

> (Mossakowski insists on an axiom of Infinity as well, but Jones argues
> that this axiom is purely ontological and is only necessary in a
> metaphysical sense.

No that is not my view.  My view on the axiom of infinity is:

Whether or not the axiom of infinity is true in some language depends upon
the semantics of the language, and specifically on the domain of discourse
of the language (which may be loosely defined).
If the domain of discourse is abstract and the axiom of infinity is true in
that domain (or in all instances of a loosely defined domain) then it will
be necessarily true and a logical truth (in the given language).
The existence of the domain of discourse (or the consistency of a loosely
specified domain) of a language is presupposed by the definition of the
language, but not asserted by the sentences of the language.
Whether or not the domain of discourse of this language exists in any
absolute sense is close to metaphysical.
It will be false if it can be shown to be inconsistent, but if consistent
there is no way to tell whether true or false and hence there is reason to
doubt that the claim has any content.

Hence, when the language of set theory is used to talk about the cumulative
heirarchy, the axiom of infinity is true as stated in that language and is a
logical truth.
When the axiom of infinity is asserted in some absolute sense, without
reference to some explicitly or implicitly understood domain of discourse,
its meaning, truth value and modal status are uncertain.

Roger Jones