FOM: Re: Ontology in Logic and Mathematics

[Roger Bishop Jones]

In response to Matt Insall Monday, September 25, 2000 4:28 AM


> Roger:
> You only have to accept that the axiom of infinity is true in the intended
> domain of discourse, e.g. in all models of ZFC (if that is the intended
> domain of discourse).
>
>
> Matt:
> By referring to ``all models of ZFC'' do you contend that such a person
must
> accept the existence of a model of ZFC?

No.

It all hangs on the semantics.
If the semantics of a language is specified by giving truth conditions for
sentences then these truth conditions tell you what it means to assert the
sentence (i.e. asserting a sentence is asserting that the truth conditions
hold).

In a first order language a sentence is true (simpliciter), and provable,
iff it is true in every model of the non-logical axioms.
If there are no models of the non-logical axioms then all the sentences are
true.
Clearly the sentences cannot be held to assert the existence of a model - if
they did they would be false if there were none.

You can of course use the syntax of first order set theory with some other
semantics (and I advocate that this should be done), in which case the
answer to your question might be different.

Roger Jones