FOM: Re: Ontology in Logic and Mathematics

Roger Bishop Jones rbjones at
Mon Sep 25 14:09:01 EDT 2000

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
> accept the existence of a model of ZFC?


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

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
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

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