🚀 go-pugleaf

In a previous article, jrk@sys.uea.ac.uk (Richard Kennaway) says:

>In article <1992Aug20.171630.18667@ariel.ec.usf.edu> Gregory McColm,
>mccolm@darwin.math.usf.edu. writes:
>>The standard (ie, wellfounded) models all satisfy Con(ZFC)
>
>Can someone clarify for me the term "standard model"?  Is this a concept
>with a formal definition, or is a "standard model" of a theory, simply a
>model which satisfies the intuitions which inspired the axioms of the
>theory?
>
>--
>Richard Kennaway     SYS, University of East Anglia, Norwich NR4 7TJ, U.K.
>Internet:  jrk@sys.uea.ac.uk               uucp:  ...mcsun!ukc!uea-sys!jrk
>

	I do not know what all "standard" might mean, but in a lot of
contexts including this one it means "not non-standard" where "non-
standard" is used as in "non-standard analysis".  In short, a model is
"standard" in this sense if all the sets it takes to be finite are
actually finite.

	If a model is standard in this sense then all the sets it takes 
as codes for proofs of ZF actually are such codes, and so there is no set
taken as coding a proof ofa contradiction, so the model satisfies Con(ZF).
Of course all of this assumes there are models in the first place, so ZF
is consistent.  As a previous poster said, models of ZF+not-Con(ZF) are
just models that recognize some sets as codes for proofs of contradictions.
But what these sets code are not really finite proofs--i.e. are not
proofs at all.