FOM: Correction to Montague's "Set Theory and Higher Order Logic"
RTragesser at compuserve.com
Thu Feb 26 14:08:45 EST 1998
Correction: in the orginal I had mistakenly
written 'set thepory' where I plainly should
have written 'higher order logic' (corrected now:
In doing a literature search beginning with
the vintage period for papers etc. which might help
with my question about the rationalization of ZF+X
as foundational, I've come across an abstract of
R.Montague's "Set-Theory and Higher Order Logic",
in the 1965 Crossley/Dummett FORMAL SYSTEMS AND
RECURSIVE FUNCTIONS, to which I don't have easy
access (I've order it on ILL), but I would imagine
that the paper is widely known to FOMers.
Among other things, Montague claims to
show that he proves theorems justifying set
theory with indviduals as logically true within
higher order logic.
Depending on what this really means (I
need to see the theorems or especially proofs),
it seems to me to go very far toward answering
answering my questions about why set theory,
since I can understand very well why higher order
(esp second order logic).
I am very curious, then, about the
sense of his results...and what might be
regarded as firming them, strengthening them...
Are they as fundamental as they seem to be,
that is importantly "reducing" set theory to
higher order logic?
(I don't recall Stu Shapiro discussing
this; but I might have overlooked it.)
A foot-note (or foot-question) to this.
Bill Tait had observed (I think) that topoi
correspond to theories in higher order
intuitionistic logic???? If this is right with
"intuitionistic higher order logic" in any sense
parallel to classical higher order logic,
then it would seem to me that, philosophically
at least, there would be a way here to
short-out the TOP/SET issue???
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