Re: Induction, Kant, Infinity

From: george (greeneg_at_cs.unc.edu)
Date: 12/09/04 

Date: 9 Dec 2004 15:49:43 -0800

I should just kill the new Google interface for making it
too hard to quote people.

RF> If you go to higher-order logic,
RF> there is a countable model for everything.

MUST You FLAUNT ignorance???
Begging people not to call you stupid is NOT going to help
if you refuse to take some basic logic and insist on excreting
turds like this one. You have this EXACTLY backwards:
the ACTUAL truth is that IF YOU STAY DOWN at FIRST-
order logic, THEN there is a countable model for, if not
"everything", at least "everything that you can consistently
define in first-order logic". In fact, tragically, as Godel
showed, there are usually TWO DIFFERENT models of it.
This is tragic because you sometimes know which one you
want, but, in FOL, can't actually SAY which, or what's worse,
even if you can, then after you have, you will THEN have
2 clashing models of THAT (ad inf).

RF> If to get the reals you need to resort to the
RF> infinitely-iterated slums of higher-order logic,

Well, YOU DON'T,dumbass. There is a decent FIRST-
order theory of the reals: it is even decidable!
Again, you have your head on EXACTLY ass-backwards:
it is to get to the NATURAL numbers that you have to
go to higher-order logic. First-order PA has non-standard
models, and Godel's theorem shows that NO recursively
axiomatizable first-order theory aspiring toward "getting"
all the way to N is ever actually going to get there. To
get there, you need to re-phrase the induction-schema at
higher-order.

RF> ever deeper into the hells of offputting
RF> that infinite sets are equivalent,

Just because something invites you personally to sit down
and shut up does not make it "hells". ZF *proves* that there
are different orders of infinity, by proving that every set is
smaller than its powerset. You don't have to believe that
powersets have to exist and you don't have to believe (for
that matter) that infinite sets exist; you could have a perfectly
decent finite set theory without either of those axioms.
And indeed, if you want a universe in which infinite sets are
equivalent,that is probably the best you can do: go to a finite
set theory where all infinite sets are equivalent because they
all DON'T EXIST, because all infinite classes are PROPER as
OPPOSED to being sets.

RF> then you get countable models for those things because
RF> infinite sets are equivalent.

In classical first-order logic, you get countable models
because first-order LANGUAGES are countable, so if
you have a model made out of anythingOTHER than terms,
you can construct a term model just by making every predicate
true of those term-tuples that your model interprets as the
things that it interprets your predicates as true of. If (as in
the case of modeling ~Con(PA) in PA, ala Godel) you have
some anonymous model-element that no term is interepreted
to, you can just grow the language to include a term for it
and try again. You do NOT need higher-order logic (except
to the extent that you are presuming it in the language
in which you are stating your interpretation and the mappings
IT uses).

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