Re: Implementable Set Theory and Consistency of ZFC

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:


Infinity "defines" something "new", namely the actual infinite set of
would-be Naturals. While Foundation defines something "old", namely a
property of the already constructed finite sets.
Perhaps there is a simple miscommunication here: by ~Infinity, I mean
the *denial* of the axiom of Infinity.

Why should someone feel the need for explicitly _denying_ Infinity?

I did not ask about your needs. I asked whether the formula ~Infinity
is also a theorem of (1)-(4), where ~Infinity is the negation of the
axiom of Infinity.

When considering only the first nine axioms of ZFC (if we denote
Infinity as the last, tenth (X) axiom), Infinity is simply _not
there_. Moreover, if ZFC had been designed in that way, infinite
sets wouldn't have come into existence: HOW can somebody conceive
the completed infinity of naturals, without the axiom of Infinity
coming into play? HOW can the transfinite ordinals be "defined" if
we don't already "have" the completed infinity of the naturals and -
by means of the Substitution axiom - a "means" to "build" those
beasties from THE naturals? It's somewhere in Halmos' book how to
"do" it. So the mere _absence_ of Infinity makes a lot impossible of
the things I've been finding suspect for years ..

It is remarkably difficult to get an answer from you sometimes.

--
Jesse F. Hughes
"Besides, discoverers are too proud to kiss ***. Indiana Jones would
never kiss some academic's ass to get published, and neither will I."
--James Harris
.

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