Re: Implementable Set Theory and Consistency of ZFC

lwalke3@xxxxxxxxx wrote:

On Oct 22, 12:21 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx>
wrote:

It's impossible to have infinity without Infinity. If not, show us such
a model, please.

It appears that the confusion stems from HdB's
failure to distinguish a model from a theory,
once again.

It appears that the confusion stems from mainstream mathematics failure
to distinguish a childish model from any serious implementation, once
again. Look, my Implementable Set Theory e.g. is covering all Database
Applications on Earth, which is .. a billion dollar business!

HdB apparently believes that his bitmap model
is the _unique_ model of ZFC-Infinity (or
indeed, the unique model of the theory of the
four Abian axioms). Much of what he writes is
based on this assumption -- especially his
proof that ZFC is inconsistent. In other
words, HdB believes that since Infinity is
false in the bitmap model, therefore Infinity
cannot be an axiom of the theory without
introducing a contradiction.

Meanwhile, I've _withdrawn_ my claim that ZFC is "inconsistent". Because
that bothers me less than the fact that Infinity is a suspect axiom from
the start. I'm not yet finished with Infinity, though ..

I believe that many people would take HdB and
the other so-called "cranks," who are also
guilty of this error, by realizing that one
must distinguish between model and theory.

Between the implementation and the theory. Yes. And the implementation
is a judge for the theory, at least as much as the theory is a judge for
the implementation.

Notice that if T is a theory and phi is any
axiom, then if M is a model of T+phi, then
M is a model of T. In other words, any model
of a theory is a model of any subtheory.

ZF-Infinity is a subtheory of ZF, since the
former is a subset of the latter. And so any
model of ZF is a model of ZF-Infinity.

Yes. And a "model" of (ZF-Infinity) is not per se a model of ZFC, which
is the very difference between an "if" and an "iff".

Han de Bruijn

.

Relevant Pages

  • Re: Implementable Set Theory and Consistency of ZFC
    ... It appears that the confusion stems from mainstream mathematics failure ... therefore Infinity ... that bothers me less than the fact that Infinity is a suspect axiom from ... ZF-Infinity is a subtheory of ZF, ...
    (sci.math)
  • Re: Some basic set theory questions
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  • Re: Ultimate debunking of Cantors Theory
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  • Re: Infinities and infinitesimals
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