Re: Implementable Set Theory and Consistency of ZFC

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.

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.

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.

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.

.

Relevant Pages

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