Re: Ultimate debunking of Cantor's Theory


"Calvin" <crice5@xxxxxxxxxxxxxx> wrote in message news:1184205380.943945.99220@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jul 11, 12:43 am, Calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Jul 11, 12:26 am, Proginoskes <CCHeck...@xxxxxxxxx> wrote:

> THE ONLY SETS WHICH EXIST ARE FINITE SETS. THE INFINITE IS ONLY A
> PRODUCT OF THE IMAGINATION.

Then what is the largest natural number?

Getting back to this original response, some have said
that there not being a largest natural number does
not imply that the natural numbers comprise a set.

First, my response was keyed on the word 'finite', not
on the word, 'set'. At that point I did not know that
the definition of 'set' was an issue. So I wanted to
produce a counterexample, a set that was infinite.

The most basic infinite set, it seems to me, is the set
of natural numbers, but if it was finite, there would
have to be a largest natural number. Hence my response.

But if the original post meant that, by the poster's
definition of 'set', only finite sets may exist, then
I can't refute that, because one cannot refute a
definition. One can only express interest or lack of
interest in a particular definition, and in my
case it is lack of interest, since I see no use or
justification for such a definition of set.

If anyone would care to enlighten me ...


Indeed.

But set theory minus the Axiom of infinity is a perfectly valid set theory. The advantage for the poster is that Cantor's diagonal construction of the Reals doesn't exist, or indeed any form of the diagonal argument applied to infinite sets. Indeed, if you consider that the existence of more than one type of infinity to be "absurd", you can use this to show that the Axiom of infinity must be false through a reductio-ad-absurdum argument. I suspect this is what the OP believes.

It is interesting to see what parts of ZF you can construct without the Axiom of Infinity, just as its interesting to see what parts of ZFC you can construct without AxC. In neither case is the answer trivial (at least not to me); whether you personally find it interesting is a matter of taste.

Does anybody actually know what can be constructed in ZF-I (ZF minus the axiom of infinity)? Can {pi} be constructed? I suspect so, but its a long way from ZF to the construction of the Reals, and you would need to check that Axiom of Infinity isn't needed for the construction of a particular Real number. (Its certainly needed to construct the set of *all* Real numbers).

My suspicion is that ZF-I can be used to construct exactly the same things as ZF, excepting infinite sets. I would be interested if anybody actually knew if this is true or false. I would also be interested to know if ZF-I plus ~I can prove its own consistency.

All pretty valid questions for a set theorist, I would have thought.







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