Re: Ultimate debunking of Cantor's Theory

On Jul 11, 7:20 am, Calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Jul 11, 9:40 am, "Peter Webb" wrote:

"Calvin" wrote:
On Jul 11, 12:26 am, Proginoskes 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?

I can't see that restricting set theory to finite sets
means there has to be a largest natural number.

The original post doesn't restrict set theory to finite
sets. It says that only sets which are finite exist.

So, we could easily take 'all sets are finite' to be an axiom of a set
theory (without the axiom of infinity). ZF-infinity+~infinity is a
consistent theory.

Obviously natural numbers exist, and can be used for
example to count the words in this post. Since they
exist, and there are supposedly only a finite number
of them,

"Only finite sets exist" may not entail "there are only finitely many
natural numbers" since the set of natural numbers would not exist even
to SAY anything about the cardinality of such a set.

then there must be a largest one. Else what
the original post stated is not true.

That doesn't follow. It is consistent to have arbitrarily large
natural numbers but no set of them.

The analogy with ordinals in even ZFC was given and it is appropriate.
There are arbitrarily large ordinals but no set of them. And there can
be arbitrarily large natural numbers but no set of them.

MoeBlee

.

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