Re: Cantor Confusion

In article <1164446020.298851.293790@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

Virgil schrieb:


By induction one can even prove that always a finite number of words
has been mapped on the natural numbers.

By induction one can also map an infinite set of words onto the natural
numbers.

Then you think the natural numbers are covered by induction? You read
my heart. I knew they were not actually infinite.

If your heart says that induction, in the sense of Peano, is limited to
the members of a finite set, your heart is stupid.


Now, after you have "completed" this finite list of words, take it and
construct a diagonal number.

I prefer to take the infinite set of words.

I know, but you don't get it by induction.

Then maybe you don't know how to use induction, as those who do know how
to use it can get get an infinite set of results from it.

The form of induction I use is based on having, as one has within ZF and
NBG, an infinite set of finite naturals. If one has the first natural
and one has the successor to every natural, then one has all of the
members of that infinite set of finite naturals.

If WM's version of induction is less effective, it is not our fault.


If the set 1,2,3,...,n is finite, then the set 1,2,3,..., n+1 is finite
too.
Therefore, by induction we (that is: those who can think logically)
prove that the generated set is always finite.

Provided one assumes that that process has a last step, one can conclude
that that process has a last step, but absent that assumption, one
cannot reach that conclusion.

And I, for one, do not make that assumption.
.

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