Re: An uncountable countable set


Franziska Neugebauer schrieb:


Therefore there are not infinitely many difference[s] of 1
between natural numbers.

Your consequent is proven false (see below). Therefore your
implication is false, too.

You are in error.

Where precisely is the error?

The assertion that infinitely many differences of 1 can be provided by
finite natural numbers.

You just proved it to be true.

You may "just" as well prove the opposite. Please do so.

The set of natural numbers (i.e., finite numbers n, i.e., numbers
with finitely many differences of 1 between 1 and n) does not yield
infinitely many differences of 1.

This is a reiteration not a proof.

There are not infinitely many differences, if we consider the finity of
each number.
There are infinitely many differences, if we consider infinitely many
numbers.
Therefore one of the assumptions is wrong: Either there are not
infinitely many numbers or there is at least one number which is
infinite by size.


Do you agree that P ~ omega?

P has the same cardinality as |N. But either there are not infinitely
many numbers or there is at least one number which is infinite by size.

Do you agree that card (omega) /e omega?

card (omega) = aleph_0 = the number of natural numbers and that cannot
be infinite, as we have seen.

Regards, WM

.

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