Re: Cantor Confusion

MoeBlee wrote:
mueckenh@xxxxxxxxxxxxxxxxx wrote:
The countability of the set W of all the words of a finite alphabet is
proved by other means than the usual countability proof. The latter
consists in constructing a bijection with N, i.e., in constructing a
list. But it is impossible to construct a list of all words. It is
impossible to construct a bijection between W and N.

WRONG. It is a theorem that from the set of finite sequences of members
of a finite alphabet there is an injection into N.

Why should it be
possible to construct a bijection between N and N?

Because ANY x is 1-1 with x, by the identity function on x.

Note however that in order to attach a definite cardinal number to
every set, well-ordering must be possible for every set. That is why
Cantor insisted on well-ordering of all sets.

Piffle. Well-ordering has nothing to do with the existence or not of
at least one bijection.

You are wrong. If a set cannot be well-ordered, it cannot be assigned a
definite cardinal number. In order to assign a cardinal, at least one
bijection to an ordinal is required. This requires at least one
well-ordering.

WRONG. By using Scott's method (which requires the axiom of
reqularity), we don't need the well ordering theorem to prove that
every set has a cardinality.

MoeBlee

Hey Moe,

It does, then. There are reasons well-ordering is called a principle.

So, if every set has a cardinal then it has an initial ordinal and
every set is well-ordered?

I can see why you would prefer that to non-trichotomy in the cardinals,
because the lack of trichotomy would erase the meaning of those
transfinite cardinals, they could not be distinct or transitive loops
would exist and they could not be distinct.

I have some suggestions how you can argue against that. I also know
why.
You might say collections with cardinals aren't sets, they generally
are, collections defined by their elements.

So, is choice not a dependent axiom? If cardinals are trichotomous
they're ordinals. Fraenkel, Bar-Hillel, and Levy have a few
interesting paragraphs to the effect of considering the _existential
character_ of the "axiom."

Wolfgang, I thought it was funny when you were translating Georg into
English and described his ruminations of counting BACKWARDS from
infinity. Today, you see, omega's a limit ordinal and can't, there is
no mechanism for the status quo to do that. Another thing I thought
was interesting was a quote about sets of numbers being basically
ignorant of the soi-disant numbers they contain, where the translation
in the English edition instead put it in terms that they were
"abstracted." Translations definitely show the translater.

The universe is infinite, and that's an example that infinite sets are
equivalent. A counterexample to the powerset result is everywhere.

Look at infinitesimals, like iota or dx, with checks and balances, sum
them to integrate.

With warm regards,

Ross

.

Relevant Pages

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