Re: Relative Cardinality



Jiri Lebl wrote:
> mueckenh@xxxxxxxxxxxxxxxxx wrote:
> > (Alhough I assume you are not yet ready to hear the truth, I'll tell it
> > you: There is not even a single irrational number. But rationals do
> > exist. Therefore, your equality cannot be true.)
>
> Yes, and so the square root of 2 is rational. But of course! Or does
> it not exist as a number?


Correct. It is an idea but not a number. It was introduced as a nunber
by Cantor and his contemporaries. And Cantor had already recognized
quite right: Set theory stands and falls with the irrational numbers.

The diagonal length of a square of side 1
> does not exist? Surely it must.

What is the 10^100 digit of the diagonal of the unit square?
>
> I think this totally explains your quest against cardinality/infinity.
> But I think you are a couple of thousand years late in your
> "irrationals don't exist" arguments.

You are in error, twice.
1) Irrational "numbers" are not older than Cantor. Meray was the first
one who in 1869 introduced fictitious limits of convergent series like
nombres incommensurables. Cantor did independently the same in 1871.
2) I am not thousand years late but about 10 years too early.

>
> > > This will require that the power set (set of all subsets of a given set)
> > > of an infinite set be the same size as the original set,
> >
> > Or it will show that ZFC is inconsistent by the only conclusion to be
> > drawn, namely: actually infinite sets do not exist.
>
> Actually infinite sets exist by axiom in ZFC, you do not prove their
> existance, you ASSUME their existance.

I do not assume their existence. No, they do not exist.

> And no, your arguments do not
> show any inconsistency in ZFC, because you are not even using ZFC for
> your arguments.

But I contradict its results.

You are using naive intuition and a firmly held belief
> that infinite sets do not exist. I'm sure there are people out there
> doing math in "ZF minus infinity axiom" (you don't even need axiom of
> choice there). But I think you'll find it rather restrictive on what
> you can get done that would in fact be useful.

Abraham Robinson (1964): (i) Infinite totalities do not exist in any
sense of the word (i.e., either really or ideally). More precisely, any
mention, or purported mention, of infinite totalities is, literally,
meaningless. (ii) Nevertheless, we should continue the business of
Mathematics 'as usual', i.e., we should act as if infinite totalities
really existed."

Let us be aware of (i) and obey (ii).

Regards, WM

.

Loading