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Topic: § 524 Are finite cardinal numbers natural numbers?
Replies: 54   Last Post: Jul 22, 2014 10:45 AM

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 wolfgang.mueckenheim@hs-augsburg.de Posts: 3,394 Registered: 10/18/08
Re: ?? 524 Are finite cardinal numbers natural numbers?
Posted: Jul 20, 2014 5:04 PM

Am Sonntag, 20. Juli 2014 22:06:36 UTC+2 schrieb Zeit Geist:

> > Every n e N is used In The Function: |s_n| = oo
>
> > Anything missing?
>
>
> Some Function E with Domain N and and Codomain Q, which Maps Each n e N Uniquely to some q e Q.
>

I have each n e N uniquely mapped on s_n, and vice versa.
>
>
> You have an Infinite Set of Finite Partial Funtions, E_n.
> You then Show, quite Correctly, that None of these Enumerate Q.
> You then Conclude, quite Incorrectly, that there is No E that can Enumerate Q.
> It is your Duty to Prove that the Former implies the Latter.
>

The proof is shown by the fact that only natural numbers n not mapped on sets s_n with |s_n| = oo can reduce sets s_n. Such natural numbers do not exist.
>
>
> You claim that the Fact that there is a Discontinity in the "Cardinality Function of the Remaining Rationals",

There is no discontinuity. If it were, how could it be constructed? There is only an error in the interpretation of the limit set.

> with a Domain of N U { N }, that is Every Natural and the Set N itself, is a Contradictions.

> It's NOT, because the "Cardinality Function" over the Domain N U { N } is Discontinuous.

Why should it? What would enumerate the rationals that are not enumerated?
>
> It would Actually be Strange if your Function was Continuous.

Not at all. My function |s_n| is oo for all n. And more than all n is not available to reduce the cardinality of |s_n|.
>
>
>

> > > Again, the Subtleties Allude you.
>
> >
>
> > Please show what subtleties are missing.
>
>
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> You beleive that "For Every n e N, P(n) implies P(N).

Cantor believes that for every q_n there is n (and vice versa) implies the bijection of N and Q. Show a better argument, if you can.
>
> You write this something like, No FIS of N can Not Enumerate Q; N is the Union of its FIS, Hence N can Not Enumerate Q.

N cannot enumerate anything because it is not a number.

>
> I would bet that Formal Examination of you "Proof" will show either Quantifier Dyslexia or the Reversal of the Premise and Conclusion in some Implication.

Insted of betting, show it.
>
> I just know that you make a False Conclusion and would like to Know what you Think the Justification for it is.

My conclusdion is that no natural numbers exists beyond all natural numbers. All natural numbers have been shown insufficient to empty the sequence of the s_n as well as of the |s_n|. So what should accomplish this task?

Regards, WM