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Topic: Irrefutable proofs that both Dedekind and Cauchy did not produce
any valid construction of the mythical "real" number

Replies: 1   Last Post: Oct 6, 2017 3:17 AM

 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: Irrefutable proofs that both Dedekind and Cauchy did not produce
any valid construction of the mythical "real" number

Posted: Oct 6, 2017 3:17 AM

On Wednesday, October 4, 2017 at 2:09:58 PM UTC-4, John Gabriel wrote:

> Dedekind Cut: A set partition of the rational numbers into two nonempty subsets L and R, such that all members of L are less than those of R and such that L has no greatest member.
>

Yes. Sets L and R are both subsets of Q. L and R are disjoint. L U R = Q.

> Any cut of the form
>
> (m, k) U (k, n) where m < k and k < n
>

Are you really suggesting that (m, k) and (k, n) comprise a Dedekind cut???

They are both subsets of Q, but they are NOT disjoint (k is common to both sets) and their union is NOT Q (n+1 is in neither set). So, they do not comprise a Dedekind cut.

> is EQUIVALENT to
>
> (-oo, k) U (k, oo) where k is not a rational number.
>

Huh? If k is not a rational number, then whatever (-oo, k) might be, it is NOT a subset of Q. Likewise for (k, oo).

Have a look at https://en.wikipedia.org/wiki/Dedekind_cut before you go any further, Troll Boy. As usual, you are in way over your head.

> So I can rewrite the cut (-oo, k) U (k, oo) as:
>
> (-oo,m] U (m, k) U (k, n) U [n, oo)
>
> Since my proof...

You have banned all axioms and the rules of logic in your goofy system, Troll Boy. Recall, you wrote:

"There are no postulates or axioms in mathematics."
-- Feb. 6, 2017

"There are no rules in mathematics."
-- March 17, 2015

How can you possibly write a proof if you believe this nonsense?

Dan

Visit my Math Blog at http://www.dcproof.wordpress.com