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Topic: Why do we need those real non-constructible numbers?
Replies: 68   Last Post: Dec 11, 2017 1:42 AM

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 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: Why do we need those real non-constructible numbers?
Posted: Nov 9, 2017 8:33 AM

On Thursday, November 9, 2017 at 1:54:32 AM UTC-5, WM wrote:
> Am Mittwoch, 8. November 2017 23:52:42 UTC+1 schrieb Dan Christensen:
>
>

> > > It convinces us of the fact that set theory can cause real brain damage. Of course you cannot be healed. But discussions like these are very helpful in convincing normal people including young students that set theory is really detrimental to clear thinking.
> >
> >
> > Brave words for someone who time and again has claimed to have found inconsistencies in set theory, but could not prove it in using the axioms of set theory as would be required.

>
> Axiom VII. The domain contains at least a set Z which contains the null-set as an element and is such that each of its elements a is related to another element of the form {a}, or which with each of its elements a contains also the related set {a} as an element.
>

You mean, the axiom of infinity? Usually stated as follows:

There exists set I: [Null in I & For all x in I: [x U {x} in I]]

where Null is the empty set.

> Axiom IV. Every set T is related to a second set ?(T) (the '"power set" of T), which contains all subsets of T and only those as elements.
>

Usually stated:

For all sets X: There exists set Y: For all Z: [Z in Y <=> For all w in Z: w in X]

> A model of a theory is a structure that satisfies the sentences of that theory, in particular its axioms.
>
> There is no countable model of axioms IV and VII.
>

There is nothing about "model" in the axioms of set theory (e.g. in ZFC). If you want to show that model theory is inconsistent, you will have to state the axioms of model theory and derive a contradiction using those axioms. Good luck with that. But then, you would STILL have to derive a contradiction using only the axioms of set theory which you have failed to do yet again, Mucke!

Dan

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

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