Either X element of M or X not element of M

Hero · science forum Guru Wannabe Joined: 29 Apr 2005 Posts: 220

There are more or less different sets of axioms for axiomatic
set-theories. As a beginner asking : what about these two axioms:
A) For all objects X and M, be it sets or elements or both,:
Either X element of M or X not element of M
B) If X element of M, then follows: X is not equal to M
Would it make You happy, to add these to Your axioms ?
Hero

Dan Christensen · science forum addict Joined: 03 May 2005 Posts: 70

"Hero" <Hero.van.Jindelt@gmx.de> wrote in message
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Jim Spriggs · science forum Guru Joined: 24 Mar 2005 Posts: 761

Hero wrote:

Lawrence House · science forum beginner Joined: 29 Apr 2005 Posts: 27

If you agree that the set of all sets exists then you have to admit that it contains itself as a member. In fact sets that contain themselves as members have a name. They are called EXTRAORDINARY sets. The others are ORDINARY sets. What about the set of all ordinary sets. Is it ordinary or extraordinary?

Hero · science forum Guru Wannabe Joined: 29 Apr 2005 Posts: 220

You need these A) and B) to avoid paradoxes.
As Jim pointed out, B) follows from the axiom of regularity (foundation
axiom). So, with Your proof-checker You could find out, if it is able
to proof this.
I'm very interested, what it finds out about this.
Hero

Hero · science forum Guru Wannabe Joined: 29 Apr 2005 Posts: 220

So in a set-theory with the axiom of regularity plus the axioms of
logic A) and B) are true. From A) and B) follows, that the expressions:

a) set of all sets
b) set of all sets, not element of itself
c) set of all sets with more then finite number of elements
are contradictionary to the theory, that is, that that, what they
express. doesn't exist (at least i hope so).
And so - i hope this is not too boring for You, Jim -finally, there are
multitudes of things, which can not be joined into a set ( a set, which
obeys the laws of this set-theory).
Thanks for straighten things out.
Hero

Hero · science forum Guru Wannabe Joined: 29 Apr 2005 Posts: 220

You really hit, what i was looking for. But i don't agree. It's not a
question, if i agree, Lawrence. I think, that from A) and B) follows,
that the sets of all sets does not exists. So not everything in the
mathematical world is a set.And if You call something an extraordinary
set, then that's a name. But does an "extraordinary set" fulfill all
the requirements for a set, given by the axioms and theorems of this
set-theory with axiom of regularity ?
Hero

Dan Christensen · science forum addict Joined: 03 May 2005 Posts: 70

"Hero" <Hero.van.Jindelt@gmx.de> wrote in message
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Jim Spriggs · science forum Guru Joined: 24 Mar 2005 Posts: 761

Hero wrote:

Hero · science forum Guru Wannabe Joined: 29 Apr 2005 Posts: 220

Thanks Jim.
So besides Cantor's paradise (expression of Hilbert) we can keep
Euclid's paradise. And there are more.
Sincerely Yours
Hero

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