Re: Small set theory.


Jesse F. Hughes wrote:
"zuhair" <zaljohar@xxxxxxxxx> writes:

Ok, let me tell you my intentions behind this axiom.

I want to currenty that there exist more than one set that is a member
of itself.

Moe Blee has suggested the following:

Exy(~x=y & xex & yey).

Do you agree that this would convey what I am aiming at. if so I will
put it instead of 7) above

so Axiom 7:) Exy(~x=y & xex & yey).

There are two problems with this approach. First, just knowing that
there are two self-membered sets doesn't tell you much of anything at
all. Instead you want a principled manner to say *which*
self-membered sets exist.

Second, being self-membered isn't really what you want anyway. You
want non-well-founded sets, but not every non-well-founded set is
self-membered.

What do you mean isn't really what you want anyway. Who is "you" in
your statement.
If you mean me , then you are wrong, I am not interested in non-well
founded sets, I am interested in sets that are members of themselfs. I
think that x is well founded means that
x!e ........Ux , for example:

x={ {} , { {} } , { { {} } } }

Ux= { {}, { {} } }

UUx= { {} }

UUUx= { }

Now x!e......Ux = x!ex /\ x!e Ux /\ x!e UUx /\ x!eUUUx

I am not interested in these set. By the way Frege's cardinals are not
well founded sets.

Anyhow , what I wanted to say is that I am not interested in such sets.
what I am interested in is_ when a set is a member of itself? and when
a set is no a member of itself? The whole of this set theory is made
with this aim in mind, so that's why I should first axiomatize in a
manner as to generate sets that are in themselves and sets that are not
in themselfs, and set the different complex interaction between them.

But I agree with you , this aim is so difficult , it needs logical and
mathematical maturity.

I think this subject should be closed.

Zuhair

But there are a lot smarter guys out there who have put some thought
into anti-well-founded set theory. You should look into it rather
than trying to haphazardly define your own axioms.

The only problem is that these theories require a bit of mathematical
maturity. It's not trivial to understand the set theory in /Vicious
Circles/, for instance. But it's a lot more profitable to at least
try to understand that theory than to come up with your own.

Anyway, Axiom 7 says there are two self-membered sets. But I don't
see that getting you very far. You have no idea what these two sets
look like.

--
"I'm the theory guy.
Other people are the experimental people.
If you push me on details I get annoyed, as I'm the theory guy.
I'm the theoretical amateur mathematician." --James S. Harris, poet

.

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