set theory

[kavlas]

how do we prove that : AεΑ is true or false ??

[Xittenn]

Are you asking for a proof with respect to membership such that A is an element of itself?

 

Prove [math] A \in A [/math]

 

If this is the case I believe you are asking about Russel's Paradox. I can post further if this is the case!

[kavlas]

Are you asking for a proof with respect to membership such that A is an element of itself?

 

Prove [math] A \in A [/math]

 

If this is the case I believe you are asking about Russel's Paradox. I can post further if this is the case!

 

I DID not define A = {x : ~xεx } SO i am not asking for the Russel's Paradox.

 

I am simply asking if we can prove that [math] A \in A [/math] is true or false

[ajb]

Are you talking about the notion of a universal set?

[Greg H.]

I've never done any in depth set mathematics, but I'm pretty sure it's an accepted definition of set theory that you can not construct a set that contains itself.

 

A quick google search turned up this: Axiom of Regularity

 

By definition, you don't prove an axiom.

 

And it does make a kind of sense (not that "common sense" is always a reasonable way to prove things) in that if I define a set, say Car Parts. I would expect it to contain elements like transmission, engine, steering (all of which are, in turn sets themselves containing still smaller members). I would not expect to find, tacked onto the end of that list, the very car I was describing. As a practical example, when iterating over an active directory listing, bad things happen to your application when you find groups that are members of each other (which in turn makes them members of themselves). It leads to things like infinite loops, application server failure, arguments with your security team over why they are flaming morons, and having to code around their inability to grasp the obvious.

 

Not that I am bitter or anything.

Edited May 14, 2012 by Greg H.

[mathematic]

how do we prove that : AεΑ is true or false ??

 

There is a problem if it is. To avoid confusion, give the A's numbers and assume they are all the same.

 

If A2εA1, then A3εA2, A4εA3, etc. How does the process ever end? Also you can up the ladder as well.

[Xittenn]

I DID not define A = {x : ~xεx } SO i am not asking for the Russel's Paradox.

 

I am simply asking if we can prove that [math] A \in A [/math] is true or false

 

I believe what you see as Russel's paradox is a weak interpretation thereof, and asking whether [math] A \in A [/math] is true or false is essentially the same. The answer depends on how you have constructed your set theory in the first place, and for the most part the question is addressed through class' or universal sets, and the axiom of choice.

 

I WASN'T sure what the question was asking, so I ASKED! : )

[kavlas]

I've never done any in depth set mathematics, but I'm pretty sure it's an accepted definition of set theory that you can not construct a set that contains itself.

 

A quick google search turned up this: Axiom of Regularity

 

By definition, you don't prove an axiom.

 

And it does make a kind of sense (not that "common sense" is always a reasonable way to prove things) in that if I define a set, say Car Parts. I would expect it to contain elements like transmission, engine, steering (all of which are, in turn sets themselves containing still smaller members). I would not expect to find, tacked onto the end of that list, the very car I was describing. As a practical example, when iterating over an active directory listing, bad things happen to your application when you find groups that are members of each other (which in turn makes them members of themselves). It leads to things like infinite loops, application server failure, arguments with your security team over why they are flaming morons, and having to code around their inability to grasp the obvious.

 

Not that I am bitter or anything.

 

 

If we accept that the the axiom of regularity doe not allow that,how do we then prove that.

 

I mean how do we prove that: ~[math]A\in A[/math]

[Greg H.]

If we accept that the the axiom of regularity doe not allow that,how do we then prove that.

 

I mean how do we prove that: ~[math]A\in A[/math]

 

As I said, by definition, you can't prove axioms. They are commonly accepted as being true without proof. If you have to prove it, it's not an axiom.

 

Edit:

Grammar failure.

Edited May 15, 2012 by Greg H.

[kavlas]

As I said, by definition, you can't prove axioms. They are commonly accepted as being true without proof. If you have to prove it, it's not an axiom.

 

Edit:

Grammar failure.

 

If you consider ~[math]A\in A[/math] as an axiom then how would you prove whether [math]A\in B[/math] and [math]B\in A[/math] is true or false?

[The Observer]

By ~A do you mean the compliment of A?

[kavlas]

By ~A do you mean the compliment of A?

 

No. By ~[math]A\in A[/math] i mean ~[math](A\in A)[/math]

[Greg H.]

If you consider ~[math]A\in A[/math] as an axiom then how would you prove whether [math]A\in B[/math] and [math]B\in A[/math] is true or false?

 

If we accept the axiom, no proof is needed. It's automatically false, because it would require a set to be an element of a subset of itself.

 

Consider the following (small) practical example.

 

Let us say that set A is the set of all counting numbers, and set B is the set of all counting numbers less than 10.

We can easily demonstrate that set B is a subset of set A. We can also just as easily see that there's no way set A can be a subset of set B, because it not only contains all of set B, but an infinite number of other members as well. I think (and this is just me thinking out loud - I don't have the chops to prove it) that in this case, if we should find a set A and a set B so that each are subsets of the other, they'd end up being the same set.

 

In the interest of full disclosure, I should point out that, according the sources I checked earlier (see my previous posts) there are set theories that do not obey this axiom - obviously the practical example I gave in my initial post in this thread demonstrates one. But we can also see that these kinds of sets lead to certain problems in practical application.

 

 

See also:

Set Theory

 

ZF Set Theory

 

Axiom of Regularity