Disproving Regularity with Transitivity of Sets

[sunmaz94]

Homework Statement

Let A, B, and C be sets. Assume the standard ZFC axioms.

Please see below for my updated question.


Thanks.

 

[Dick]

A ε B means the set A is a element of the set B, right? That's not the same thing as A being a subset of B. You have sets that are elements of other sets. Try to construct a counterexample.

 

[sunmaz94]

That's what I was afraid of. How then can I show that if A ε B and B ε C and C ε A, that C ε C (yields the contradiction I require).

 

[Dick]

Pick A={a}. Pick B={A}={{a}}. So A ε B. Correct so far? Pick C={B}={{{a}}}. So B ε C. Is A ε C?? This is a little tricky. a is not equal to the set consisting of a.

 

[sunmaz94]

Pick A={a}. Pick B={A}={{a}}. So A ε B. Correct so far? Pick C={B}={{{a}}}. So B ε C. Is A ε C?? This is a little tricky. a is not equal to the set consisting of a.

Yes I understand this. But how does the chain of set inclusions I mention lead to the fact that C ε C?

(Thanks for all your help!)

 

[Dick]

Yes I understand this. But how does the chain of set inclusions I mention lead to the fact that C ε C?

(Thanks for all your help!)

You seem to have deleted the question. I thought I suggested you try to prove that A ε B and B ε C doesn't necessarily imply A ε C? Since inclusion is not transitive?

 

[sunmaz94]

You seem to have deleted the question. I thought I suggested you try to prove that A ε B and B ε C doesn't necessarily imply A ε C? Since inclusion is not transitive?

I agree. I am now asking modified question: How then can I show that if A ε B and B ε C and C ε A, that C ε C (yields the contradiction I require).

Apologies for the confusion.

 

[Dick]

I agree. I am now asking modified question: How then can I show that if A ε B and B ε C and C ε A, that C ε C (yields the contradiction I require).

Apologies for the confusion.

I think you are reaching way too far for a contradiction. C ε C puts you squarely in Bertrand-Russell paradox territory. A simple example of A ε B and B ε C with simple sets and A not an element of C will serve nicely. I basically gave you one. Follow it up. It shows inclusion isn't transitive. That's all you need.

 

[sunmaz94]

I think you are reaching way too far for a contradiction. C ε C puts you squarely in Bertrand-Russell paradox territory. A simple example of A ε B and B ε C with simple sets and A not an element of C will serve nicely. I basically gave you one. Follow it up.

This is a separate question. I absolutely want to be in such territory. I need to show that the aforementioned set inclusions yield C ε C and then I can invoke the axiom of regularity/foundation to show it is a contradiction.

 

[Dick]

This is a separate question. I absolutely want to be in such territory. I need to show that the aforementioned set inclusions yield C ε C and then I can invoke the axiom of regularity/foundation to show it is a contradiction.

I don't think so. Inclusion ISN'T transitive. You can't say A ε B and B ε C implies A ε C. At all.

 

[sunmaz94]

I don't think so. Inclusion ISN'T transitive. You can't say A ε B and B ε C implies A ε C. At all.

Hmm...

Then how do I go about using the axiom of regularity to prove that no set membership loops like that I described exist?

 

[Dick]

Hmm...

Then how do I go about using the axiom of regularity to prove that no set membership loops like that I described exist?

C ε C violates regularity. But the statement C ε C doesn't follow from anything you've said before because inclusion isn't transitive.

 

[sunmaz94]

C ε C already violates regularity. The statement C ε C doesn't follow from anything you've said before because inclusion isn't transitive.

Yes but I want to show that A ε B and B ε C and C ε A violates regularity.

 

[Dick]

Yes but I want to show that A ε B and B ε C and C ε A violates regularity.

Mmm. I'm not all that hot with set axiomatics. But you can simplify that. Suppose A ε B and B ε A, can you show that violates regularity?? Like I say, I don't have ZFC axioms at my fingertips.

 

[sunmaz94]

Mmm. I'm not all that hot with set axiomatics. But you can simplify that. Suppose A ε B and B ε A, can you show that violates regularity?? Like I say, I don't have ZFC axioms at my fingertips.

Then I can, but that doesn't help me.

See http://en.wikipedia.org/wiki/Axiom_of_regularity

It states:

Every non-empty set A contains an element B which is disjoint from A.

I appreciate all of your help.