March 7th, 2018, 10:21 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  ZFC Axiom of regularity
"In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermeloâ€“Fraenkel set theory that states that every nonempty set A contains an element that is disjoint from A. In firstorder logic, the axiom reads: $\displaystyle {\displaystyle \forall x\,(x\neq \varnothing \rightarrow \exists y\in x\,(y\cap x=\varnothing ))} $ "* Which says every set x contains a set y (Set intersection is only defined for sets). x={1,2}. So natural numbers can't be members of a set? *https://en.wikipedia.org/wiki/Axiom_of_regularity Note: Decided to do this because I discovered I could copy it from google Latex coming through. 
March 7th, 2018, 10:47 AM  #2 
Senior Member Joined: Aug 2012 Posts: 2,082 Thanks: 595  
March 7th, 2018, 11:01 AM  #3  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
There is a difference between a natural number and its representation as a set. So is it saying every set has to contain a set? Last edited by skipjack; March 8th, 2018 at 08:08 AM.  
March 7th, 2018, 11:11 AM  #4  
Senior Member Joined: Aug 2012 Posts: 2,082 Thanks: 595  Quote:
And "it's" is a contraction of "it is." The form you wanted is "its." Please make a note of it. I must be missing your point. Are you saying the empty set violates regularity? But the definition you posted specifically says the set in question must be nonempty.  
March 7th, 2018, 11:30 AM  #5  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
EEDIT "ZFC is intended to formalize a single primitive notion, that of a hereditary wellfounded set, so that all entities in the universe of discourse are such sets. Thus the axioms of ZFC refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets)" https://en.wikipedia.org/wiki/Zermel...kel_set_theory There may be a problem with that which I brought up many moons ago. Thanks for response. Last edited by zylo; March 7th, 2018 at 11:38 AM.  
March 7th, 2018, 11:35 AM  #6  
Senior Member Joined: Aug 2012 Posts: 2,082 Thanks: 595  Quote:
And in set theory, all sets are "pure" sets. They are either empty or they contain other sets. There are no urelements (objects that are not sets) in ZFC. There are set theories with urelements.  
March 7th, 2018, 10:54 PM  #7  
Senior Member Joined: Aug 2012 Posts: 2,082 Thanks: 595  Quote:
I was inspired to read the Wiki page on urelements and I ended up learning something. Zermelo's original 1908 version of set theory had urelements. I hadn't known that. You know when it comes to the creation of set theory, Cantor gets the credit but Zermelo did most of the heavy lifting as far as I can tell. Last edited by Maschke; March 7th, 2018 at 11:00 PM.  
March 10th, 2018, 09:02 AM  #8  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
Thanks, What motivated the OP was that the OP link didn't specify what an element was. It also says that you can prove from AR that a set can't belong to itself. How? Found something in an old thread Quote:
Do you have a proof?  
March 10th, 2018, 09:05 AM  #9 
Senior Member Joined: Aug 2012 Posts: 2,082 Thanks: 595  
March 10th, 2018, 10:22 AM  #10  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote: Last edited by zylo; March 10th, 2018 at 10:43 AM. Reason: remove "fundamental mistake in set theory"  

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