Axiom of Foundation: The Limitations and Implications for Sets in Mathematics

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Why does the axiom of foundation not allow this?

a_1 \in a_2 \in a_3 \in a_4 \in a_1 ?
 
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The axiom of regularity (foundation) states that every non-empty set A contains an element disjoint from A.

If it's possible, then A = \{a_1,a_2,a_3,a_4\} is a set, so A contains an element disjoint from A. It's not a_1, since a_4 \in a_1, it's not a_2, since a_1 \in a_2, it's not a_3, since a_2 \in a_3, and it's not a_4, since a_3 \in a_4; a contradiction. Hence it's not possible.

This can be generalized to infinite sequences as shown below:
http://en.wikipedia.org/wiki/Axiom_of_regularity#No_infinite_descending_sequence_of_sets_exists
Alternatively for you case then, this expands to an infinite sequence ... \in a_3 \in a_4 \in a_1 \in a_2 \in a_3 \in a_4 \in a_1 which is impossible by the reasons given in the link.
 
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wow, I like the first argument using A = \{a_1,a_2,a_3,a_4\} !
thank you so much.
ingenius and beautiful :)
 

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