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

[quantum123]

Why does the axiom of foundation not allow this?

$$a_1 \in a_2 \in a_3 \in a_4 \in a_1$$ ?

 

[disregardthat]

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.

 

[quantum123]

wow, I like the first argument using $$A = \{a_1,a_2,a_3,a_4\}$$ !
thank you so much.
ingenius and beautiful :)