Problem understanding the Axiom of Foundation

Question

I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that "every non-empty set $x$ contains a member $y$ such that $x$ and $y$ are disjoint sets." In particular, how can $y$ be an element of a set $x$, and also be disjoint from it (I have seen this be called the epsilon-minimal element). My intuitive understanding of sets (which is obviously wrong) tells me that if $y$ is an element of $x$, then the intersection of $\{y\}$ and $x$ should be $\{y\}$. How can it be $\varnothing$?

Answer 1

The axiom isn't saying that $x\cap \{y\}=\emptyset$. It's saying $x\cap y=\emptyset$. Keep in mind that in ZFC everything is a set, including the elements of other sets.

Answer 2

You need to understand that $y$ and $\{y\}$ are two different things.

Suppose $x = \{ \{ 1,2,3,4\}, \{2,3,4,5\},\{5,6,9,10\}\}.$

Then $x$ has three members. One of those members is $\{1,2,3,4\}$. That set has four members. One of those members is $2$. That member, $2$, is not a member of $x$. Similarly each of the other four members of $\{1,2,3,4\}$ fails to be one of the three members of $x$. So the intersection of $x$ with any of its members is empty.

Now consider the set $$ x=\Big\{ \quad\varnothing,\quad \big\{\varnothing\big\},\quad \big\{\varnothing, \{\varnothing\}\big\},\quad \big\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\big\}\quad \Big\}. $$ This set has four members. One of those members is $\big\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\big\}$. Another is $\{\varnothing\}$. Those two sets do have a member in common. In this case $x$ has only one member that does not intersect $x$.