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I'm studying set theory. I understand that a set can't contain identical elements. However when I read about partially ordered sets and the introduction of "$\leq$" I get confused by some sentences like:

"Given elements $a,b$ in a set $L$ we impose the axiom: If $a\leq b$ and $b\leq a$, then $a=b$."

How can $a=b$ if the set can't contain identical elements?

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  • $\begingroup$ It's misleading to say that a set cannot contain identical elements. It's just that if an element is repeated, then the set is exactly the same. $\endgroup$
    – Wojowu
    Jan 24, 2019 at 19:08
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    $\begingroup$ Different variables can still represent the same element of a set. $\endgroup$
    – Carl Mummert
    Jan 24, 2019 at 19:11

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Here is an analogy. No two people are the same. If Person A has the same mother as Person B and they are not twins and where born on the same day then Person A and Person B are the same Person.

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  • $\begingroup$ "No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,b\in P$ then it might be the case that $a = b$. $\endgroup$
    – Mario Carneiro
    Jan 24, 2019 at 20:00
  • $\begingroup$ It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example. $\endgroup$
    – Jagol95
    Jan 24, 2019 at 20:15
  • $\begingroup$ Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $\forall x y,x\ne y\to x\ne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $\forall x y,x\ne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this. $\endgroup$
    – Mario Carneiro
    Jan 25, 2019 at 1:52

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