# If a set can't contain identical elements, how can one element be less than or EQUAL to another element

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?

• 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. Jan 24, 2019 at 19:08
• Different variables can still represent the same element of a set. Jan 24, 2019 at 19:11

• "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 $a = b$. Jan 24, 2019 at 20:00
• 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. Jan 25, 2019 at 1:52