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I believe a tie is different from equality.

The reason the above holds is because a and b must both be the same element.

A "tie" in this case assume they are different elements. Ie the author is a different person from his grandmother. If that's the case, then it's false that author == grandmother.

Ie two things can't be "tied" unless they are actually just one thing.

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Ok. But I still don't agree that the property of antisymmetry implies or requires that restriction.

Ok, how are you defining a tie?

I believe the author is defining a tie as the following:

(a <= b) && (b <= a) && (a != b)

Then a and b are "tied". Where "!=" means a and b are different.

I'm fine with that restriction. But that isn't the axiom of antisymmetry. That's the axiom of antisymmetry plus a rule that holds that equality implies identity (which would be typically described in terms of the axiom of extensionality). My problem is with the implied claim that antisymmetry alone gives you that restriction, which is incorrect. Antisymmetry is entirely consistent with collections that contain equal but distinct elements.

If we agree that the axiom of extensionality applies, then the restriction of no ties is implied. I don't think antisymmetry is necessary or sufficient for it.

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