# Talk:Axiom of extensionality

## Choice of symbols

it's been a while since I did ZFC. Should it be ⇔ for iff, or the single line arrow currently in the article? -- Tarquin 23:13 Nov 28, 2002 (UTC)

It depends on your notational conventions, and is more an issue of symbolic logic than ZFC as such. IME, when people do really symbolic logic, not trying to mix with natural language at all and restricting to a minimal set of defined symbols -- which is what I'm doing in the symbolic portions of the document, restricting to the symbols of predicate logic, including equality, and the one set-theoretic symbol ∈ -- then they tend to use the arrows with single lines. OTOH, when being less formal and especially in conjunction with natural language, people tend to use the arrows with double lines; this is especially especially true when the single-lined arrows might serve some other purpose (such as indicating functions), which is not the case here.

Some will also make a distinction in meaning between the two types of arrows; for them, ⇔ is a symbol in the metalanguage indicating that two expressions in the object language are logically equivalent (either semantically or syntactically), while ↔ remains a symbol in the object language indicating (in classical logic) the material biconditional of the two expressions. Thus:

pq if and only if pq is a tautology (semantically) or a theorem (syntactically).

Note that in the above sentence, "↔", "⇔", and "if and only if" all mean slightly different things! If I wished to express it entirely in words, I would say:

Two expressions are logically equivalent (semantically or syntactically) if and only if their material biconditional is (respectively) a tautology or a theorem.

Or from another perspective, "↔" links two propositions in the object language to get another proposition in the object language, "⇔" links two propositions in the object language to get a proposition in the metalanguage, and "if and only if" links two propositions in the metalanguage to get a proposition in the metalanguage. Of course, when I translate the axioms in this article into words, I'm not so precise about these distinctions; if I were, the resulting English would be even more incomprehensible that the symbolic statement. So the English translations must be read as from a Platonic point of view, pretending that the purely formal symbolic statements above are representing certain actual facts about some real things called "sets".

For us, there's also the practical matter that there are some web browsers (at least some versions of M\$IE) that can read "↔" but not "⇔".

Toby 04:11 Nov 29, 2002 (UTC)

My web browser cannot read ∀ but it can read ${\displaystyle \forall }$. I would assume that the ${\displaystyle \forall }$ is a better choice for standardisation as it is a part proper of Tex. As such I'll change it over on this page and wait a while for objections. I will then (further no objections) start changing over the symbols on other pages. Barnaby dawson 09:56, 31 Aug 2004 (UTC)

## With ur-elements

The section on set theory with ur-elements says that one could define ur-elements to contain themselves as unique elements. Apart from the mentioned adaptation of regularity this requires, I see another problem: on can no longer distinguish an ur-element x from {x} or {{x}} or {x,{x}} and so forth (more exactly, one can show these to be all equal). Shouldn't that be mentioned? Marc van Leeuwen (talk) 13:07, 7 April 2011 (UTC)