How many symbols are in an alphabet if there are two symbols that mean the same

Question

If you were to define a formal alphabet as the set {a, b}, but also say "a = b," then does the alphabet contain 1 symbol or 2 symbols? On the one hand, it seems like it should contain 2 symbols since they are two distinct symbols syntactically. Semantically, the 2 symbols are the same thing, so the set contains one thing, although the "thing" in mind may not be symbols per se.

Note: I'm assuming this is a different issue from token vs symbol distinction, since saying multiple tokens of a symbol doesn't mean there are more than 1 symbol is different from asking if 2 symbols being equal in some sense makes it 1 or 2 symbols.

Answer 1

Here's an answer I got from user keitamaki on Reddit that seems perfect to me.

If your formal alphabet is {a,b}, then "a=b" isn't a string in your formal language (since '=' isn't in your formal alphabet).

So if you're saying "a=b", then you're making that statement in the meta-language. And if you literally mean by that that a and b are the same symbol, then yes, your formal alphabet is just {a}.

But perhaps the better answer is: "don't do that" :)

If you define your formal alphabet as {a,b}, then it is usually understood that a and b are different symbols. If you then go on to say (in the meta-language) that a and b are the same symbol, this isn't an issue of semantics at all (again, because "a=b" doesn't have a semantic meaning inside your formal language, you won't be writing truth tables for "a=b" and you won't be talking about models of that statement because it isn't a statement in your language). It's literally you saying that: "oh, by the way, when I wrote "a" and "b" earlier in my definition, I didn't mean to."