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I've been learning about ZFC set theory on my own for a moment now, and since it's expressed in the language of first order logic I thought I might as well look into that and language theory too. But when I did I immediately ran into a confusion: the notion of a formal language requires the notion of an alphabet, which is understood to be a set of symbols.

So if the notion of a language requires the previous notion of a set, then how can any theory that tries to formalise the notion of sets come after a theory of languages? To me this looks like a catch-22...

Basically what I'm asking is if there is any way to make a theory of formal languages that doesn't requires the notion of sets, of is somehow the fact that languages imply the previous notion of sets is not a problem for set theory.

Can someone with a deeper and more educated understanding of all this enlighten me? Is there some crucial subtelty I'm missing? Or is there no satisfactory answer to this question yet? My head is starting to run in circles...

Asaf Karagila
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Thomas.M
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    "an alphabet, which is understood to be a set of symbols" but not necessarily in the sense of mathematical set theory. A "collection" of symbols will suffices. – Mauro ALLEGRANZA Mar 26 '20 at 15:50
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    See e.g. this post – Mauro ALLEGRANZA Mar 26 '20 at 15:51
  • Naive set theory works fine for finite sets. You can work with "a bunch of symbols" or "a list of symbols written down on a Post-It" instead, if you like. – mjqxxxx Mar 26 '20 at 15:51
  • I have read the posts linked in the comments and the ones that have been associated to my post. So if I understand correctly: first of all, ZFC does't really define sets as collection of anything, just as objects that behave according tho the axioms and no more,which I was quick to forget, second of all this is all related to model theory and metamathematics and is a very hairy ball of complicated stuff :( but apparently this is not a problem in the way I thought it was. I think I'll just accept it for now, until I can fully understand the scope of my own question. Thanks for the comments. – Thomas.M Mar 26 '20 at 16:27

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