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I'm trying to learn some logic to understand different kinds of foundations of mathematics. However, most of the logic texts I've seen will define things like formal languages, valuations and models using the word set repeatedly, before any mention of ZFC or other foundations is made. Then, ZFC (for example) is defined based on the previous discussion. I can't help feeling uncomfortable about this, as if there is some kind of circularity. Is it the case that set within ZFC and set within the general logician's vocabulary mean different things, and if so, could someone make this more clear to me?

Btw, the texts I've been looking at are W. Rautenberg - A Concise Introduction to Mathematical Logic, and W. Hatcher - The Logical Foundations of Mathematics.

l3nb3n
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    Also relevant: https://math.stackexchange.com/questions/1334678/does-mathematics-become-circular-at-the-bottom-what-is-at-the-bottom-of-mathema – Eric Wofsey Jun 07 '20 at 22:37
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    I strongly recommend you to read the origins of set theory of Zermelo and ZFC theory from the establishment of the Naive set theory (e.g., https://plato.stanford.edu/entries/set-theory/index.html) – John M-D94 Jun 07 '20 at 22:45
  • @EricWofsey I guess it answers the first part, although it's admittedly a rather vague question which I guess doesn't have a clear-cut answer. Putting circularity aside, I'm still interested in what logicians mean by set for the purposes of model theory - do they necessarily mean "ZFC-set" or is it a kind of "foundation-agnostic" usage of the word? – l3nb3n Jun 07 '20 at 23:05
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    When logicians use the word "set", they mean the same thing as what any other mathematician means by that word! – Alex Kruckman Jun 07 '20 at 23:52

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