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For example, I want to know the meaning(semantic) of "=" in FOL for the set theory. In structure $\mathscr{A}$, "$c_{1}=c_{2}$" is interpreted as $c_{1}^\mathscr{A}=c_{2}^\mathscr{A}$, i.e., the "=" in FOL for the set theory is interpreted as the = in meta language, which says "=" means =, and it does not make any sense. I still do not know the meaning of =.

Let's talk about the FOL with equality as it seems the one most logic books talk about. For FOL without equality, it seems easier to understand the meaning of "=" which is interpreted as a predicate.

William
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  • There are two flavors of FOL: one with equality (treated as a logical symbol), and one without. Which one are you talking about? – Alex Jun 25 '23 at 15:36
  • @Alex, let's talk about the FOL with equality as it seems the one most logic books talk about. – William Jun 25 '23 at 15:40
  • OK, don't forget to add this to the question . – Alex Jun 25 '23 at 15:41
  • Yes, in FOL with equality we understand the "=" symbols as a binary predicate that we interpret as identity between objects of the domain of interpretation. – Mauro ALLEGRANZA Jun 25 '23 at 15:53
  • @MauroALLEGRANZA but what does the "identity between objects of the domain of interpretation" mean? – William Jun 25 '23 at 15:57
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    If you don't have a metatheory to fall back on (e.g., you are talking about your foundational set theory), then = is just an undefined primitive notion. In general, it doesn't make sense to talk about "semantics" without a metatheory in which you are working. – Eric Wofsey Jun 25 '23 at 16:59
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    It is what we usually use in our daily life: we are able to speak (syntax and semantics), to count (arithmetic), to identify our car in the car park (identity). – Mauro ALLEGRANZA Jun 25 '23 at 17:27
  • @EricWofsey Notice that I'm researching set theory, while the metatheory I fall back on happens to be the set theory itself. There seems a loop in it. If the metatheory were something more fundamental, it would be more reasonable. Since almost everything in math falls back on set theory finally, if there is no more foundation that set theory can fall back on, does that mean I can only rely on my institution about the facts such that the two things are equal in a set because they look the same? – William Jun 26 '23 at 02:16
  • Well, if you want to interpret what equality means intuitively, then that's what you fall back on. But for actually reasoning about equality, you just follow the rules of logic. The point is that there is no rigorous "semantics" for your foundational theory--it's just syntactic. – Eric Wofsey Jun 26 '23 at 02:24
  • @EricWofsey However, the logic is proved to be complete and sound by the relation between its syntax and semantics, from which math is considered to have a solid foundation. Now, the set theory is just syntactic, how can we rely confidently on it? – William Jun 26 '23 at 02:35
  • I would dispute your first sentence: the completeness and soundness theorems have little to do with the solidity of foundations, precisely because they require a strong metatheory. There is no way to avoid taking some amount of foundation "on faith". – Eric Wofsey Jun 26 '23 at 03:27
  • You may find this answer helpful for getting a clearer picture of the whole story. – Eric Wofsey Jun 26 '23 at 03:30
  • @EricWofsey If the set theory, as a foundation, has no semantics, why do we need to consider the semantic of FOL that is based on it? I can say: just follow the rules I give to you to deduce, don't ask me why, that is what human being uses for thousands of years. don't believe me? ok, I will prove its soundness and completeness using its semantic and set theory but don't ask further, otherwise, I have nothing to say. – William Jun 27 '23 at 13:08
  • You don't need to consider semantics for foundational purposes. Semantics is mainly useful as a method for doing mathematics in its own right--models of first-order theories are interesting mathematical structures to study. – Eric Wofsey Jun 27 '23 at 13:11
  • @EricWofsey ok, maybe I expected too much from math, which may be eventually nothing but some tricks of string transformation. Thank you very much! – William Jun 27 '23 at 13:20

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The structure $\mathscr{A}$ has an underlying set $A$. Given two elements $a,b\in A$, it makes sense to ask whether $a = b$, i.e., whether $a$ and $b$ are the same element of $A$.

If you agree that this question makes sense, then there's no problem: we define $\models$ so that, for elements $a,b\in A$, we have $\mathscr{A}\models (a = b)$ if and only if $a = b$.

If you don't think this question makes sense, then you have a pretty nonstandard view of the nature of sets and their elements, which is going to make it hard to communicate about mathematics.

Alex Kruckman
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