What is the standard first-order language to formalize ZFC?

Question

I was hoping to find an easy, viusally intuitive first-order formalization of ZFC.

I have tried to look in several books, but it's often something that is understimated.

I think a schematic presentation of the language would be straightforward and helpful.

Variables:
Non-logical Symbols:
Logical Symbols:

Signature, Structure, Vocabulary.....

Just an overview that might remain in your head.

Moreover, sometimes I find the equality symbol included, other times it's not.

What is the conventional choice?

You can see Takeuti, Introduction to Axiomatic Set Theory-Springer (1982), Ch.2 Language and Logic. — Mauro ALLEGRANZA, Feb 25 '19 at 16:55
Regarding equality, both options are available; see e.g. the post Why include equality in FOL for ZFC? — Mauro ALLEGRANZA, Feb 25 '19 at 16:56
Almost a duplicate, https://math.stackexchange.com/questions/916072/what-axioms-does-zf-have-exactly I think. — Asaf Karagila, Feb 25 '19 at 17:03
@AsafKaragila The question you linked is about axioms. My question is just about the synctactic formalization of ZFC in first-order logic. — Gabriele Scarlatti, Feb 25 '19 at 17:40
@MauroALLEGRANZA Thank you for the resources. I was hoping for an answer that would make this information accesible to the community. — Gabriele Scarlatti, Feb 25 '19 at 17:41
So your question is "what is the language of ZF" rather than "What the formulation of ZF"? Perhaps it would help to clarify that in your post. — Asaf Karagila, Feb 25 '19 at 17:43
@MauroALLEGRANZA Just looked to the resource. It just give a basic overview of what a language is. I was asking for the particular case of ZFC — Gabriele Scarlatti, Feb 25 '19 at 18:02
I'm not sure what texts you've looked at, but there are plenty which give the details. E.g. Kunen's Set theory: an introduction to independence proofs describes all this in the bottom of page $2$ to the top of page $3$ (and then goes on to give general comments about syntax/semantics): "The basic symbols of our formal language are $\wedge,\neg\exists,(,),\in,=,$ and $v_j$ for each natural number $j$. [etc.]" (cont'd) — Noah Schweber, Feb 26 '19 at 15:39
And while not a textbook, the wikipedia page for ZF covers it as: "Formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted $\in$." So I disagree with the implicit claim that this information is inaccessible to the community. — Noah Schweber, Feb 26 '19 at 15:42
@NoahSchweber I'm not contesting what you say. Probably to an expert eye it may seem easy to notice it. For this reason I claimed for a visual intuitve representation. On wikipedia is left as a footnote, I would say. In many texbooks it's just left as an exercise or assumed. That's what I felt, at least. — Gabriele Scarlatti, Feb 26 '19 at 15:56
@GabrieleScarlatti On the wikipedia article it's in the main text before the introduction; I'd hardly call that a footnote. I'm not downvoting obviously, but I do think you're seriously overstating the situation. — Noah Schweber, Feb 26 '19 at 15:58

Mauro ALLEGRANZA · Answer 1 · 2019-02-26T08:45:18.203

See e.g. Enderton, page 70.

FOL Logical symbols

0) Parentheses: $( , )$.

1) Sentential connective symbols: $→, ¬$

2) Variables: $v_1, v_2,\ldots$

3) Equality symbol (optional): $=$

4) Quantifiers: $\forall, \exists$.

Non-logical symbols

Predicate symbols: $\in$ (binary)

Function symbols: none (or occasionally the constant symbol $\emptyset$).

If equality is not part of the underlyig logic, we have to add it to the list of non-logical symbols.

As you can see from the literature, both options are used:

Formally, $\mathsf {ZFC}$ is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted $\in$.

See e.g. Takeuti, page 7 for the set theoretic definition of equality:

$a=b \leftrightarrow_{def} (\forall x) [x \in a \leftrightarrow x \in b]$.

score 5 · Answer 2 · answered Feb 25 '19 at 18:09

There is a not a single theory 'ZFC'; there is a family of closely related, equivalent theories, which vary slightly from one author to the next. There's not a single standard.

In every case I can think of, the language of ZFC has a non-logical binary relation '$\in$', and possibly an equality relation '$=$', which can be a logical or nonlogical symbol depending on taste. Apart from that, there are no more non-logical symbols, and everything is the same as the underlying first-order logic.

Fraenkel, Bar-Hillel, and Levy [1, p. 25ff.] talk at some length about the equality relation. As they point out, there are essentially three options:

(a). Equality is treated as a logical symbol, which is always interpreted by the actual equality relation. The axioms of equality are then viewed as logical axioms.

(b). Equality is treated as a second, undefined binary relation, and appropriate axioms are added to the theory.

(c). Equality is taken as a defined relation within the theory. In this case, the axiom of extensionality is tweaked.

In older texts, (c) was somewhat common, because it allowed set theory to have only one undefined symbol ($\in$). In modern texts, (a) is almost universal, because people are not worried anymore about including the equality axiom as part of the logic.

1: A.A. Fraenkel, Y. Bar-Hillel, and A. Levy (1973), Foundations of Set Theory, 2nd ed., Studies in Logic v. 67, North Holland, 978-0-0808-8705-0

What is the standard first-order language to formalize ZFC?

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