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First of all, I'll give some context, just to clarify my mind, and hopefully some others as well. If you are familiar with this stuff you can skip to the quoted main question.

In order to start doing math, we need a common formal language, lets say we have the set theory language, which (for example) consists on the following special symbols

  • $(,),\neg,\to,\vee,\wedge,\leftrightarrow,\forall,\exists,=,\in$

and an unlimited list of variables. Once we have defined the alphabet of our formal language, we need to define the syntax rules to build up valid formulas. For example

  1. If $x$ and $y$ are variables, then $(x=y)$ and $(x\in y)$ are formulas.
  2. If $\alpha$ and $\beta$ are formulas, then $\neg\alpha$, $(\alpha\to\beta)$, $(\alpha\vee\beta)$, $(\alpha\wedge\beta)$ and $(\alpha\leftrightarrow\beta)$ are formulas.
  3. If $\alpha$ is a formula and $x$ is a variable, then $\forall x\alpha$ and $\exists x\alpha$ are formulas.

So, once we all speak in the same language, is not (still) our business to ask questions about our language itself.

Finally, if we want to prove stuff, we need two things

  1. A list of sentences we assume to be true. This will be, for example, the ZFC axioms.
  2. A list of inference rules, being each inference rule a list of descriptions of sentences (acording to the previously defined rules) which we call premises and a description of a sentence, which we call conclusion.

So, the general rule is that, if we have a list of sentences which fits all of some inference rule premises descriptions, and we know they are all true, then, the sentence built up from the conclusion description and our original sentences is also true.

Finally, the main question is,

What are the "standard" inference rules for set theory?

Álvaro G. Tenorio
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  • See First order logic : usual Hilbert-style axioms with Modus Ponens. Alternatively, Natural Deduction. – Mauro ALLEGRANZA Oct 07 '19 at 12:05
  • See e.g. The axioms of set theory : "$\mathsf {ZFC}$ is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $\in$ for membership." – Mauro ALLEGRANZA Oct 07 '19 at 12:11
  • See similar post. – Mauro ALLEGRANZA Oct 07 '19 at 12:12
  • Perhaps one way to summarize the links in the previous comments is that the standard inference rules are formulated for first order logic, without regard to set theory (or group theory, or any other kind of theory). Then, whatever theory one wishes to investigate, one adds new symbols and axioms that are special to the theory, but the rules of inference do not change. – Lee Mosher Oct 07 '19 at 12:31
  • Are you quoting from some textbook ? If so, perhaps going on you will find the def of first-order tehory with example : arithmetic, set theory. – Mauro ALLEGRANZA Oct 07 '19 at 13:15
  • @MauroALLEGRANZA No, I'm not quoting from any textbook, this are just my (probably wrong) thoughts in the topic. – Álvaro G. Tenorio Oct 07 '19 at 14:33
  • @MauroALLEGRANZA I'm not even a newcommer into logic, and I only wanted a complete list of inference rules for set theory, but, if I have understood your previous comments, the set theory language defined above is just a particular case of first order language, and these languages appear to have several equivalent inference rules defined on them. – Álvaro G. Tenorio Oct 07 '19 at 14:42
  • @MauroALLEGRANZA So, is this list complete? – Álvaro G. Tenorio Oct 07 '19 at 14:45
  • Maybe… See here for a sound and complete one. – Mauro ALLEGRANZA Oct 07 '19 at 14:52
  • See this excellent book for issues like this. –  Oct 07 '19 at 15:35
  • What does $x|\alpha$ stand for? – celtschk Oct 07 '19 at 16:16
  • @celtschk Informally, it stands for the set $x$ that verifies $\alpha$ for example, $\emptyset$ will be $x|\forall y y\not\in x$. We must only use it if the set $x$ is guaranteed to exist and to be unique. – Álvaro G. Tenorio Oct 07 '19 at 17:47
  • @ÁlvaroG.Tenorio: Normally, if you just want to know the axioms of ZFC you can just go to Wikipedia and find them all. But given that you included the definite expression syntax "$x|a$", I presume that you actually want to have a practical formal system whose underlying theory is ZFC, rather than a pure FOL system. In that case, take a look at this Fitch-style system based on many-sorted FOL, which includes an axiomatization that is as strong as ZFC but is much more intuitive. – user21820 Oct 11 '19 at 06:19
  • I think you would know how to add the definite expression syntax if you wish to. But note that it may be better to add full abbreviation power (also called definitorial expansion), as it is more powerful. In particular, if "$x|a$" is well-defined in terms of some parameters then you have a definable function and can use rule (2) in my post. Secondly observe that even in absence of any choice principle rule (3) is conservative and stronger than your definite expression syntax (since it does not require uniqueness). – user21820 Oct 11 '19 at 06:26
  • In fact, full abbreviation power rule (3) is already totally subsumed by the Fitch-style rule ∃elim (see my first linked post), and it is entirely natural for Fitch-style. In addition, it is much more natural to have a generalized definite expression syntax, allowing statements like "∀k∈N ( k>1 ⇒ ( p≠1 ∧ ∀x,y∈N ( x·y=p ⇒ x=1 ∨ y=1 ) where p>1 ∧ ∃d∈N ( p·d=k ∧ ∀q,e∈N ( q·e=k ⇒ p≤q ) ) ) )". This can be made even more elegant using rules (1) and (2), by first defining Prime(p) :≡ p≠1 ∧ ∀x,y∈N ( x·y=p ⇒ x=1 ∨ y=1 ), and LeastFactor := ( N k ↦ p | p>1 ∧ ∃d∈N ( p·d=k ∧ ∀q,e∈N ( q·e=k ⇒ p≤q ) ) ), – user21820 Oct 11 '19 at 06:44
  • yielding "∀k∈N ( k>1 ⇒ Prime(LeastFactor(k)) )". So much cleaner! If you are interested in discussing further on foundations of mathematics, you're welcome to the logic chat-room. – user21820 Oct 11 '19 at 06:46

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