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Kind of a dumb question but, is the Null Set Axiom necessary in the ZF axiomatization of set theory?

As I understand it, the NSA states: $\exists x \ \forall y \ (y \notin x)$. But is the explicit stating of the existence of such a set truly necessary? Can one not derive its existence from the Axiom (Schema) of Separation and the fact that a model's universe cannot be empty?

Perhaps such a question is —in solely thinking of the ZF axiomatization as a set of sentences to be modelled by (possibly) a certain model— too constrained by the framework of logic and model theory, and the explicit stating of the existence of some set is indeed necessary outside such contexts; but I suppose I would be interested to know if, indeed, within FOL (at the very least), its inclusion is redundant.

That is, given that $\mathrm{Mod}(\lambda_1)$ is the (proper) class of all models (i.e., nonemptiness is a requirement for any model's universe), we can know that $\mathrm{ZF} \vdash \exists x \ (x = x)$ (this also by FOL's Completeness).

Hence, by the SAS ($\forall u_1 \dots u_k \ [\forall w \ \exists v \ \forall r \ (r \in v \leftrightarrow r \in w \land \psi(r, u_1, \dots, u_k))]$), we have, given $\psi \equiv x \neq x$, that $\forall x \ [\forall w \ \exists v \ \forall r \ (r \in v \leftrightarrow r \in w \land x \neq x)] \vdash \exists v \ \forall r \ (r \notin v)$ (this a simple exercise in NK), and thus $\mathrm{ZF} \vdash \exists x \ \forall y \ (y \notin x)$.

I do not know if such a derivation could be carried in a different manner (perhaps via the existence stated in the Axiom of Infinity? Although such seems dependant on a previous characterization of $\emptyset$...), or even if the above stated is actually valid, so any comments clarifying the matter asked above would be much appreciated!

Sho
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    Why must a model's universe not be empty? (The empty set is a fine model for the axioms of a semigroup...) You can derive the empty set with separation once you know a set exists; e.g., from the Axiom of Infinity. And AI doesn't need a definition of $\varnothing$, as you can replace its mention by saying it contains an $x$ such that for all $y$, $y\notin x$. – Arturo Magidin Sep 11 '23 at 02:09
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    @ArturoMagidin You are correct as to your last point regarding the possibility of deducing NSA off of AI (seems like I overlooked the obvious solution). As to the possibility of an empty universe... I did not know of it. I've been working with van Dalen's "Logic and Structure," and there it is explicitly stated that a model's universe must not be empty... I've certainly thought of the possibility of "empty" models before, but its inclusion seems to do away with a lot of nice properties of FOL, no? For starters, Completeness immediately goes out the window, as $\vdash \exists x \ (x = x)$... – Sho Sep 11 '23 at 02:20
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    It's a matter of convention whether structures need to be non-empty in the context of first-order logic. Some authors require it and some authors don't. – James Hanson Sep 11 '23 at 02:24
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    My experience is that people usually include a "there exists at least one set" or "the empty set exists" axiom (sometimes explicitly labeled as temporary) because they want to develop some of the basics of set theory before introducing inductive sets and the Axiom of Infinity. But as I recall, Jech, for example, does not have any such axiom. (Also, even if you disallow empty models, we don't know that set theory has a model, because we cannot prove its consistency... so how do you justify the existence of a model, empty or non-empty?) – Arturo Magidin Sep 11 '23 at 02:25
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    @JamesHanson I see... that is very interesting! May I ask, how do such authors treat Completeness in FOL (in light of my above comment about the derivability/theoremicity of the existence of an $x$ such that $x = x$ (at least given the Gentzen derivation rules I was taught in van Dalen...))? – Sho Sep 11 '23 at 02:56
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    @Sho Van Dalen's system is sound and complete for the class of nonempty structures, but not sound for the class of all structures including the empty structure. If you want something sound and complete for the latter, you need to slightly weaken the system so that it can't prove, e.g., $\exists x ; x = x$ but is still complete for the class of all (possibly empty) structures. Unfortunately I don't know of any textbook ND system that does this, but see here – spaceisdarkgreen Sep 11 '23 at 04:19
  • @spaceisdarkgreen Thank you! You are always really helpful :) – Sho Sep 11 '23 at 04:42
  • What does not having empty model have in common with having empty set? The empty set in particular model doesn't has to be empty model in meta sense. Also yes, NSA might be derived from diffent axiom. For example from axiom of regularity. – Antares Sep 11 '23 at 14:21
  • @JamesHanson what is an example of such a book? – Vivaan Daga Sep 11 '23 at 18:21
  • @VivaanDaga I believe Hodges' model theory textbook does not require that structures are non-empty. – James Hanson Sep 11 '23 at 19:11
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    @Antares regularity doesn't imply NSA... starting a statement with 'for every nonempty set' is not the same thing as asserting the existence of a nonempty set. Regarding the first part: "having an empty model" means there is an empty model of the theory, not that the theory proves there exists an empty model (which is the sense in which a theory 'has an empty set'). If empty models/structures aren't allowed then it is logically true that a set exists, and hence separation implies an empty set exists. – spaceisdarkgreen Sep 11 '23 at 23:04

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If you are working with a form of first-order logic where $\vdash \exists x (x = x)$, and if you moreover postulate the axiom scheme of separation, then you are correct that the null set axiom is redundant. Alternately, if you postulate the axiom of infinity, you get the empty set more directly.

For a look at the benefits and drawbacks of using first-order logic which assumes $\exists x (x = x)$, see this question. The bottom line is that first-order logic which does not automatically prove $\exists x (x = x)$ is complete for all models (including possibly empty models), while first-order logic which does prove $\exists x (x = x)$ is complete for non-empty models.

Mark Saving
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