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Fix a language $\mathcal{L}:=\{=\}$, and let $T$ be the $\mathcal{L}$-theory of singleton collections. Namely the theory axiomatized by the $\mathcal{L}$-sentence $$\sigma:=\exists x((x=x)\wedge(\forall y(y=y\implies y=x))).$$

From what I can tell, there are no meaningful consequences of this theory. Aside from the size of the collection being restricted, it doesn't seem to me that I can conclude anything else universal about any potential model of $T$ that is meaningful. Hence, my question:

Are there any universal $\mathcal{L}$-sentences in $T$, aside from the obvious ones like $\forall x(x=x)$ or $\forall x\forall y(\neg y=x\implies \neg x=y)$?

(By universal $\mathcal{L}$-sentences, I'm talking about sentences of the form $\forall x\phi(x)$.)

Context: I'm trying to see if there is anything restricting a two element collection from modelling the universal theory $T_{\forall}$ of $T$, where $T_{\forall}$ is the set of $\mathcal{L}$-sentences in $T$ which are universal and their consequences.

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    $\begingroup$ Are you talking about first-order logic with equality (so the usual axioms of equality are part of the background logic) or not? Your discussion at the start makes it sound like you aren't, but $\forall x(x=x)$ is not in $T$ if you aren't... $\endgroup$
    – Eric Wofsey
    Commented Sep 13, 2019 at 20:52
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    $\begingroup$ Also, $\forall x(\neg y=x\implies \neg x=y)$ is not a sentence (and if you add $\forall y$ to the start, it again would not be in $T$ if you are not working in first-order logic with equality). $\endgroup$
    – Eric Wofsey
    Commented Sep 13, 2019 at 20:55
  • $\begingroup$ @EricWofsey Yes, I am talking about fol with equality. And oops, it should be $\forall x\forall y...$ Will edit that. Thanks for pointing it out. I just threw equality into the language to make it clear that it was fol with equality, but I guess I should just exclude it since it's implicitly there? $\endgroup$
    – quanticbolt
    Commented Sep 13, 2019 at 21:00
  • $\begingroup$ Yes, in first-order logic with equality, $=$ is not considered part of the signature--instead, it is part of the logical language, like $\wedge$. Also, a simpler statement of you $\sigma$ would be just $\exists x\forall y(y=x)$, since $x=x$ and $y=y$ are always true automatically. $\endgroup$
    – Eric Wofsey
    Commented Sep 13, 2019 at 21:05

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There are tons of consequences of $\sigma$; indeed, $T$ is a complete theory, so for every sentence $\varphi$, either $\varphi$ or $\neg\varphi$ is in $T$. For instance, one very meaningful universal statement in $T$ is $\forall x\forall y(x=y)$. Or, for any formula $\varphi(x)$, $\forall x\forall y(\varphi(x)\leftrightarrow \varphi(y))$ is in $T$. The theory $T$ is in some sense "trivial" (since everything is equal!), but it has plenty of meaningful consequences.

Note also that your definition of "universal sentences" is nonstandard and uninteresting. In particular, if $\varphi$ is any sentence, $\forall x\varphi$ will be a universal sentence by your definition and is equivalent to $\varphi$ in any nonempty structure. The usual definition of universal sentence is one of the form $\forall x_1\forall x_2\dots\forall x_n\varphi$ where $\varphi$ is quantifier-free, so that the universal quantifiers at the start are the only quantifiers.

With this standard definition, models of $T_\forall$ must be either singletons or empty (assuming you allow empty models), in order to satisfy $\forall x\forall y(x=y)$. (Of course, the empty set will satisfy any universal sentence vacuously.)

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