# Does the theory of singleton collections have any universal consequences?

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.

• 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... Commented Sep 13, 2019 at 20:52
• 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). Commented Sep 13, 2019 at 20:55
• @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? Commented Sep 13, 2019 at 21:00
• 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. Commented Sep 13, 2019 at 21:05

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.)