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I was on physics stackexchange and came across an unusual answer where it was stated that the axiom,

$$\forall x ((x \in x) \land (x \notin x))$$

Creates an axiom system where "nothing" exists in it. I kind of misquoted here, here's an exact,

"From that single axiom we can conclude that nothing exists, that the universe of discourse of this axiom system is exactly the empty universe of discourse."

Needless to say, I got into a heated debate on the self-consistency of the axiom. My position is that it can't create a universe or universe of discourse because there are no entities within it. See wikipedia's definition

In English, I interpret the axiom to say,

For all X, X is an element of itself and X is not an element of itself.

This seems self contradicting, akin to Russell's paradox, but the Op insists that it isn't.

I'm open to either take on the issue, but a simple yes/no response will not cut it. A definition review might also help quantify the dialogue.

If it helps, from a semi philosophical-mathematical I consider nothing to be the null set.

Community
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Zach466920
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3 Answers3

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Working in classical logic, the rules of logic dictate that $x\in x\land x\notin x$ is a false statement.

Therefore the only structure satisfying $\forall x(x\in x\land x\notin x)$ is the empty structure. However due to superficial reasons, we choose not to accept the empty structure as a valid interpretation of a first-order language, so it is not a model of this axiom.

In particular it means that any theory including this axiom is inconsistent. If you want to argue in non-classical logic, perhaps paraconsistent logics, that is a whole other things which should be explicitly mentioned.

(I took a quick look at the Phy.SE answer, and I'd probably stay away from that answer and as a rule of thumb largely ignore any claims about mathematical logic made on Phy.SE without references.)

Asaf Karagila
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  • A structure for a non-empty language cannot be empty! I will elevate the comment I was just typing up into an answer. (And as an aside, $\phi \land \lnot\phi$ false in intutionistic logic too.) – Rob Arthan Aug 18 '15 at 23:02
  • @Rob: Arguably a structure to a language without constants can be an empty set. But it is easier to accept that the empty set is not permitted to be a structure at all. – Asaf Karagila Aug 18 '15 at 23:03
  • To make things clearer, I should have "structure for a non-empty signature". The language here is over a signature with a predicate symbol $\in$ and a structure for this signature cannot be empty. – Rob Arthan Aug 18 '15 at 23:08
  • I have a question. Is it simple to determine whether or not ZFC, or any logical structure, falls into the category of classical logic? – Zach466920 Aug 18 '15 at 23:14
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    @Zach: Classical logic is a collection of basic axioms and inference rules. In mathematics, unless stated otherwise, the underlying logic is classical logic. In particular ZFC is using classical logic. There are systems of set theory which use variants of ZF with non-classical logic (specifically, intuitionsitic and constructive logics) called IZF and CZF. But ZFC and CZF are very far from one another. :-) – Asaf Karagila Aug 18 '15 at 23:16
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    @AsafKaragila: I fail to see why you say "superficial" above. A structure has to interpret all the symbols in the signature. You could equally well say that for superficial reasons we exclude the empty set from being a complex number. – Rob Arthan Aug 18 '15 at 23:17
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    @Rob: I was on the other side of this before, but still this answer is interesting: http://math.stackexchange.com/a/45208 – Asaf Karagila Aug 18 '15 at 23:20
  • There is a misunderstanding here. When you say "empty model/structure" you mean "model/structure with an empty universe". I was thinking that you meant that the empty set could somehow be viewed as a model or structure of $\phi \land \lnot\phi$. I retract my comments. – Rob Arthan Aug 18 '15 at 23:27
  • So, can saying "paraconsistent logic" to "classical logic" as "nonstandard" is to "standard analysis" a good analogy? – Zach466920 Aug 18 '15 at 23:29
  • @Zach: I don't know, but I don't think so. – Asaf Karagila Aug 18 '15 at 23:30
  • Ok, would you say non-classical logic is an uncommon topic or occurrence in set-theory? – Zach466920 Aug 18 '15 at 23:31
  • @Zach: Depends on who you ask. I cannot think of many people I'd call set theorists that work in non-classical settings. Most of those, I'd probably brand as "logicians" instead. Maybe other people feel differently. – Asaf Karagila Aug 18 '15 at 23:34
  • Thank you for engaging with my questions. I definitely learned some interesting things :) – Zach466920 Aug 18 '15 at 23:35
  • If we allowed empty models, would $\exists x\land(\forall x(x\in x)\land(x\not\in x))$ be contradictory? – Akiva Weinberger Aug 19 '15 at 00:14
  • @AsafKaragila "However due to superficial reasons, we choose not to accept the empty structure as a valid interpretation of a first-order language"* I don't think the reasons are entirely superficial. We'd typically like universal instantiation to let infer $\phi(\tau)$ from $\forall x \phi(x)$ to $\phi(\tau)$ for any term $\tau$, and to infer $\exists x.\phi(x)$ from any ground formula $\phi(\tau)$. That lets us do $\forall x.x=x$ to $a=a$ to $\exists x.x=x$. With those inference rules, we'd need to reject $\forall x.x=x$ if the domain can be empty. – Joshua Taylor Aug 19 '15 at 01:09
  • @Zach466920 Paraconsistent logics are an interesting field, but not necessarily what you need in order to consider non-empty universes of discourse. The field of logics that permit empty universes of discourse are free logics. There's a bit of discussion in this answer (disclaimer, it's mine) and the comments on it. – Joshua Taylor Aug 19 '15 at 01:11
  • @Joshua ha, ha I kind of feel like that theorem could be stated as the "begging the question theorem". – Zach466920 Aug 19 '15 at 01:22
  • In order for us to be able to "choose not to accept the empty structure" and still have the completeness theorem hold, we have to choose axioms of logic for which $\exists x (x=x)$ is a theorem. Indeed, such logical axioms are included in "classical logic". Then it's not hard to show directly that this contradicts $\forall x (x \in x \land x \notin x)$, so that any set of sentences containing this one is inconsistent. – Nate Eldredge Aug 19 '15 at 03:43
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    @JoshuaTaylor: Yes, but in a certain sense those inference rules are also superficially chosen. The intended semantics of the universal quantifier is that if you have an object of the type quantified, then you can instantiate the quantified statement for that object. There is no problem even if the universe is empty, because with a consistent formal system you would not be able to obtain any object at all and so cannot instantiate any universally quantified statement. The rule you mention, on the other hand, assumes a non-empty universe. – user21820 Aug 19 '15 at 05:18
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This is an example of how mathematical conventions are often divided on the status of degenerate cases.

Some more modern approaches to logic do indeed allow empty universes of discourse. For example, you'd likely see this in any source that works with first-order logic from the point of view of category theory.

However, more traditional approaches reject empty universes of discourse. In the corresponding formulation of first-order logic, your formula would indeed be a contradiction.

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A formula of the form $\phi \land \lnot\phi$ has no models (other than in the model theory of some rather obscure substructural logics). This is unconnected with Russell's paradox and not at all deep. There is no such thing as an "empty model" for a non-empty language, e.g., a language including he membership symbol $\not\in$.

[I should point out that this answer does not conflict with Asaf Karagilia's answer: if you allow models that have empty universes, then $\forall x(\psi(x)\land\lnot\psi(x))$ does not have the form $\phi\land\lnot\phi$ and may hold in a model with an empty universe. However, allowing models with empty universes puts you in the far from classical realm of free logic.]

Rob Arthan
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    I think that's why it was on PSE. No comment on the correspondence to reality needs to be drawn. – Zach466920 Aug 18 '15 at 23:03
  • @Zach466920: OK: I relegate the following observation to this comment and emphasise that it is not a criticism: "In my opinion, to draw conclusions about the physical world from these mathematical facts is outside the scope of MSE." – Rob Arthan Aug 18 '15 at 23:11
  • @Rob: And outside the scope of any reasonable person who knows anything about mathematics... :-) – Asaf Karagila Aug 18 '15 at 23:12
  • I was honestly skeptical of using set theory to form a physical theory from the get go. Nevertheless, this subject seems very interesting so I'm interested in it for curiosity's sake. – Zach466920 Aug 18 '15 at 23:19
  • @RobArthan - I'm not sure allowing models with an empty universe gets you outside classical logic. Hodges, for instance, allows models with an empty universe in his A Shorter Model Theory, yet I don't think he's working outside classical logic. – Nagase Aug 19 '15 at 00:05
  • Hodges doesn't present a logical system (and doesn't seem to care much for logic). $\exists x (x = x)$ is a theorem of classical logic that does not hold if you allow empty universes. – Rob Arthan Aug 19 '15 at 00:09