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After reflecting on the answers in Why does set theoretic union and intersection operate on reverse logic, a follow-up question arose concerning the definition of predicates.

In the definition of $A \cup B := \{x : x \in A \lor x \in B\}$, and in the definition of $A \cap B :=\{x : x \in A \land x \in B\}$, there is some unspecified, nebulous ambient space that $x$ comes from and then becomes a bound variable in the predicate portion of set builder notation.

What is this?

Anne Bauval
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ArtIntoNihonjin.
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  • I cannot find a good reference (and way to put it), but the underlying universe of discourse is that of predicate logic, which is predicates over (unqualified) individuals. –  Apr 28 '24 at 06:42
  • @JulioDiEgidio Aha! A real answer! Thank you . Now I know why ZF Set Theory is so borked. Individual is a much strictly inferior concept to personhood. – ArtIntoNihonjin. Apr 28 '24 at 07:22
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    Are you being ironic? I don't know what "borked" you are referring to, and especially I don't know what has "personhood" here to do with anything. –  Apr 28 '24 at 07:25
  • According to your reference, one of the design principles used to originate ZF Set Theory is individualism. And from the cloud of pure logic, personhood is always superior (as an abstract idea) to individualism. And borked is just a moralistic way of saying fouled up. – ArtIntoNihonjin. Apr 28 '24 at 07:27
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    Hm, I am not aware of that (and one could even object that "personhood" comes from "persona", i.e. rather a cultural/social construct): maybe it's a matter of terminology but I don't think I have ever met it, would you have any reference? -- P.S: I did not give any references, although if you look up any article on predicate logic you at least find mention of individual constants. –  Apr 28 '24 at 07:30
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    @JulioDiEgidio Because mathematics is a human endeavor, it can be argued that everything except the counting numbers originates from a social or cultural construct. "God made natural numbers; all else is the work of man." - Leopold Kronecker. – ArtIntoNihonjin. Apr 28 '24 at 07:40

3 Answers3

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There is no need for any "ambient space": $$A\cap B=\{x\in A:x\in B\}$$ and for $A\cup B$, see https://en.wikipedia.org/wiki/Axiom_of_union#Relation_to_Pairing: The axiom of union allows one to unpack a set of sets and thus create a flatter set. Together with the axiom of pairing, this implies that for any two sets, there is a set (called their union) that contains exactly the elements of the two sets. More explicitely, given two sets $A$ and $B$, the axiom of pairing produces the set $$C:=\{A,B\}$$ whose elements are exactly $A$ and $B$, and then the axiom of union produces the set $$A\cup B:=\bigcup C$$ whose elements are the elements of elements of $C$.

Anne Bauval
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  • This looks like a good work around. – ArtIntoNihonjin. Apr 28 '24 at 06:16
  • The intuitive argument for why we have the axiom of union (and why there's no symmetrical axiom of intersection) is that we need an operation to make sets bigger and only union can do that. But it's hard to formulate this in a way that doesn't sound ignorant. – ArtIntoNihonjin. Apr 28 '24 at 17:20
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Frame challenge: these operations can be defined without reference to any ambient space of possible elements.

The defining feature of a specific set is a way of telling when a thing is or is not an element of a set. Given a thing $x$, how do we decide whether it's an element of $A\cup B$ or of $A\cap B$? We check whether it's an element of $A$ and whether it's an element of $B$ and then use "or" or "and", respectively. This is the case regardless of what thing $x$ we're given or where it came from; we don't need to know what "all possible things" are to make this decision for any specific given thing, and that's all the power we need to work with these sets.

Greg Martin
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  • In A.D. 1882, Moritz Pasch discovered that there was a tacit assumption (Pasch's axiom) required to make Euclid's axiomatic system flow. I think that there's a similar kind of tacit assumption in set-builder notation that allows one to talk about an $x$ without any reference to a set. – ArtIntoNihonjin. Apr 28 '24 at 06:11
  • This answer seems very close to arguing for the existence of Russell's paradoxical set. "Given a thing $x$, how do we determine if it's an element of $R$? We check whether it doesn't contain itself. ... We don’t need to know where $x$ came from." – Mark S. Jul 06 '24 at 07:36
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The other answers are correct, but I would like to add a couple of points.

  1. There is no trouble imagining that there is an ambient “universe of discourse”, in which all this is taking place. It often isn't mentioned explicitly because the details don't really matter, as long as the universe of discourse is big enough that it has the things we expect it to such as $\emptyset, \Bbb N$, ordered pairs, and so on.

  2. In a few contexts where this universe of discourse is actually important, there are standard ones that set theorists study. See Von Neumann universe for details. The universes called $V_\omega$ and $V_{\omega+\omega}$ are particularly important.

MJD
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