"{ø} = {N sub a}
This is a false statement.
{ø} = N sub a
This is a meaningless statement." [Necessary_Truths]
What does ‘N sub a’ stand for?
"[…] it’s hard to see what it could mean to say that the empty set is equinumerous with a natural number. "
The empty set is a natural number, and every set is equinumerous with itself, therefore the empty set is equinumerous with a natural number.
“I reject the concept of cardinality on the basis that you can’t have different measurements (levels) of immeasurability (infinity).”
First, the empty set is not infinite. It’s finite. Do you object to speaking of the cardinality of finite sets?
Second, I grant that defintion of infinite cardinality depends on the axiom of choice. But even if you reject the axiom of choice, infinite cardinality would be meaningful in the conditional sense of the axiom being an antecedent in conditional statements.
Third, even without the axiom of choice, and without a definition of cardinality, it is still a theorem that for any set x, there exists a set y, such that a proper subset fo y dominates x (informally speaking, for any set, there’s always a larger set).
“The empty set is something. It is the set having no members.” [LauricAcid]
“This is a contradiction.” [Necessary_Truths]
No, it is not.
First, you have to understand that the word ‘the empty set’ is a defined term and that it is defined by the only primitive term of set theory (other than identity), which is the 2-place predicate symbol we informally think of as the predicate symbol for the element relation.
The empty set axiom asserts that there exists at least one thing that has no members. (Recall that not even ‘set’ is an official term of Z set theory). What this thing is is not specified. The axiom of extensionality ensures that there is exactly one such set that has no members. Then we define the empty set to be that set.
This is not contradictory since we can easily show models of the those two axioms.
“A set is only a collection of elements, it is not the collection of elements plus the box you’ve put them in.”
I haven’t mentioned any box, nor does set theory.
I recognize that in the everyday sense of ‘set’, a set is a colllection of elements, so one would think that there is no set that has no elements. But set theory does diverge from this everyday sense. If this bothers you, then you should recognize that the intent of set theory is not to overturn anyone’s everyday use of the word ‘set’ nor to convince anyone of a particular ontology. In set theory (as opposed to class theory), the word ‘set’ is not part of the official system and is used just to easily convey things about the theory that can be conveyed without using the word ‘set’ but only at the price of awkardness or lack of informal visualization.
In other words, instead of ‘an empty set’, we could say ‘a memberless object’. Then we agree to countenacne, for the purpose of a theory, a domain in which there is only one memberless object, so in those domains there is ‘the memberless object’. That’s all there is to it.
“The empty set is nothing, literally.”
What do you think the number zero is? Is it a number, which is something, or is it not a thing at all?
/
“Indeed, it seems like “cardinality” is arbitrary.” [Felipe]
I suspect that you don’t take the notion of cardinality of finite sets to be arbitrary. So it’s the notion of cardinality of infinite sets that you think is arbitrary.
Is the notion of equinumerosity arbitrary? Is the notion of well order arbitrary? Is the notion of a choice function arbitrary? If not, then why would infinite cardinality, which is defined on the basis of those concepts, be any more arbitrary?