> That universe would be deprived from the bottomless wellspring of dryness that is the set theoretic foundations of mathematics. Unthinkable!
"Wellspring of dryness" is quite a metaphor, and I take it from that metaphor that this outcome wouldn't much bother you. I'll put in a personal defense for set theory, but only an appeal to my personal taste, since I have no expert, and barely even an amateurish, knowledge of set theory beyond the elementary; but I'll also acknowledge that set-theoretic foundations are not to everyone's taste, and that someone who has an alternate foundational system that appeals to them is doing no harm to themselves or to me.
> That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth.
In this alternate universe, the ZF or ZFC axioms (where C becomes, of course, "the product of sets is a set") would certainly involve, not the axiom of the empty set, but rather some sort of "axioms of sets", declaring that there exists a set. Because it's not empty, this set has at least one element, which we may extract and use to make a one-element set. Now observe that all one-element sets are set-theoretically the same, and so may indifferently be denoted by *; and then charge ahead with the construction, using not Ø, Ø ∪ {Ø}, Ø ∪ {Ø} ∪ {Ø ∪ {Ø}}, etc. but *, * ∪ {*}, * ∪ {*} ∪ {* ∪ {*}}, etc. Then all that would be left would be to decide whether our natural numbers started at the cardinality 1 of *, or if we wanted natural numbers to count quantities 1 less than the cardinality of a set.
"Wellspring of dryness" is quite a metaphor, and I take it from that metaphor that this outcome wouldn't much bother you. I'll put in a personal defense for set theory, but only an appeal to my personal taste, since I have no expert, and barely even an amateurish, knowledge of set theory beyond the elementary; but I'll also acknowledge that set-theoretic foundations are not to everyone's taste, and that someone who has an alternate foundational system that appeals to them is doing no harm to themselves or to me.
> That's an interesting thought, but I think that'd break the usual trick of building up objects from the empty set, a set containing the empty set, then the set containing both of those and so forth.
In this alternate universe, the ZF or ZFC axioms (where C becomes, of course, "the product of sets is a set") would certainly involve, not the axiom of the empty set, but rather some sort of "axioms of sets", declaring that there exists a set. Because it's not empty, this set has at least one element, which we may extract and use to make a one-element set. Now observe that all one-element sets are set-theoretically the same, and so may indifferently be denoted by *; and then charge ahead with the construction, using not Ø, Ø ∪ {Ø}, Ø ∪ {Ø} ∪ {Ø ∪ {Ø}}, etc. but *, * ∪ {*}, * ∪ {*} ∪ {* ∪ {*}}, etc. Then all that would be left would be to decide whether our natural numbers started at the cardinality 1 of *, or if we wanted natural numbers to count quantities 1 less than the cardinality of a set.