This reduces the problem of explaining “standard integers” to the problem of explaining “subsets”, which is not easier. I don’t think there’s any good first-order explanation of what a “subset” is. For example, your definition fails to capture “finiteness” in some weird models of ZFC. More generally, I think “subsets” are a much more subtle concept than “standard integers”. For example, a human can hold the position that all statements in the arithmetical hierarchy have well-defined (though unknown) truth values over the “standard integers”, and at the same time think that the continuum hypothesis is “neither true nor false” because it quantifies over all subsets of the same integers. (Scott Aaronson defends something like this position here.)
Yes, but Subsets(x,y) is a primitive relationship in ZFC. I don’t really know what cousin_it means by an explanation, but assuming it’s something like a first-order definition formula, nothing like that exists in ZFC that doesn’t subsume the concept in the first place.
No, it isn’t. The only primitive relations in ZFC are set membership and possibly equality (depending on how you prefer it). “x is a subset of y” is defined to mean “for all z, z in x implies z in y”.
Can I downvote myself? Somehow my mind switched “subset” and “membership”, and by the virtue of ZFC being a one-sorted theory, lo and behold, I wrote the above absurdity. Anyway, to rewrite the sentence and make it less wrong: subsets(x,y) is defined by the means of a first-order formula through the membership relation, which in a one-sorted theory already pertains the idea of ‘subsetting’. x E y --> {x} ⇐ y. So subsetting can be seen as a transfinite extension of the membership relation, and in ZFC we get no more clarity or computational intuition from the first than from the second.
A finite number is one that cannot be the cardinality of a set that has a subset with an equal cardinality.
This reduces the problem of explaining “standard integers” to the problem of explaining “subsets”, which is not easier. I don’t think there’s any good first-order explanation of what a “subset” is. For example, your definition fails to capture “finiteness” in some weird models of ZFC. More generally, I think “subsets” are a much more subtle concept than “standard integers”. For example, a human can hold the position that all statements in the arithmetical hierarchy have well-defined (though unknown) truth values over the “standard integers”, and at the same time think that the continuum hypothesis is “neither true nor false” because it quantifies over all subsets of the same integers. (Scott Aaronson defends something like this position here.)
Well, ZFC is a first-order theory...
Yes, but Subsets(x,y) is a primitive relationship in ZFC. I don’t really know what cousin_it means by an explanation, but assuming it’s something like a first-order definition formula, nothing like that exists in ZFC that doesn’t subsume the concept in the first place.
No, it isn’t. The only primitive relations in ZFC are set membership and possibly equality (depending on how you prefer it). “x is a subset of y” is defined to mean “for all z, z in x implies z in y”.
Can I downvote myself? Somehow my mind switched “subset” and “membership”, and by the virtue of ZFC being a one-sorted theory, lo and behold, I wrote the above absurdity. Anyway, to rewrite the sentence and make it less wrong: subsets(x,y) is defined by the means of a first-order formula through the membership relation, which in a one-sorted theory already pertains the idea of ‘subsetting’. x E y --> {x} ⇐ y. So subsetting can be seen as a transfinite extension of the membership relation, and in ZFC we get no more clarity or computational intuition from the first than from the second.
Set theory is not easier than arithmetic! Zero is a finite number, and N+1 is a finite number if and only if N is.
Yes, that is a much better definition. I don’t know why this one occurred to me first.
Sewing Machine’s previous comment isn’t really a definition, but it leads to the following:
“n is a finite ordinal if and only if for all properties P such that P(0) and P(k) implies P(k+1), we have P(n).”
In other words, the finite numbers are “the smallest” collection of objects containing 0 and closed under successorship.
(If “properties” means predicates then our definition uses second-order logic. Or it may mean ‘sets’ in which case we’re using set theory.)
Though with the standard definitions, that requires some form of choice.