Why not call the set of all sets of actual objects with cardinality 3, “three”, the set of all sets of physical objects with >cardinality 2, “two”, and the set of all sets of physical objects with cardinality 5, “five”?

Because that’s how naive class theory works, not how consistent formal mathematics works.

The closest thing to a canonical approach these days is to start from what you have, nothing, and call that the first set. Then you make sets from those sets in a very restrictive, axiomatic way. Variants get as exotic as the surreal numbers, but the running theme is to avoid defining sets by intension unless you’re quantifying over a known domain.

For the record, I don’t think any of these things “exist” in any meaningful sense. We can do mathematics with inconsistent systems just as well, if less usefully. The law of non-contradiction is something I don’t see how to get past (ie I can’t comprehend such a thing), and there is nothing much else distinguishing the consistent systems as being anything other than collections of statements to the effect that this & that follow if we grant these or those axioms. (Fortunately, it’s more interesting than that at the higher levels.)

You’ve misunderstood me. It’s really not at all conspicuous to allow a none-empty “set” into your ontology, but if you’d prefer we can talk about heaps; they serve for my purposes here (of course, by “heap”, I mean any random pile of stuff). Every heap has parts: you’re a heap of cells, decks are heaps of cards, masses are heaps of atoms, etc. Now if you apply a level filter to the parts of a heap, you can count them. For instance, I can count the organs in your body, count the organ cells in your body, and end up with two different values, though I counted the same object. The same object can constitute many heaps, as long as there are several ways of dividing the object into parts. So what we can do, is just talk about the laws of heap combination, rather than the laws of numbers. We don’t require any further generality in our mathematics to do all our counting, and yet, the only objects I’ve had to adopt into my ontology are heaps (rather inconspicuous material fellows in IMHO).

I should mention that this is not my real suggestion for a foundation of mathematics, but when it comes to the challenge of interpreting the theory of natural numbers without adopting any ghostly quantities, heaps work just fine.

(edit):
I should mention that while heaps, requiring only for you to accept a whole with parts, and a level test on any gven part, are much more ontologically inconspicuous than pure sets. Where exactly is the null set? Where is any pure set? I’ve never seen any of them. Of course, i see heaps all over the place.

Because that’s how naive class theory works, not how consistent formal mathematics works.

The closest thing to a canonical approach these days is to start from what you have, nothing, and call that the first set. Then you make sets from those sets in a very restrictive, axiomatic way. Variants get as exotic as the surreal numbers, but the running theme is to avoid defining sets by intension unless you’re quantifying over a known domain.

For the record, I don’t think any of these things “exist” in any meaningful sense. We can do mathematics with inconsistent systems just as well, if less usefully. The law of non-contradiction is something I don’t see how to get past (ie I can’t comprehend such a thing), and there is nothing much else distinguishing the consistent systems as being anything other than collections of statements to the effect that this & that follow if we grant these or those axioms. (Fortunately, it’s more interesting than that at the higher levels.)

You’ve misunderstood me. It’s really not at all conspicuous to allow a none-empty “set” into your ontology, but if you’d prefer we can talk about heaps; they serve for my purposes here (of course, by “heap”, I mean any random pile of stuff). Every heap has parts: you’re a heap of cells, decks are heaps of cards, masses are heaps of atoms, etc. Now if you apply a level filter to the parts of a heap, you can count them. For instance, I can count the organs in your body, count the organ cells in your body, and end up with two different values, though I counted the same object. The same object can constitute many heaps, as long as there are several ways of dividing the object into parts. So what we can do, is just talk about the laws of heap combination, rather than the laws of numbers. We don’t require any further generality in our mathematics to do all our counting, and yet, the only objects I’ve had to adopt into my ontology are heaps (rather inconspicuous material fellows in IMHO).

I should mention that this is not my real suggestion for a foundation of mathematics, but when it comes to the challenge of interpreting the theory of natural numbers without adopting any ghostly

quantities, heaps work just fine.(edit): I should mention that while heaps, requiring only for you to accept a whole with parts, and a

level teston any gven part, are much more ontologically inconspicuous than pure sets. Where exactly is the null set? Where is any pure set? I’ve never seen any of them. Of course, i see heaps all over the place.