I don't think it is a "funny" notion of inclusion, but rather that once ℚ is defined, ℤ is silently redefined to be that one subset of ℚ that was used in the construction.
In the end you still need to maintain a correspondence between the embedding and the original set as in the typical way where you do that and consider the subset notation as a shorthand that requires you to "lift" or "wrap" through the correspondence wherever necessary.
Collapsing a countably infinite set to 0 doesn't seem useful or reversable.
I'm not sure what you're trying to disagree with here.
Not heaps sure what this really means with respect to whether 1 the integer is really completely related (as in, equal, or the-exact same-thing) to 1.0 the real though. Kinda seems like it might still need a bit more information to fully identify a real, even when it happens to be infinitely-close to an integer?
There is an infinite chain of supersets of the rational numbers, real numbers, etc.
Think of it like this… we can define 0, and then define 1 as the successor. Repeating this, we can have a definition for every finite number. But we cannot do this the other way around. We cannot start with ∞, define the predecessor to ∞, and then somehow get back to 0.
In other words, if you want to work backwards and say that smaller sets (like the natural numbers) are a subset of the bigger sets (like complex numbers), then you have to pick a “biggest set” containing all numbers, which is unsatisfactory. Somebody always wants a bigger set.
I have thought about this too, and I'd initially agree with you. but I thought at some point how mathematical history is not extremely dissimilar from this. put in very rough terms:
at first humans discovered/invented numbers (i.e. the counting numbers); these started at number one — the first number. later on, at some point we had to go back and realize that there was a zero before number one which "silently" redefined the first number as zero and this created the natural numbers a the modern set-based N
edit: adding this alternative rendering of my intended comment triggered by a condescending reply: "mathematics silently redefines stuff all the time. deal with it"
[I believe this YouTube video goes into more detail in its discussion of why 1 was not considered Prime in the ancient world: https://youtu.be/R33RoMO6xeA]
No you only do it for all the sets (of numbers) that you are currently working with.
Exactly what we did in the Analysis I course I attended during my bachelor: defined the reals axiomatically, and the N as smallest inductive(?) subset containing 0.
Satisfactory or not, it worked well for the purpose. And I actually liked this definition, if anything because it was original. Mathematical definitions don't need to have some absolute philosophical value, as long as you prove that yours is equivalent to everyone else's it's fine.
That’s exactly the point I was making in the first place.
“Unsatisfactory” just means “unsatisfactory” in the sense that some mathematicians out there won’t be able to use your definitions and still get the subset property. This means that you are, in all realities, forced to deal with the separate notions of “equivalence” and “equality”. Which is what the article is talking about—all I’m really saying here is that you can’t sidestep equivalence by being clever.
Q = Z u Q’, where Q’ is the set of all rational numbers that aren’t integers.
Redefining the smaller set can’t work because there may be more than one larger set, e.g. split complex numbers vs regular complex numbers. But you can define a larger set to strictly extend a smaller set.