4 ms·

There’s a very deep problem with that—every time you invent a “superset”, do you then have to redefine the subset to be a subset of that “superset”? There is a

There’s a very deep problem with that—every time you invent a “superset”, do you then have to redefine the subset to be a subset of that “superset”? 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.

ysofunny 2y ago
> There’s a very deep problem with that—every time you invent a “superset”, do you then have to redefine the subset to be a subset of that “superset”? 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"
- klodolph 2y ago
  There is a different answer here which is more satisfactory… which is to use notions of equality other than set theoretic equality. Which is what the article is talking about.
- threatofrain 2y ago
  I have the naive feeling that early counting may have included 0 as a primitive right from the get-go. The need to say "nothing" seems pretty primal.
  - Izkata 2y ago
    Zero entered western writing systems through India, with limited usage in math before that. It seems like it was invented/borrowed as part of switching from additive numbers (such as Roman numerals) to positional numbers.
  - tsimionescu 2y ago
    "Nothing" as a concept always existed of course. But it wasn't considered a number, generally. Certainly no one counted "nothing, one, two", and even today natural language doesn't include "nothing" or some equivalent as a numeral noun.
    - NeoTar 2y ago
      You need to be careful about the phrase “considered a number” since I believe one, or unity, was also not considered a number by some ancient civilisation - i.e. a number was only multiple copies of unity. [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 https://youtu.be/R33RoMO6xeA]
    - threatofrain 2y ago
      That's where I'm quite skeptical. Imagine you are in charge of trade or rationing important village resources in the winter. It just seems to me almost necessary that people would have a way to symbolically indicate that all the sheep are gone. As opposed to just not having any symbol for that at all.
Jaxan 2y ago
> There is an infinite chain of supersets of the rational numbers, real numbers, etc. No you only do it for all the sets (of numbers) that you are currently working with.
- klodolph 2y ago
  You can work with infinitely many such sets at a time.
sebtron 2y ago
> 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. 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.
- klodolph 2y ago
  > 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.
tuatoru 2y ago
You're not redefining anything; you're adding a relationship.

ysofunny 2y ago

> There’s a very deep problem with that—every time you invent a “superset”, do you then have to redefine the subset to be a subset of that “superset”? 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"

klodolph 2y ago
There is a different answer here which is more satisfactory… which is to use notions of equality other than set theoretic equality. Which is what the article is talking about.
threatofrain 2y ago
I have the naive feeling that early counting may have included 0 as a primitive right from the get-go. The need to say "nothing" seems pretty primal.
- Izkata 2y ago
  Zero entered western writing systems through India, with limited usage in math before that. It seems like it was invented/borrowed as part of switching from additive numbers (such as Roman numerals) to positional numbers.
- tsimionescu 2y ago
  "Nothing" as a concept always existed of course. But it wasn't considered a number, generally. Certainly no one counted "nothing, one, two", and even today natural language doesn't include "nothing" or some equivalent as a numeral noun.
  - NeoTar 2y ago
    You need to be careful about the phrase “considered a number” since I believe one, or unity, was also not considered a number by some ancient civilisation - i.e. a number was only multiple copies of unity. [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 https://youtu.be/R33RoMO6xeA]
  - threatofrain 2y ago
    That's where I'm quite skeptical. Imagine you are in charge of trade or rationing important village resources in the winter. It just seems to me almost necessary that people would have a way to symbolically indicate that all the sheep are gone. As opposed to just not having any symbol for that at all.

Jaxan 2y ago

> There is an infinite chain of supersets of the rational numbers, real numbers, etc. No you only do it for all the sets (of numbers) that you are currently working with.

klodolph 2y ago
You can work with infinitely many such sets at a time.

sebtron 2y ago

> 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. 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.

klodolph 2y ago
> 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.

tuatoru 2y ago

You're not redefining anything; you're adding a relationship.