Comment by CJefferson
6 days ago
To me, the least real thing in maths is, ironically, the real numbers.
As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite.
At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.
how large is the set of all possible subsets of the natural numbers?
edit: Just to clarify -- this is a pretty obvious question to ask about natural numbers, it's no more obviously artificially constructed than any other infinite set. It seems to be that it would be hard to justify accepting the set of natural numbers and not accepting the power set of the natural numbers.
Only countably many of those subsets can be distinguished from other ones. So "anything I could ever imagine caring about" is surely still a countable set.
I think the argument you are trying to make rests on a pretty serious fallacy generalizing "I care about some subsets of natural numbers" (and maybe " I care about subsets of natural numbers, in general") to "I care about all subsets of natural numbers, including undefinable ones".
One could argue that infinite subsets of the natural numbers are not really interesting unless one can succinctly describe which elements are contained in them. And of course there is only a countable number of such sets.
I don't agree, but I agree it's an interesting discussion to have.
When is the set of all possible subsets of natural numbers worth considering more than the set of all sets which don't contain themselves (which gets us Russell's paradox of course), once we start building infinite sets non-constructively?
The naturals to me are a clearly separate category, as I can easily write down an algorithm which will make any natural number given enough time. But then, I'm a constructionist at heart, so I would like that.
You can construct a real number by using an infinite series so it's no less constructive than a rational function on the naturals.
Non-constructive arguments are things like proof by contradiction i.e., the absence of the negative implies the existence of the positive.
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Some people (not me) would consider only countably many of those subsets to be “possible”.
You might appreciate this video where Matt Parker lays out the various classes of numbers and concludes by describing the normal numbers as being the overwhelmingly vast proportion of numbers and laments "we mathematicians think we know what's what, but so far we have found none of the numbers."
https://www.youtube.com/watch?v=5TkIe60y2GI
You can encompass them all by talking about numbers that can be described. Since you can trivially enumerate all possible descriptions, this is countably infinite. By definition, it is impossible to describe a number outside that set.
The entirely opposite perspective is quite interesting:
The "natural numbers" are the biggest mis-nomer in mathematics. They are the most un-Natural ones. The numbers that occur in Nature are almost always complex, and are neither integers nor rationals (nor even algebraics).
When you approach reality through the lens of mathematics that concentrates the most upon these countable sets, you very often end up with infinite series in order to express physical reality, from Feynman sums to Taylor expansions.
The natural numbers are 'natural' because they are definite quantities that can be used for counting.
Taylor expansions about a point of a function requires that the function has a derivative defined at that point.
The derivative itself is the point at which an infinite sequence (say, of incrementally closer approximations) converges.
So derivatives and Taylor series are really more of an arbitrary precision approximation of a value rather than a concrete exact quantity.
Arbitrary precision approximation just happens to be a very elegant way to model the physical world around us.
For truly exact solutions, you still have to work with the naturals (and rationals, etc.)
But you can't really have chemistry without working with natural numbers of atoms, measured in moles. Recently they decided to explicitly fix a mole (Avogadro's constant) to be exactly 6.02214076×10^23 which is a natural number.
Semiconductor manufacturing on nanometer scales deals with individual atoms and electrons too. Yes, modeling their behavior needs complex numbers, but their amounts are natural numbers.
I agree. Had humanity made turning the more fundamental operation than counting that would have sped up our mathematical journey. The Naturals would have fallen off from it as an exercise of counting turns.
The calculus of scaled rotation is so beautiful. The sacrificial lamb is the unique ordering relation.