does this also imply that, for example, ℝ and ℕ have the same cardinality because ℕ can be expressed as a subset of ℝ? it's hard for me to accept this

they do not, because while all elements of N can be assigned a number from R, not all elements of R can be assigned a number from N, which makes R bigger than N
this is proven by the somewhat famous Cantor's diagonal argument

also please assume i am using the fancy doublestruck characters

how would this be different from selecting a subset of ℕ, like the even numbers, in which the cardinality of both sets is the same? you have the same mapping issue unless i'm misunderstanding something

the cardinality of even numbers is the same as the natural numbers, because you can also map them back
just as four apples and four oranges are an equal amount (every apple has an orange to go with it), but four apples and five oranges aren't - every apple has an orange, but not every orange has an apple

i have just dug out the video which first explained these concepts to me, i used to watch vihart a fair bit a long time ago. not sure if you'll like the style and the source's "unofficiality", but it definitely has the right concepts

alright i understand now, i was confusing myself because i was assuming the reals were a countable infinity and got stuck trying to figure out what kind of exterior information about the sets was making them unmappable; however, i'm now thinking about a corollary to this which i would be curious to know the answer to:

consider a set which is the square roots of all rational numbers. would this set still be countably infinite? the domain would be constructed on the rational numbers but the range would be an infinite set of real numbers for all instances where the square root of the rational number is irrational

this set would still be countably infinite, because you'd be able to list them just as you can list the rationals, or equivalently as you could map them to rationals (or to naturals) and back
just because the domain you're picking from is larger doesn't mean there has to be "more" of them, just like the mapping of rationals to naturals shows

my purpose in constructing that set was more intending to emulate at least in part the ambiguity within the continuity of the reals which affects its cardinality. the decimal portion of 1/3 is no more or less infinite than the decimal portion of √2 but rather the construction of the continuities of their respective sets determines the sets' cardinalities, therefore it seemed at least somewhat possible to me that in mapping the rationals to reals it could also affect the cardinality of the resulting set

a final question before i stop pretending that i know what i'm talking about (for now :3): is it possible to construct an uncountably infinite set based on a countably infinite set, or is this impossible by definition?

there exists a function known as the power set (P) which takes a set as an input and returns all subsets of that set. for example P({1, 2, 3}) returns {{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}. cantor's theorem states that the cardinality of a set is less than the cardinality of its power set. this can be understood by demonstrating how the power set of the countably infinite natural numbers yields an uncountably infinite set in a manner very similar to cantor's diagonal argument, except instead of the decimal digits of the real numbers it uses the boolean values of whether or not an element is included in a subset. one can always construct a new subset which differs from every other subset in at least one place (that is to say in whether or not it includes a specific element). another consequence of cantor's theorem is that one may take the power set of the real numbers to yield a set of cardinality higher than 𝔠.

I'm only here because szymszl notified me lol.

With regards to obrado's last post, there are a number of constructions in math that involve the idea of "constructing an uncountable set out of a countable set" but the problem here is that your question is a bit vague to work with, so I'm just going to name a few that popped into my mind. But these are a bit technical, so I don't know if you're going to get much out of them.

The first thing I thought of is the construction of (models of) reals from the rationals as cauchy sequences of rationals numbers and "forcing" each rational cauchy sequence to converge by "adding" new points into the rationals (as a certain kind of "completion"), using fancy machinery like imposing equivalence classes on them to get rid of redundancy, and then defining order and arithmetic on those sequences. The second construction I thought of is the dedekind cut construction where you think of each irrational number as the set of all rational numbers that are strictly less than that irrational number. A subtle problem here is that how would you even define "the set of all rational numbers that are strictly less than a particular irrational number" when you haven't even proven they exist, much less give a definition of an irrational number being greater than some rational numbers. As an example, you can think of the dedekind cut representing the square root of 2 as simply the set of all rational numbers whose square is strictly less than 2. Then this gets rid of the problem. Afterwards, you define arithmetic and order. In a sense, this is also a certain kind of "completion" (but I'm not sure if it's exactly the same as the first).

The second thing I thought of is just taking the power set of a given set which is defined to be the set of all subsets of that given set. It's fairly easy to prove that for every finite set S of cardinality n, the power set P(S) of S has cardinality 2ⁿ (using induction). But in fact, this holds true for every cardinal, not just the finite ones.

I'll not be accepting follow-up questions such as "Wait, how do you define a dedekind cut for the transcendental numbers?" as I know set theory very superficially and also really hate it.

I have a question for Kou because you mentioned "a power set function". Is this based off of some other foundations that I'm unaware of because as far as I know (in ZFC, I don't know much set theory.) the existence of a power set of a given set is just an existence claim for a set and not a function. Hope I'm not coming off too condescending with this question, I'm just confused.