Algebra Math Help

Do you need topology or algebra to come to a conclusion about the cardinality of a set?

For example, suppose you want to show R (the reals) is uncountable. One famous proof is based on Cantor’s diagonal reasoning. But to use it you must know every real number has a decimal representation, actually you have to work not on R but on the field (R, +, *). You must know there are relationships between real numbers. Another proof is based on a sequence of nested closed intervals. It doesn’t require algebra, but you must know what the closed sets of R are, so you need a topology defined on R. This proof won’t work if, instead of the topology induced by the Euclidean metric, you pick the discrete metric.
So, my question is: Does the cardinality of a set depend on topology, or on algebraic structures, or on the relationships between its elements or on something other than the pure nature of such elements? I think so, because if the only information you have about a set A is that A is a set, then I don’t see how anyone can affirm anything about its cardinality.
Thank you.

There is something that needs to be pointed out here: all topological and algebraic notions can be enterely formalized within a formal Set Theory.

This implies that, for determining the cardinality of a given set (if it can be determined: it’s possible to show that the most common Set Theory, ZFC, is too weak to decide several “natural” questions about sets, but this is not relevant for the present matter, because no amount of Topology or Algebra will settle them either), Algebra and/or Topology (or any other common mathematical field) is not, strictly speaking, necessary, for those are just abbreviations for set-theoretical notions.

Of course, no one will attempt to formalize any proof of, say, the uncountability of the reals entirely within ZFC (after all, medical costs are high); instead, we use abbreviations that we know will do the job. But let it be noted that all different proofs of this fact (Cantor diagonalization, Cantor’s nested intervals, the impossibility of a surjection that maps N onto R, etc.) may be done entirely from the axioms of ZFC and any formal system of first-order logic (by the way, Cantor proved this before having a rigorous definition of the reals).

So, the impression that we need other subjects to decide set-theoretical questions is mainly a result of:

(1) Forgetting that these subjects are themselves set-theoretical notions, and not something that stands outside of it.

(2) Forgetting that a set is not, strictly speaking, given to us fully formed, but it’s a result of a cumulative process, in which sets are formed from simpler sets, starting with the empty set. When you speak of R with the discrete topology, you are no longer speaking about the set of reals, but of another set: a topological space that, as a set, is P(R) (R with the usual topology is also not R, but a subset of P(R)).

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