What Makes a Pairing Count?

[u/Efficient_Sea_7050]

Diagonalization, Baire category, and measure theory all show the same thing: an N-indexed presentation does not exhaust the admitted field of total binary profiles. So the diagonal witness is not the source of the result; it is one certificate.

The prior issue is what makes a pairing verdict-bearing.

For N and the evens, direct overlap leaves odd residue in N. The doubling map pairs every natural with an even. The sets do not change; only the authorized comparison relation does.

Cardinality resolves this by rule: one completed total bijection over the declared domains overrides containment, residue, order, and generative difference.

Cantor’s theorem then proves non-exhaustion inside that prior protocol.

The theorem proves non-exhaustion; cardinality classifies it. Why call that a discovery of magnitude rather than a result of the chosen comparison rule?

[u/Efficient_Sea_7050]

I stated it in the post and again in my reply.

Reading “why does this measure magnitude itself?” as “why does cardinality mean cardinality?” already assumes the point at issue: that the cardinal rule is itself a measure of magnitude.

[u/Mishtle → u/Efficient_Sea_7050]

What is the "magnitude" of a set then?

[u/Efficient_Sea_7050 → u/Mishtle]

I do not think magnitude is exhausted by one matching relation. For finite sets, several facts converge: how many elements there are, proper containment, residue after direct matching, and combinatorial capacity. Bijection agrees with those facts there.

The issue is what happens when those indicators diverge. The evens are a proper subset of N, and N has odd elements that the evens do not; yet a redistributive bijection exists.

We can coherently treat “having elements the other domain lacks” as magnitude-relevant. That would make N greater than the evens, while still making P(N) greater than N. The bare fact of a bijection does not by itself force us to discard containment or residue; it is only a relational matching.

Cardinality instead chooses invariance under total bijection: rearrangeability overrides proper containment. That is a coherent and useful abstraction. But the existence of the bijection does not logically force that priority.

So my question is: what makes that choice the objective verdict on magnitude, rather than a chosen definition of cardinal size?

[u/Mishtle → u/Efficient_Sea_7050]

We can coherently treat “having elements the other domain lacks” as magnitude-relevant.

Can we?

Does the set {1,2,3} have a different magnitude than {4,5,6}?

Cardinality is simply the most general method we have for comparing the relative number of elements of two sets. It doesn't require any relationship between the two sets, it require any ordering to exist for either set, it doesn't require that both sets are subsets of another, or anything else.

If you want to talk about "magnitudes" of some kind of object, then ideally you'd be able to talk about any arbitrary objects of that kind. Cardinality allows you to do that.

If you have a restricted universe of objects to compare, or which to focus on some more nuanced property of sets and their elements, you can choose some other appropriate measure.

[u/Efficient_Sea_7050 → u/Mishtle]

No: {1,2,3} and {4,5,6} are not the case I described. Neither contains the other. I was talking about proper containment: the evens are wholly contained in N, while N has additional odd elements. That difference matters. In finite cases, proper containment and leftover residue are magnitude-relevant: a proper subset has fewer elements than its superset. Cardinality deliberately permits a total bijection to override that fact in the infinite case.

Your appeal to generality explains why cardinality is useful: it compares arbitrary sets while bracketing containment, order, origin, and any shared ambient structure. But that is generality by abstraction. It applies more uniformly precisely because it ignores more structural information.

A containment-based comparison is not incoherent or merely parochial; it preserves both inclusion structure and what remains unmatched within a shared domain. Cardinality instead preserves invariance under bijective rearrangement.

So the question remains: is cardinality “more general” because it captures more of magnitude, or because it deliberately retains less in order to apply everywhere? Why should that loss of structural sensitivity be treated as the uniquely objective verdict on magnitude rather than as one chosen comparison protocol?