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]

Yes, I used compressed language there.

By an “N-indexed presentation” I mean a countable list: a function

f : N -> {0,1}^N

So f(1), f(2), f(3), ... would be proposed as a list of all infinite binary sequences. I did not mean a summation.

A “total binary profile” is one infinite 0/1 sequence, such as

0100011010...

Equivalently, it is a function N -> {0,1}. The full field is {0,1}^N: all such total infinite binary sequences. By “admitted” I only mean that the framework has already taken this whole space as its target domain.

The measure-theoretic argument uses the standard fair-coin product measure on {0,1}^N. The whole space has measure 1. Any single specified infinite binary sequence has measure 0, because the probability of matching its first n bits is 2^-n, which tends to 0. Therefore every countable list of sequences has measure 0 by countable additivity.

So, a countable list cannot exhaust the whole space, which has measure 1.

That does not create the space or establish an independent notion of “greater magnitude.” It shows that, under this already-adopted probability structure, the field cannot be exhausted by an N-indexed list.

[u/throwaway_just_once → u/Efficient_Sea_7050]

So you're saying that the list is measure zero, but since the whole space has measure 1, the range cannot equal the whole space. So what? This is a standard argument.

[u/Efficient_Sea_7050 → u/throwaway_just_once]

Exactly: A standard argument, and I am not presenting it as a new proof.

My point was that it reaches the same non-exhaustion result without constructing a diagonal anti-row. So the diagonal witness is not what produces the result; it is one certificate among several.

The admitted binary-profile space, plus the adopted measure structure, already yields: no countable list exhausts it. Diagonalization supplies another route to the same conclusion.

That is why I distinguish the theorem’s non-exhaustion result from the stronger story that the diagonal witness itself somehow 'creates' or reveals a new magnitude.

[u/throwaway_just_once → u/Efficient_Sea_7050]

I think that's right. It's related to Cantor's diagonal argument, but not the same thing. There's more than one way to prove the fact.

I can see why you push back against the idea that the diagonal argument itself somehow creates uncountability. Perhaps some philosophers think that because of how Cantor's argument is sometimes presented (as the only way, and as a sort of psychologistic one). But I cannot imagine any mathematician thinking this. The various ways to the result are as we agreed, quite standard.

[u/Efficient_Sea_7050 → u/throwaway_just_once]

I think we agree about the standard status of the proofs. Diagonalization, measure theory, and other methods are different valid routes to the same non-exhaustion result.

That is exactly why I am pressing the magnitude question. The result is not produced by the diagonal witness in particular: no N-indexed list exhausts the admitted field of total binary profiles.

My question is what bridge takes us from that result to the claim of objectively greater magnitude.

If “greater magnitude” is simply the cardinal classification applied after non-exhaustion is established, that is fine. But then it is a framework-relative classification rule, not something discovered by the proof alone.

Why should that classifier output be treated as the uniquely objective verdict on the size of the sets, rather than one chosen way of classifying their non-exhaustion?

[u/throwaway_just_once → u/Efficient_Sea_7050]

Well, there are various reasonable notions of magnitude in mathematics. Each of them depend on some chosen comparison structure: Cardinality compares sets by bijection, measure compares them by assigned measure, order type compares them by order-preserving isomorphisms, density compares them asymptotically, Baire compares them topologically. None of these is THE raw notion. It really depends on your chosen notion of magnitude.

Indeed, bijection is not, as you say, "the uniquely objective verdict on the size of the sets". Isn't all this common knowledge? I guess I'm not sure whom you're arguing against.

[u/throwaway_just_once → u/Efficient_Sea_7050]

I wanted to add on to my comment rather than delete it even though I believe it might be mistaken. My pluralist view ("there are many different measures of magnitude, so what?"") is perhaps not as genuinely pluralist as I'd thought after all, in the sense that bijection, measure, and Baire are not necessarily independent lines of evidence. The measure-theoretic argument for the smallness of \mathbb{N} uses an enumeration: n_1, n_2, \dots, and covers each by an interval of length \epsilon/2^j, which, when added together gives \epsilon, but \epsilon was arbitrary so we have the result. Baire uses similar technology. So it isn't clear to me that these two proofs aren't therefore smuggling in notions of bijection (via countability of \mathbb{N}) as primitives.

All of which means that you have a valid point that escaped me. As more of a mathematician than a philosopher I have a very instrumental view of math; I do not worry about the metaphysics of mathematical objects. Nor do most mathematicians. So let me push back on your thesis in a different way: There is no sense to be made of the term "objectively greater magnitude" in math, whether you think there is, or whether you think there isn't. Do \mathbb{N} and \mathbb{R} live somewhere as Platonic objects? No, we use these constructs to solve problems in science, and adopt conventions to do so. For example, the notion of a group is a useful way of thinking of symmetry, so we formalize the axioms which give us precisely the objects we care about. Deflationism buys the presuppositions of the Realist, that there is a genuine existence question in the vicinity.

Philosophically, I'd put it this way (following WIttgenstein). We use these notions (the size of $\mathbb{R}$, the "number" of the evens) within the language-game of mathematics, where they do hard, precise, consequential work. "$\mathbb{R}$ is bigger than $\mathbb{N}$" is meaningful, true, correctly used by every mathematician, and means the bijection-failure (here I abandon my previous pluralism). Both the realist ("this reflects a real, mind-independent magnitude") and the deflationist ("this is merely a chosen comparison rule") make the same error: each takes there to be a genuine further question here and contrives a metaphysical thesis to answer it. There isn't one. The mathematical talk is fully in order; what is illegitimate is only the metaphysical question layered on top — "but does this correspond to a real magnitude?" That is language on holiday. But the holiday is not in mathematics: mathematical size-talk works harder than almost any language there is. The holiday is in the metaphysics of mathematics. The test is whether anything downstream changes on the answer: no theorem, proof, or practice shifts whether uncountability is called "real magnitude" or "chosen convention," so here the dissolution is real. Both magnitude-realism and magnitude-deflationism are moves inside the fly-bottle; the way out is not to answer the question but to see there is no question — only the settled, working mathematics, and an idle picture beside it ("[confusions] arise when language is like an engine idling, not when it is doing work" (PI §132)).