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?

This follows from Part I: On Gödel: What Exactly Is Incomplete?.

My earlier post separated the mathematical domain, a complete truth-set about it, and a fixed effective proof-generator. Gödel directly limits the third.

Several replies accepted that a complete truth-set could exist while remaining non-enumerable and non-computable. So suppose, conditionally, that every arithmetical sentence has a determinate truth-value in the intended structure.

What is missing when Gödel applies?

Not necessarily another truth or axiom. What is missing is a uniform effective access rule: a finite mechanical procedure that takes any sentence and always returns its correct truth-value.

A fixed theory may fail to decide every sentence. No algorithm may decide every arithmetical truth. Neither point alone shows that the target domain lacks determinate truth-values.

So when people say Gödel shows mathematics is incomplete, do they mean:

• truth itself is absent or indeterminate; or
• no single effective formal method can derive or decide every truth of the intended domain?

The first is a claim about truth. The second is a claim about formal access.

Gödel gives a formal incompleteness result. What additional argument would establish the first claim?

I am not assuming that a completed truth-set exists. I am asking what, beyond Gödel’s theorem, would justify denying it.

Gödel’s incompleteness theorems are often presented as showing that mathematics is incomplete, that there can be no mathematical “theory of everything,” or that truth outruns proof. I want to separate the precise technical theorem from those broader conclusions.

There are three different objects here: the mathematical domain, the total ledger of truths about that domain, and a fixed effective proof-generator. Gödel limits the third. It does not by itself show that the domain is unfinished or that the complete truth-ledger fails to exist.

So when the broader slogan is used, is it only shorthand for the limits of effective formal systems, or is a further claim about mathematical truth itself intended? If the latter than what warrants the move from limits on proof to limits on truth?

I accept Cantor’s formal result:

|A| < |P(A)|

as a cardinal classification.

My objection is to the stronger claim that this proves greater infinite magnitude, rather than merely a higher cardinal index.

The issue seems to be this.

First, cardinality is selected as the ruler for comparing infinite size. Then the diagonal argument shows that no A-indexed enumeration exhausts P(A). So, under the cardinal ruler, P(A) receives a higher index.

I accept all of that.

But then the higher index is described as “greater magnitude,” and that description is used to make cardinality look like the natural sovereign measure of infinite size.

That seems circular unless “magnitude” has already been defined as “cardinality.”

The Roman-foot analogy makes the worry clearer.

Suppose the standard Roman foot is around 296 mm, shorter than the modern foot, while the pes Drusianus, associated with the Germanic provinces, is around 332-335 mm, longer than the modern foot.

If we first select the longer Germanic/Roman standard and define that as “the Roman foot,” then yes, formally, we can say:

>the Roman foot is longer than the modern foot.

But that would be true by convention. It would not prove that the Roman foot, as an unqualified historical magnitude, was simply longer.

It would prove only that we selected one standard and then used “longer” according to that selection.

That is my worry about “bigger infinity.”

If “bigger” means “higher cardinal index,” then Cantor’s theorem proves bigger infinity by definition.

But if “bigger” means unqualified greater magnitude, then the framework needs an additional bridge:

cardinal-larger => magnitude-larger

That bridge cannot be supplied merely by applying cardinality and then calling its result “magnitude.”

The asymmetry appears in the contrast between N vs 2N and A vs P(A).

In N vs 2N, surplus is denied magnitude-force. 2N is a proper subset of N, and N contains the odds outside 2N, but cardinality says this does not matter because there is a bijection.

In A vs P(A), surplus is treated differently. The diagonal witness, omitted profile, or “more combinations” language is allowed to carry hierarchy-force.

So the question is not whether the formal cases differ. They do.

The question is whether the same framework may deny magnitude-force to surplus in one case, then grant magnitude-force to surplus in another case, while presenting both verdicts as neutral discoveries of magnitude.

If pairability is the sovereign ruler of magnitude, then surplus should not become magnitude-producing without an earned exception.

If surplus is magnitude-relevant, then the natural/even surplus should not be dismissed without an earned exception.

And that exception cannot be:

>because P(A) is already bigger.

That is the conclusion under dispute.

So my question is:

Does Cantor’s theorem prove only a cardinal ordering, or does it also prove unqualified greater magnitude?

And if “greater magnitude” just means “larger cardinality,” then is the stronger “bigger infinity” language definitional rather than discovered?

Longer version of the argument here:

https://www.reddit.com/u/Efficient_Sea_7050/s/QJquRaY4Lj

I am trying to understand the exact status of the common statement that Cantor proved one infinity is “bigger” than another.

I am not denying the formal theorem that there is no bijection between N and P(N), or that |N| < |P(N)| follows under the usual cardinal framework.

My question is: when people say this proves a genuinely bigger infinity, is “bigger” being used only as a technical synonym for cardinal-larger, or is there an additional interpretation from cardinal-status to magnitude-language?

In other words, is this bridge just definitional:

cardinal-larger -> bigger in size

or is it supposed to carry a stronger magnitude claim?

I wrote out the longer version here, but the core question is the one above:

https://www.reddit.com/u/Efficient_Sea_7050/s/QJquRaY4Lj

Where exactly would this reasoning go wrong?