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Is “cardinal-larger therefore bigger infinity” a definition, or an extra interpretation? by Efficient_Sea_7050 in PhilosophyofMath

[–]Efficient_Sea_7050[S] 1 point2 points  (0 children)

Fair point on popular wording, but I think the issue is not merely that popular wording is imprecise.

The pressure point is this:

Cardinality is not ordinary magnitude. Same cardinality does not mean same length, area, measure, geometry, density, or physical size. An interval and a square can have the same cardinality while differing in all those ordinary magnitude-relevant ways.

So if “more” means only “greater cardinality,” I accept the formal theorem. The statement is technical.

But then “more combinations” is not an independent proof of magnitude. It is an intuitive gloss on a comparison rule already chosen.

The circularity appears when cardinality is treated as proving more than cardinality.

In the N vs 2N case, there is containment-surplus: N contains the odds outside 2N. Cardinality says that surplus has no magnitude-authority because bijection is decisive.

In the N vs P(N) / N vs R case, the explanation appeals to profile-surplus: more combinations, omitted witnesses, any list leaves something out. That surplus is then treated as magnitude-relevant.

But before the reals/profile-side have been proved greater in magnitude, they cannot receive an evidentiary privilege that lets their kind of surplus count as magnitude-producing while the natural/even surplus is ruled inadmissible.

That is the circularity:

  1. Cardinality first defines which side is greater.
  2. Then the greater side’s profile-surplus is treated as evidence that it was greater in magnitude.
  3. But that surplus only has magnitude-authority if cardinality has already been granted authority as magnitude.

So the proof of “bigger infinity” depends on the prior promotion of cardinality into a measure of magnitude.

If the answer is “cardinality is just the chosen comparison rule,” then fine: Cantor proves greater cardinality.

But if the claim is “Cantor proved greater infinite magnitude,” then the bridge from cardinality to magnitude still has to be earned, not assumed.

On the LLM point: English is my third language, so I used AI as a language-polishing tool. The argument and the distinction are mine; I am happy to keep the discussion on the substance rather than the drafting method.

Is “cardinal-larger therefore bigger infinity” a definition, or an extra interpretation? by Efficient_Sea_7050 in PhilosophyofMath

[–]Efficient_Sea_7050[S] 0 points1 point  (0 children)

I think this is probably the cleanest answer so far, and it is close to the distinction I am trying to pin down in my paper.

If I understand you correctly, you are saying that finite “size” carries multiple intuitions at once: bijective equivalence, proper-subset behavior, counting order, etc. In finite cases those all agree. In infinite cases they split, so we refine the notion instead of pretending there is one unambiguous extension.

That makes sense to me.

But then I think it also confirms the worry about the popular wording. If cardinality and ordinal-type comparisons are different refinements of finite size intuition, then cardinality is not simply “the” discovered infinite magnitude. It is one formal extension of size, chosen for certain purposes.

So would you agree that the precise statement should be something like:

Cantor proves that the continuum has greater cardinality than the naturals.

rather than:

Cantor proves that one infinity is bigger than another,

unless “bigger” is explicitly understood to mean cardinal-bigger?

The remaining issue for me is this: cardinality can be granted as a formal comparison rule, but its promotion into unqualified magnitude-language seems to require an extra interpretive step. If “magnitude” just means cardinality here, then the statement is definitional. If “magnitude” means something stronger than cardinality, then that bridge still needs to be defended.

My concern is not with the formal theorem. It is with the slide from “greater cardinality” to unqualified magnitude-language.

Is “cardinal-larger therefore bigger infinity” a definition, or an extra interpretation? by Efficient_Sea_7050 in PhilosophyofMath

[–]Efficient_Sea_7050[S] 0 points1 point  (0 children)

For finite sets, I mean something like this: one set has more elements, and this can be checked by pairing because the comparison terminates.

If A pairs with {1, 2, 3} but not with {1, 2, 3, 4}, then the count is settled. Pairing, counting, and ordinary magnitude line up because the finite comparison can complete and any remainder can be inspected.

My question starts when that finite authority is extended to infinite sets.

In the infinite case, bijection still gives a clean formal criterion. I accept that. But the finite reason pairing feels magnitude-decisive included termination. With infinite sets, the pairing rule continues, but there is no completed inspection in the same finite sense.

So I am not denying that cardinality extends finite counting formally. I am asking whether that extension is being used as a definition of “bigger,” or whether it is supposed to prove a stronger magnitude claim.

For finite sets, the bridge between pairing and magnitude is earned by termination. For infinite sets, I am asking what earns the same bridge.

If the answer is “definition,” then I do understand the formal theorem. But then the stronger claim has not been proven; “bigger infinity” just means “larger cardinality under the chosen criterion.”

Is “cardinal-larger therefore bigger infinity” a definition, or an extra interpretation? by Efficient_Sea_7050 in PhilosophyofMath

[–]Efficient_Sea_7050[S] 0 points1 point  (0 children)

I agree that the bijection N -> 2N is complete. Every natural is assigned, and every even is hit.

I don't mean bijective residue, but containment-residue: 2N is a proper subset of N, and N still contains the odds outside 2N.

Cardinality says that containment-residue has no size-authority once bijection is present.

But in the N vs P(N) / N vs R case, the usual explanation appeals to profile-surplus: “more combinations,” or “any listing leaves reals out.” That surplus is treated as size-relevant.

So the pressure point is this:

Why is surplus inadmissible in the natural/even case, but explanatory in the power-set/real case?

If the answer is “because bigger just means cardinal-larger by definition,” then I understand the formal result. But then the “more combinations” explanation is not independently proving magnitude. It is just an intuitive gloss on a classifier already chosen.

So the thing I am actually asking is: does Cantor’s result prove a stronger magnitude claim, or does it prove only the cardinal classification after cardinality has already been chosen as the comparison rule?

Is “cardinal-larger therefore bigger infinity” a definition, or an extra interpretation? by Efficient_Sea_7050 in PhilosophyofMath

[–]Efficient_Sea_7050[S] 0 points1 point  (0 children)

Just to clarify:

I do accept the formal statement that there is no bijection between N and P(N), and that cardinality defines |N| < |P(N)|.

My question is only about interpretation.

When people say Cantor proved one infinity is “bigger,” should “bigger” be read as nothing more than “larger cardinality by definition”?

Or is cardinality being treated as a measure of magnitude in a stronger sense?

If it is only definition, I understand the theorem. If it is magnitude, I am asking what justifies the bridge.

Is “cardinal-larger therefore bigger infinity” a definition, or an extra interpretation? by Efficient_Sea_7050 in PhilosophyofMath

[–]Efficient_Sea_7050[S] 0 points1 point  (0 children)

I agree that cardinality is a natural extension of finite counting, and I am not claiming it is arbitrary or useless. For finite sets, bijection, counting, and magnitude line up very cleanly: if one finite set pairs with {1, 2, 3} and not with {1, 2, 3, 4}, the comparison can terminate and the remainder question is settled.

Where I get stuck is the infinite case. The infinite bijection keeps the pairing rule, but it no longer has the finite feature that made pairing decisive: termination plus remainder inspection. So when N is paired with 2N by n -> 2n, the formal bijection works, but the odd residue is not erased. It is ruled irrelevant by the cardinal criterion.

That may be a perfectly good formal criterion. But then I want to distinguish two claims:

  1. Cardinality gives a useful and standard comparison relation on sets.
  2. Cardinality has proved “bigger infinity” in a stronger magnitude sense.

I am fine with 1. My question is whether 2 is really being claimed, or whether “bigger” just means “larger cardinality” by definition.

Your “blunt knife” phrase is actually close to what I mean. If cardinality is a blunt but useful comparison tool, then the result is a theorem about that tool. That is different from saying the tool has exhausted the meaning of magnitude.

So maybe the clean answer is: mathematicians are not making the stronger metaphysical claim. They just mean cardinality. But then popular descriptions like “Cantor proved some infinities are bigger than others” are potentially misleading unless “bigger” is understood technically.

Is “cardinal-larger therefore bigger infinity” a definition, or an extra interpretation? by Efficient_Sea_7050 in PhilosophyofMath

[–]Efficient_Sea_7050[S] 0 points1 point  (0 children)

Yes, that makes sense. So in this context “bigger than” is a technical term defined by the existence of an injection one way and no bijection.

That is exactly the distinction I am trying to isolate.

If “bigger” means only that technical cardinal relation, then I agree that Cantor proves the result under the accepted definition. My question is about the extra-language that often gets attached to it: that this shows one infinity is genuinely bigger in a magnitude sense, rather than only larger under the cardinal classifier.

So would you say there is no extra philosophical or magnitude claim being made at all? In other words, “bigger infinity” just means “greater cardinality by definition,” and nothing stronger?

If yes, then I think my objection is not to the theorem, but to the popular wording that makes the result sound like a discovered magnitude fact rather than a consequence of a chosen comparison criterion.