Comment by soulofmischief
3 days ago
I don't think you can just call Cantor's diagonal argument trash without providing a very strong, self-consistent replacement framework that explains the result without invoking infinity.
This sounds like ranting from someone who doesn't deeply understand the implications of set theory or infinite sets. Cardinality is a real thing, with real consequences, such as Gödel's incompleteness theorems. What "weird results" are tripping you up?
> when he was going through stressful personal situations
Ad hominem. Let's stick to arguing the subject material and not personally attacking Cantor.
https://en.wikipedia.org/wiki/Cantor's_diagonal_argument#Con...
The Löwenheim-Skolem theorem implies that a countable model of set theory has to exist. "Cardinality" as implied by Cantor's diagonal argument (which happens to be a straightforward special case of Lawvere's fixed point theorem) is thus not an absolute property: it's relative to a particular model. It's internally true that there is no bijection as defined within the model between the naturals and the reals, as shown by Cantor's argument; but externally there are models where all sets can nonetheless be seen as countable.
You're referring to Skolem's paradox. It just shows that first-order logic is incomplete.
Ernst Zermelo resolved this by stating that his axioms should be interpreted within second-order logic, and as such it doesn't contradict Cantor's theorem since the Löwenheim–Skolem theorem only applies in first-order logic.
The standard semantics for second-order logic are not very practical and arguably not even all that meaningful or logical (as argued e.g. by Willard Quine); you can use Henkin semantics (i.e. essentially a many-sorted first-order theory) to recover the model-theoretic properties of first-order logic, including Löwenheim-Skolem.
I just gave the reason - The notion of comparison and 1-to-1 mapping has an underlying assumption about the subjects being quantifiable and identifiable. This assumption doesn't apply to something inherently neither quantifiable nor is a cut in the continuum, similar to a number. What argument are you offering against this?
I'm not the person you replied to, and I doubt I'm going to convince you out of your very obviously strong opinions, but, to make it clear, you can't even define a continuum without a finite set to, as you non-standardly put it, cut it. It turns out, when you define any such system that behaves like natural numbers, objects like the rationals and the continuum pop out; explicitly because of situations like the one Cantor describes (thank you, Yoneda). The point of transfinite cardinalities is not that they necessarily physically exist on their own as objects; rather, they are a convenient shorthand for a pattern that emerges when you can formally say "and so on" (infinite limits). When you do so, it turns out, there's a consistent way to treat some of these "and so ons" that behave consistently under comparison, and that's the transfinite cardinalities such as aleph_0 and whatnot.
Further, all math is idealist bullshit; but it's useful idealist bullshit because, when you can map representations of physical systems into it in a way that the objects act like the mathematical objects that represent them, then you can achieve useful predictive results in the real world. This holds true for results that require a concept of infinities in some way to fully operationalize: they still make useful predictions when the axiomatic conditions are met.
For the record, I'm not fully against what you're saying, I personally hate the idea of the axiom of choice being commonly accepted; I think it was a poorly founded axiom that leads to more paradoxes than it helps things. I also wish the axiom of the excluded middle was actually tossed out more often, for similar reasons, however, when the systems you're analyzing do behave well under either axiom, the math works out to be so much easier with both of them, so in they stay (until you hit things like Banac-Tarsky and you just kinda go "neat, this is completely unphysical abstract delusioneering" but, you kinda learn to treat results like that like you do when you renormalize poles in analytical functions: carefully and with a healthy dose of "don't accidentally misuse this theorem to make unrealistic predictions when the conditions aren't met")
About the 1-to-1 mapping of elements across infinite sets: what guarantees us that this mapping operation can be extended to infinite sets?
I can say it can not be extended or applied, because the operation can not be "completed". This is not because it takes infinite time. It is because we can't define completion of the operation, even if it is a snapshot imagination.
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