When you talk about infinity, you are no longer talking about numbers. Mix it with numbers, you get all sorts of perplexing theories and paradoxes.
The reason is simple - numbers are cuts in the continuum while infinity isn't. It should not even be a symbolic notion of very large number. This is not to say infinity doesn't exist. It doesn't exist as a number and doesn't mix with numbers.
The limits could have been defined saying "as x increases without bound" instead of "as x approaches infinity". There is no target called infinity to approach.
Cantor's stuff can easily be trashed. The very notion of "larger than" belongs to finite numbers. This comparitive notion doesn't apply to concepts that can not be quantified using numbers. Hence one can't say some kind of infinity is larger than the other kinds.
Similarly, his diagonal argument about 1-to-1 mapping can not be extended to infinities, as there is no 1-to-1 mapping that can make sense for those which are not numbers or uniquely identifiable elements. The mapping is broken. No surprise you get weird results and multiple infinities or whatever that came to his mind when he was going through stressful personal situations.
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.
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.
It's an axiom (the axiom of choice, actually). A valid way of viewing an axiom is not dissimilar to a "modeling requirement" or an "if statement". By that I mean, for example with the axiom of choice: it is just a formal statement version of "assume that you can take an element from a (possibly infinite) collection of sets such that you can create a new set (the new set does not have to be unique)." It makes intuitive sense for most finite sets we deal with physically, and, for infinite sets, it can actually make sense in a way that actually successfully predicts results that do hold in the real world and provides a really convenient way to define a lot of consistent properties of the continuum itself.
However, if you're dealing with a problem where you can't always usefully distinguish between elements across arbitrary set-like objects; then it's not a useful axiom and ZFC is not the formalism you want to use. Most problems we analyze in the real world, that's actually something that we can usefully assume, hence why it's such a successful and common theory, even if it leads to physical paradoxes like Banac-Tarsky, as mentioned.
Mathematicians, in practice, fully understand what you mean with your complaint about "completion," but, the beauty of these formal infinities is the guarantee it gives you that it'll never break down as a predictive theory no matter the length of time or amount of elements you consider or the needed level of precision; the fact that it can't truly complete is precisely the point. Also, within the formal system used, we absolutely can consistently define what the completion would be at "infinity," as long as you treat it correctly and don't break the rules. Again, this is useful because it allows you to bridge multiple real problems that seemingly were unrelated and it pushes "representative errors" to those paradoxes and undefined statements of the theory (thanks, Gödel).
If it helps, the transfinite cardinalities (what you call infinity) you are worried about are more related to rates than counts, even if they have some orderable or count-like properties. In the strictest sense, you can actually drop into archimedian math, which you might find very enjoyable to read about or use, and it, in a very loose sense, kinda pushes the idea of infinity from rates of counts to rates of reaching arbitrary levels of precision.
The difference between infinities is I can write every possible fraction on a piece of A4 paper, if the font gets smaller and smaller. I can say where to zoom in for any fraction.
You can't enumerate the real numbers, but you can grab them all in one go - just draw a line!
The more I learn about this stuff, the more I come to understand how the quantitative difference between cardinalities is a red herring (e.g. CH independent from ZFC). It's the qualitative difference between these two sets that matter. The real numbers are richer, denser, smoother, etc. than the natural numbers, and those are the qualities we care about.
Countability is the whole point, there's no need to apologize. I was merely offering the perspective that "towers of infinity" is possibly the least useful consequence that comes from defining the notion of countability. To my mind, what we really reap from Cantor's work is a better understanding of the topology of the real numbers. But you have to define countability first in order to understand what uncountability really implies.
That doesn't make one set "larger" than the other. You need to define "larger". And you need to make that definition as weird as needed to justify that comparison.
The fact that I can't even fit the real numbers between 0 and 1 on a single page, but I can fit every possible fraction in existence, doesn't mean anything?
I don't think this definition is that weird, for example by 'larger' I might say I can easily 'fit' all the rational numbers in the real numbers, but cannot fit the real numbers in the rational numbers.
It doesn't mean anything because, with arbitrary zooming for precision, every real number is a fraction. You can't ask for infinite zooming. There is no such thing.
So, let's inspect pi. It's a fraction, precision of which depends on how much you zoom in on it. You can take it as a constant just for having a name for it.
If infinite zooming/precision is ruled out, all real numbers can be written as fractions. Problem is with the assumption that some numbers such as pi or sqrt(2) can have absolute and full precision.
.. because that is where you are allowed to challenge some biblical stories of the math without the fear of expulsion from the elite clubs.
Most of math history is stellar, studded with great works of geniuses, but some results were sanctified and prohibited for questioning due to various forces that were active during the times.
Application of regular logic such as comparison, mapping, listing, diagonals, uniqueness - all are the rules that were bred in the realms of finiteness and physical world. You can't use these things to prove some theories about things are not finite.
I'm aware that I'm ignorant of many things, just like anyone else on this Earth. Some are less ignorant and some are more.
Could you be kind enough to explain the phrase "set of all integers" when the word all can not apply to an unbounded quantity? I think the word all is used loosely to extend it's meaning as used for finite sets, to a non-existent, unbounded set. For example, things such as all Americans, all particles in universe have a meaning because they have a boundary criterion. What is all integers?
I think one need to first define the realm of applicability or domain for the concepts such as comparison, 1-to-1 mapping, listing, diagonals, uniqueness, all etc.
The reason is simple - numbers are cuts in the continuum while infinity isn't. It should not even be a symbolic notion of very large number. This is not to say infinity doesn't exist. It doesn't exist as a number and doesn't mix with numbers.
The limits could have been defined saying "as x increases without bound" instead of "as x approaches infinity". There is no target called infinity to approach.
Cantor's stuff can easily be trashed. The very notion of "larger than" belongs to finite numbers. This comparitive notion doesn't apply to concepts that can not be quantified using numbers. Hence one can't say some kind of infinity is larger than the other kinds.
Similarly, his diagonal argument about 1-to-1 mapping can not be extended to infinities, as there is no 1-to-1 mapping that can make sense for those which are not numbers or uniquely identifiable elements. The mapping is broken. No surprise you get weird results and multiple infinities or whatever that came to his mind when he was going through stressful personal situations.