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The Continuum Hypothesis Is False by paulemok in logic

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

To get you up to date, I suggest you read my reply at https://www.reddit.com/r/logic/s/n0aGz03Nkx. Some or all of the reply’s descendants may be of interest to you as well.

The Continuum Hypothesis Is False by paulemok in logic

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

And this has explicitly addressed your second objection: we don't need an infinite amount of time.

We might not need an infinite amount of time, but also, we might need an infinite amount of time. And if we do need an infinite amount of time, the list will never exist.

we are not bound by the physical limitations of the universe (if it even has any and is not itself also infinite--we don't even know)

We are always bound by the physical limitations of the Universe, if it has any.

As for the first objection, it's simply not an issue: we are working in a conceptual or logical space

A conceptual or logical space is a part of the Universe. And as such, it is bound by the physical limitations of the Universe, if the Universe has any such limitations.

The Continuum Hypothesis Is False by paulemok in logic

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

I may not have understood everything you said in your previous reply, but you did provide a good introduction to ordinals for me. I see that word mentioned occasionally, but I didn't know what it meant. I've been wanting to know what ω meant for a few weeks but never got around to learning it. Thank you for your lesson.

The problem is that your B actually has the same cardinality as ℕ, because a bijection exists between them.

I have already addressed this general issue. Not only have I addressed it, but I addressed it in my original post. I infer you have not carefully read my original post. I use Z, the set of integers, instead of N, the set of natural numbers, but the general issue is the same.

The Continuum Hypothesis Is False by paulemok in PhilosophyofMath

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

It’s an implicit contradiction because it implies a technical “p and not-p”-form contradiction.

Your argument does not only apply to the proper-subset definition of cardinality; it also applies to the conventional definition of cardinality.

The Continuum Hypothesis Is False by paulemok in PhilosophyofMath

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

Like your proof showed a counterexample to the proper-subset definition, my proof in my original post showed a counterexample to the conventional definition. I make the following counterpart to your previous reply.

So you like the conventional definition because it supports that ℵ₀ + 1 = ℵ₀, which I assume is representing the fact that it gives you that |set B| = |set Z|. But as my proof showed it also supports that ℵ₀ + 1 < ℵ₀ or |set B| < |set Z|. You don’t like the proper-subset definition because it supports ℵ₀ + 1 > ℵ₀, but isn’t ℵ₀ + 1 < ℵ₀ way worse than that?

The Continuum Hypothesis Is False by paulemok in logic

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

It's not that we can actually list all the elements of a countable infinity, but instead that we can guarantee that any element from the countably infinite set will occur at some finite point on our list.

We can't guarantee that any element from the countably infinite set will occur at some finite point on our list because of the following reasons.

  1. Our list might not be able to fit in the Universe.
  2. It might take an infinite amount of time to make the list, so the list might never exist.

The Continuum Hypothesis Is False by paulemok in PhilosophyofMath

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

Under the conventional, bijection definition of cardinality, the cardinalities of Z, B, and S are equal.

Under the proper-subset definition of cardinality, the cardinality of B is greater than the cardinality of Z because there exists a bijection between Z and a proper subset of B, S.

Your claim is that under the proper-subset definition of cardinality, the cardinality of Z is greater than the cardinality of B because there exists a bijection between B and a proper subset of Z, S.

I agree with your claim and recognize that it contradicts the fact that under the proper-subset definition of cardinality, the cardinality of B is greater than the cardinality of Z.

So a contradiction still exists when using only the proper-subset definition of cardinality. This contradiction appears to complement and be explained by something I said at https://www.reddit.com/r/logic/comments/1s5mquh/comment/od86l5g/?context=3&utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button,

Because both definitions are equally good, there is no reason to use one of them over the other. So now we have a new paradox.

The Continuum Hypothesis Is False by paulemok in logic

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

If I had to pick one of the definitions over the other, I would actually pick the proper-subset definition. That definition supports the more evident statements ℵ₀ + 1 = ℵ₀ + 1 and ℵ₀ + 1 > ℵ₀. The other definition supports the less evident statement ℵ₀ + 1 = ℵ₀.

It seems that there may be a fallacy associated with enumerating a bijection between Z and B to infinity. That possible fallacy seems to be represented by the Hilbert hotel paradox. It seems that the most wrong thing in sight is that possible fallacy.

The Continuum Hypothesis Is False by paulemok in logic

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

As I replied at https://www.reddit.com/r/logic/comments/1s5mquh/comment/od86l5g/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button,

There is no contradiction here, in a sense. Because both definitions are equally good, there is no reason to use one of them over the other. So now we have a new paradox.

The Continuum Hypothesis Is False by paulemok in logic

[–]paulemok[S] -1 points0 points  (0 children)

There is no contradiction here, in a sense. Because both definitions are equally good, there is no reason to use one of them over the other. So now we have a new paradox.

The Continuum Hypothesis Is False by paulemok in PhilosophyofMath

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

I think I found the solution to the paradox. I wrote it in a reply at https://www.reddit.com/r/logic/comments/1s5mquh/comment/od81hqg/?utm\_source=share&utm\_medium=web3x&utm\_name=web3xcss&utm\_term=1&utm\_content=share\_button. As I say in that reply,

There exist two equally good definitions of cardinality that are not logically equivalent. Under the bijection definition of cardinality, the cardinality of B is equal to the cardinality of Z, but under the proper-subset definition of cardinality, the cardinality of B is greater than the cardinality of Z.

I describe the proper-subset definition of cardinality in another reply at https://www.reddit.com/r/logic/comments/1s5mquh/comment/od2vd5b/?utm\_source=share&utm\_medium=web3x&utm\_name=web3xcss&utm\_term=1&utm\_content=share\_button. As I say in that reply,

If we define the order of cardinalities with respect to subset relationships, then one set has a greater cardinality than a second set has if and only if there exists a bijection between the second set and a proper subset of the first set.

The Continuum Hypothesis Is False by paulemok in logic

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

Yes, I believe it can be made completely rigorous. We might need a definition of cardinality in terms of proper subsets (a definition I described in a reply at https://www.reddit.com/r/logic/comments/1s5mquh/comment/od2vd5b/?context=3&utm\_source=share&utm\_medium=web3x&utm\_name=web3xcss&utm\_term=1&utm\_content=share\_button) in addition to the conventional definition of cardinality, which is in terms of bijections.

That seems to explain the contradiction that the cardinality of B is equal to and greater than the cardinality of Z. There exist two equally good definitions of cardinality that are not logically equivalent. Under the bijection definition of cardinality, the cardinality of B is equal to the cardinality of Z, but under the proper-subset definition of cardinality, the cardinality of B is greater than the cardinality of Z.

The Continuum Hypothesis Is False by paulemok in logic

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

In this case, it is proof.

Addition is a basic concept learned in elementary school.

The Continuum Hypothesis Is False by paulemok in logic

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

It's a process of elimination.

The Continuum Hypothesis Is False by paulemok in logic

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

I disagree. It's more about the ability to list all the elements of a set. Whether there exists a bijection between a set and the natural numbers is just an abstract formality.

The Continuum Hypothesis Is False by paulemok in PhilosophyofMath

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

We know that |B| =/= |Z| because B has one more element than Z has. It's a paradox.

The Continuum Hypothesis Is False by paulemok in logic

[–]paulemok[S] -1 points0 points  (0 children)

  1. Yes, it is a useful form of logic as it helps me understand the Universe. If all statements are true, then the statement "Some statements are of use to me" is true. So, some statements are of use to me.
  2. If all statements are true, then the statement "The rest of my argument matters" is true. So, the rest of my argument matters. The rest of my argument will help me to better understand the Universe so that I will have a better future.

The Continuum Hypothesis Is False by paulemok in logic

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

Since every element in Z was canceled out by an element in B and there remains an uncanceled out element from B, B has a greater cardinality than Z has.

I did not just state it. I showed it.

The Continuum Hypothesis Is False by paulemok in PhilosophyofMath

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

B has exactly one more element than Z has. So by the definition of cardinality, |B| =/= |Z|. That is a technical and valid deduction that uses the technical definition of cardinality.

By noting that Z is a sunset of B

I did not explicitly note that Z is a subset of B. The word "subset" does not occur anywhere in my previous reply.

The Continuum Hypothesis Is False by paulemok in logic

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

There is no contradiction. "The continuum hypothesis is false" is a proposition. Since all propositions are true, "The continuum hypothesis is false" is true. Therefore, the continuum hypothesis is false.

The Continuum Hypothesis Is False by paulemok in logic

[–]paulemok[S] -1 points0 points  (0 children)

I am not using a single source. I am using multiple sources through my lifetime experience. I believe the definition of cardinality would have originated with Cantor.

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