The cardinality of the set of real numbers is aleph_1 — TonesInDeepFreeze

it's easy to give a name to an infinite cardinality (aleph_1 for example), just like we might name it "an infinity", but naming it in no way demonstrates that it is countable.

Correct, alelph_1 is not countable.

The cardinality of the set of real numbers is aleph_1 — TonesInDeepFreeze

By the way, I did not say that the cardinality of the set of real numbers is aleph_1. I said that "The cardinality of the set of real numbers is aleph_1" is the continuum hypothesis.

I think we are simply talking differently about the subject. But as I'm no mathematician, I acknowledge that my wording might not be rigorous.

Do notice that Cantors system is the sequence of cardinal numbers: aleph_0. aleph_1, aleph_2, aleph_3 and so on. The question is if this hierarchial system holds and if there is a cardinality or not between the naturals or the reals. The continuum hypothesis is that the reals is the next aleph, that there isn't anything else.

Cantor's proof was not by reductio ad absurdum — TonesInDeepFreeze

Your somewhat correct. The first proof in 1874 wasn't. But he did give this in 1891 with the diagonal argument, which I find more simple.

The incoherency is quite clear, and I'll explain it to you. You can deny that it exists, and call me whatever name you like, but that doesn't address the problem.

In simple terms, counting is a task. To be "counted" implies that the task is completed. To be "countable" implies that the task may be completed. In such common terms, no infinite number is countable, because the task to count an infinite number can never be completed.

Now, it may be possible to define "countable" in a way such that completion of the task would not be required as a criterion for being countable. If mathematicians have successfully done this, then an infinite number would be countable.

However, mathematicians have not successfully done this. They have defined "countable" in relation to another task, bijection, and proper bijection would also require completion of the task, just like counting. Since bijection is a task which cannot be completed in the case of an infinite number, they have not defined "countable" in a way such that an infinite number can be truthfully said to be countable.

So mathematicians pretend that an infinite number, the cardinality of the set of natural numbers, is countable when it really is not. "Countable" as defined by mathematicians is not consistent with "infinite" as commonly used in reference to the natural numbers, so the idea that an infinite set is countable is a pretense.

This pretense produces a new meaning for "infinite", one which is consistent with the pretense. However, this meaning of "infinite" is not consistent with how "infinite" is commonly used by mathematicians, hence the word "transfinite" is sometimes employed. I will call this concept of "infinite" a phantom infinite because it's a completely imaginary concept, totally distinct from actual usage, created solely for the purpose of making it appear like it is possible to do the impossible task, count the infinite natural numbers. The phantom infinite concept is a product of that pretense.

Now we have a concept, the phantom infinite, which hides the inconsistency between "countable" and "the natural numbers". The natural numbers are not really countable, (being infinite as implying a task which cannot be completed), but the phantom infinite makes it appear like they are by changing the meaning of "infinite". The phantom infinite is a false concept because this sense of "infinite" is not consistent with how "infinite" is actually used in relation to the natural numbers. We do not allow that one can actually complete the task of counting the natural numbers. The result being a false representation of the natural numbers, having been created by the phantom concept of infinite. The pretense requires a false representation of the natural numbers, for its support. So the phantom infinite is imposed onto the natural numbers, as if this is the real way that "infinite" is used in relation to the natural numbers, but this is not a true representation of how the natural numbers are actually used, and how they are said to be "infinite".

Keeping all that in mind, the problem with the continuum hypothesis ought to become crystal clear to you. The idea that an infinite set is countable, or has a specifiable cardinality, is a product of the phantom infinite concept. This is not consistent with "infinite" as used by mathematicians. Therefore there is an inconsistency between the concepts of "infinite set", and "countable" inherent within the continuum hypothesis. The "infinite set" of natural numbers takes the traditional meaning of "infinite" (implying not countable as the task cannot be completed), and the designation that such a set is "countable" uses the phantom infinite concept which is inconsistent with the traditional concept.

The inconsistency between the traditional concept "infinite", and the phantom concept of infinite makes the two completely incompatible. Because of this, any attempt to establish commensurability between the two fails.

my wording might not be rigorous — ssu

Not merely a lack of rigor. Rather, your statement "the continuum hypothesis is that from aleph_0 the next is aleph_1" is plainly false.

the sequence of cardinal numbers: aleph_0. aleph_1, aleph_2, aleph_3 and so on. The question is if this hierarchial system holds — ssu

There is no question of whether it "holds". It doesn't even make sense to say that a sequence holds or not. What hold or not are statements.

if there is a cardinality or not between the naturals or the reals — ssu

Yes, the continuum hypothesis is the claim that there is no cardinality between the set of naturals and the cardinality of the set of reals. That's another way of saying what I've been telling you.

The continuum hypothesis is that the reals is the next aleph, that there isn't anything else. — ssu

The continuum hypothesis is that the cardinality of the set of reals is aleph_1. That is equivalent to saying that there is no uncountable subset of the set of reals that is not 1-1 with the set of reals. Of course, no matter the continuum hypothesis, there are cardinals greater than aleph_1.

the diagonal argument — ssu

The diagonal argument given by Cantor was not a reductio ad absurdum.

The continuum hypothesis is that the cardinality of the set of reals is aleph_1. That is equivalent to saying that there is no uncountable subset of the set of reals that is not 1-1 with the set of reals. Of course, no matter the continuum hypothesis, there are cardinals greater than aleph_1. — TonesInDeepFreeze

Cantor proved that you cannot make a bijection between the natural number to the reals, hence the reals aren't aleph_0 like for example rational numbers.

The diagonal argument given by Cantor was not a reductio ad absurdum. — TonesInDeepFreeze

When you first assume that there is a bijection between the natural numbers and reals, then show that there is a real that cannot be in this bijection, that is a reductio ad absurdum.

Cantor proved that you cannot make a bijection between the natural numbers to the reals, hence the [cardinality of the] reals [is not] aleph_0 like for example [the cardinality of the] rational numbers. — ssu

Correct.

When you first assume that there is a bijection between the natural numbers and reals — ssu

Cantor didn't make that assumption.

What was then his working hypothesis?

Here's the argument, which is not reductio ad absurdum:

Let f be a function from the set of natural numbers to the set of denumerable binary sequences.

Construct a denumerable binary sequence not in the range of f.

Conclude there is no function from the set of natural numbers onto the set of denumerable binary sequences.

/

He also could have used reductio ad absurdum, but he didn't:

Assume there is function from the set of natural numbers onto the set of denumerable binary sequences.

Derive a contradiction.

Conclude there is no function from the set of natural numbers onto the set of denumerable binary sequences.

To the layman (as me) this is rather scholastic, but I understand that math is precise. So a question:

When you construct "a denumerable binary sequence not in the range of f", aren't you deriving that contradiction? There's the negative reference to f.

After all, when let's say one asks if the set of natural numbers and rational numbers have the same cardinality, there is a direct proof (the set of rational numbers can be well ordered and Cantor showed this).

When you construct "a denumerable binary sequence not in the range of f", aren't you deriving that contradiction? — ssu

When you use reductio ad absurdum, you construct a denumerable binary sequence not in the range of f, which contradicts the assumption that f is a bijection between the set of natural numbers and the set of denumerable binary sequences. But Cantor didn't do it that way.

the set of natural numbers and rational numbers have the same cardinality, there is a direct proof (the set of rational numbers can be well ordered — ssu

The proof doesn't rest on showing a well ordering. Showing a well ordering of the set of rational numbers is not adequate for showing that the set of rational numbers is countable. (Even uncountable ordinals are well ordered.)

Showing a well ordering of the set of rational numbers is not adequate for showing that the set of rational numbers is countable. — TonesInDeepFreeze

Countable, right. Thanks for the correction.

You've picked up the scent mon ami but que sais-je.

Very easy: aleph-1. The the infinite cardinal of the real numbers

Because I think there's still something for us to understand with infinity, it isn't so easy that to use finite logic. And more interestingly, bigger infinities seem not to be usefull for example in physics, computing, etc. — ssu

What's $\aleph_1$?

Are we talkin' about the reals?

aleph_1 is the least cardinal greater than aleph_0.

That is the case by definition.

"aleph-1. The the infinite cardinal of the real numbers"

That is the continuum hypothesis. The continuum hypothesis is "aleph_1 is the cardinality of the set of real numbers" (or equivalently, "aleph_1 = 2^aleph_0") and it is not a theorem of ZFC nor is its negation a theorem of ZFC.

Very easy: aleph-1 — ssu

So is ssu right?

You want a greatest practical natural number. aleph_1 is not a natural number; it's an infinite cardinal.

A cardinal number is, well, a number. When we talk about finite sets, the cardinal number is a natural number.

And even if this will irritate the mathematicians here, we do use infinity quite a lot, we just don't talk about it as infinity. Just like with limit sequences and limit points etc.

You're right, but only on a technical point and it's reassuring to see people like yourself with experience in philosophy around here.

As to the topic at hand, I was in search of a finite number N_max such that no physical calculation ever exceeds N_max.

Nonetheless, it seems you're on the mark, $\aleph_1$ (the reals?) is the only level of $\infty$ required for our universe.

Yes, x is a natural number iff x is a finite cardinal.

And aleph_1 is not a finite cardinal.

And the poster is asking about finding a certain natural number, so aleph_1 is not an answer.

ℵ1 (the reals?) — Agent Smith

I addressed that about half a dozen times in posts above.

The statement "aleph_1 is the cardinality of the set of real numbers" is the continuum hypothesis. It is not derivable in ZFC and its negation is not derivable in ZFC.

Ok!

Cardinality as in the cardinality of the set {a, &} is 2?

I addressed that about half a dozen times in posts above. — TonesInDeepFreeze

Apologies ... patience is a virtue. :smile:

thm: n is a natural number <-> (n is finite & n is an ordinal)

dfn: card(x) = the least ordinal k such that x is 1-1 with k

dfn: c is a cardinal <-> there exists an x such that c = card(x)

thm: n is a natural number <-> (n is finite & n is a cardinal)

thm: x is finite <-> card(x) is a natural number

thm: x is infinite <-> card (x) is infinite

/

So, yes, if x is finite, then its cardinality is a natural number.

But if x if infinite, then its cardinality is not a natural number but rather is an infinite cardinal.

In either case, every set is 1-1 with its cardinality.

/

dfn: aleph_0 = the set of natural numbers

dfn: aleph_1 = the least infinite cardinal that is strictly greater than aleph_0

dfn: R = the set of real numbers

dfn: x is denumerable <-> (x is 1-1 with the set of natural numbers & x is infinite)

thm: card(R) = 2^aleph_0 (in other words, the cardinality of the set of real numbers is the cardinality of the set of denumerable binary sequences)

The great question of set theory: Is the cardinality of the set of real numbers aleph_1? Put another way: Is it the case that there is no set that has a cardinality strictly greater than the set of natural numbers but strictly less than the cardinality of the set of real numbers? Put another way: Is aleph_1 = 2^aleph_0?

The assertion "the cardinality of the set of real numbers is aleph_1" is called 'the continuum hypothesis' ('CH'). Cantor thought CH was true, but he couldn't prove it. Hilbert proposed that finding a proof, one way or the other, was a priority of mathematics. Later, Godel proved that ZFC does not prove the negation of CH, then later Cohen proved that ZFC does not prove CH.

So the great question of set theory does not have an answer per merely ZFC. So mathematicians have been proposing and studying axioms that they consider to be intuitively true and could be added to ZFC to settle CH. Some mathematicians propose axioms that prove CH, and other mathematicians propose axioms that prove the negation of CH. There is no consensus.

So it is silly just to say "the cardinality of the set of real numbers is aleph_1", without citing a context for belief in the assertion, when it is a profound open question.

/

PS: The generalized continuum hypothesis (GCH) is that for any x, aleph_(x+1) = 2^aleph_x.

Of course, GCH implies CH, and the negation of CH implies the negation of GCH. Godel actually proved that ZFC does not prove the negation of GCH. And Cohen, by proving that ZFC does not prove CH perforce proved that ZFC does not prove GCH.

I thought the Continuum Hypothesis was about the existence/nonexistence of an infinity (I) such that cardinality-wise, set of naturals ($\aleph_0$) < I < set of reals ($\aleph_1$).

What would be the implications of there being an I?

No. The way you wrote it is wrong. The continuum hypothesis is that the cardinality of the set of reals is aleph_1. This point keeps getting lost. Don't just take it for granted that the cardinality of the set of reals is aleph_1, when that is the very point that is in question with the continuum hypothesis.

But this part that is suggested (though mangled) by you is correct: The continuum hypothesis is equivalent to the assertion that there is no set that has cardinality strictly greater than the cardinality of the set of naturals and strictly less than the cardinality of the set of reals.

One more time. Here are three equivalent ways of saying the continuum hypothesis [here 'N' stands for the set of naturals]:

(1) aleph_1 = 2^aleph_0

(2) card(R) = aleph_1

(3) There is no set that has cardinality strictly greater than card(N) and strictly less than card(R)

Those are three ways of saying the continuum hypothesis.

If you say card(R) = aleph_1, then you are asserting the continuum hypothesis, which is an assertion that can't be proven in ZFC nor disproven in ZFC.

Or, since this point seems not to be getting through, I'll say it this way:

Don't just assume that card(R) = aleph_1. "Is card(R) = aleph1 ?" is the QUESTION. Don't just assume its answer is 'yes'.

But this part of what you wrote is correct: The continuum hypothesis is equivalent to the assertion that there is no set that has cardinality strictly greater than the cardinality of the set of naturals and strictly less than the cardinality of the set of reals. — TonesInDeepFreeze

Thank god! :pray:

So we're looking for an infinity bigger than N and smaller than R. What are the ramifications of the existence/nonexistence of such an infinity? You said or someone else did that Hilbert wanted the CH proven/disproven ASAP.

So we're looking for an infinity bigger than N and smaller than R. — Agent Smith

If you think the continuum hypothesis is false, then you think there is a set with cardinality strictly greater than the cardinality of N and strictly less than the cardinality of R.

For more about the ramifications, you should read more about set theory, to gather the whole context. But perhaps the more pertinent matter is what are the ramifications of added set theoretic principles that would prove CH and of added set theoretic principles that would disprove CH. This is at the heart of contemporary set theory research. I couldn't do the subject justice in posts (and I'm not expert enough anyway).

Yeah, Hilbert enjoined mathematicians to either prove or disprove CH. Godel did half by proving that ZFC doesn't disprove CH and Cohen did the other half by proving that ZFC doesn't prove CH. So a kind of "stalemate": Hilbert's challenge can't be answered on the terms Hilbert had in mind. This is a fascinating, profound and remarkable intellectual turn of events.

Danke!

So it's like I can't prove whether Jesus existed or not from the fact that green light has a wavelength of 555 nm.

No, it's not like that. You say it is only because you think being a wiseracre suits you.