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Mathematicians Measure Infinities, Find They’re Equal

quantamagazine.org

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Re: Mathematicians Measure Infinities, Find They’re Equal

#91

Post by clintonc »

samoright wrote: post #86
clintonc wrote: post #84

Earlier quoted context omitted.

The article is misleading on this point. The Continuum Hypothesis says that it is undecidable in the standard axioms whether there is a cardinality between countable and the continuum (i.e., the cardinality of the real). This work shows that p and t are different, which also means that t is of greater cardinality than the reals.

What is p and t? You say that this work shows that p and t are different. What two entities are proven to be equal then in this work?

p and t are implicitly defined cardinalities (i.e., the smallest cardinalities with given properties). See Definition 1.1 of the source article (https://arxiv.org/pdf/1208.5424.pdf)

Re: Mathematicians Measure Infinities, Find They’re Equal

#92

Post by spi »

abnry wrote: post #46

I am having trouble grasping the result. As far as I understand it, any proof of any result has to be with respect to some axiom system. The continuum hypothesis result is that you cannot prove a infinity exists between the naturals and the real numbers under the usual axiom system A. How that was proved, I don't know, and I also don't know what axiom system was used in that proof. If the axiom system was also A, the…

Same disclaimer as v64: my background is also mathematics, but not in this field.

As he says, what was proved so far is that: a_0 The continuum hypothesis claims that they are all equal; that's been proved by Cohen to be independent of what you call "A" (and is generally called ZFC), by "Forcing" which is not easy, technical, and has won Cohen a Fields medal... And you're right in saying that what is proved in an axiom set is true in any larger axiom set.

However, the main point is that "we cannot have pAnd no, the statement "The continuum hypothesis is undecidable in ZFC" is proved within ZFC, otherwise it wouldn't have been proved at all... To give a trivial counter-example, as a corollary of Cohen's proof, the axiom choice "ZFC+CH" (ZFC + continuum hypothesis) is as valid an axiom set as ZFC is. Within it, you can certainly prove that "ZFC implies CH" since... CH is simply true.

Re: Mathematicians Measure Infinities, Find They’re Equal

#93

Post by abnry »

clintonc wrote: post #84
abnry wrote: post #46

I am having trouble grasping the result. As far as I understand it, any proof of any result has to be with respect to some axiom system. The continuum hypothesis result is that you cannot prove a infinity exists between the naturals and the real numbers under the usual axiom system A. How that was proved, I don't know, and I also don't know what axiom system was used in that proof. If the axiom system was also A, the…

The article is misleading on this point. The Continuum Hypothesis says that it is undecidable in the standard axioms whether there is a cardinality between countable and the continuum (i.e., the cardinality of the real). This work shows that p and t are different, which also means that t is of greater cardinality than the reals.

Oh, thank you. So the article just threw in Continuum Hypothesis because different infinities 'n stuff.

But what is confusing to me is that the article suggested p and t are defined as collections of subsets of the natural numbers. The power set of the natural numbers has the same cardinality as the reals, right? Since p and t are both subsets of this power set, they must have cardinalities below the cardinality of the reals.

Re: Mathematicians Measure Infinities, Find They’re Equal

#94

Post by abnry »

samoright wrote: post #79

> In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size I thought there are only two types of infinity and Cantor already proved that they are different. * Uncountable infinity which is the cardinality of the set of real numbers * Countable infinity which is the cardinality of the set of integers Cantor has alr…

If you take the power set of an infinite set, you /always/ get a cardinality bigger than the original set. So there are an infinite number of infinities.

Re: Mathematicians Measure Infinities, Find They’re Equal

#95

Post by SilasX »

xamuel wrote: post #59
burkaman wrote: post #49

Earlier quoted context omitted.

Why do you think this is clickbait? The headline accurately describes the content of the article, which as far as I can tell accurately describes one result of this paper. Clickbait doesn't mean any interesting headline, it means a misleading and purely attention-grabbing headline.

It's like if someone proved P=NP and the magazine title was: "Mathematicians Measure Difficulty Levels, Find They're Equal."

Clickbait refers to vague titles that also overhype the story. Your example, if anything, underhypes it.

Re: Mathematicians Measure Infinities, Find They’re Equal

#96

Post by abnry »

spi wrote: post #92
abnry wrote: post #46

I am having trouble grasping the result. As far as I understand it, any proof of any result has to be with respect to some axiom system. The continuum hypothesis result is that you cannot prove a infinity exists between the naturals and the real numbers under the usual axiom system A. How that was proved, I don't know, and I also don't know what axiom system was used in that proof. If the axiom system was also A, the…

Same disclaimer as v64: my background is also mathematics, but not in this field. As he says, what was proved so far is that: a_0 The continuum hypothesis claims that they are all equal; that's been proved by Cohen to be independent of what you call "A" (and is generally called ZFC), by "Forcing" which is not easy, technical, and has won Cohen a Fields medal... And you're right in saying that what is proved in an axi…

Thanks.

I think the idea I had in my head was this: If p=t or p!=t is decidable in ZFC+AoC, then it must be that p=t because otherwise that contradicts that the continuum hypothesis is undecidable.

So a better way to sum up this result is: "The question of whether p=t or p!=t is decidable in ZFC+AoC. And oh yeah, they happen to be equal."

How does that sound?

Re: Mathematicians Measure Infinities, Find They’re Equal

#97

Post by [deleted] »

kgwgk wrote: post #55
threatofrain wrote: post #41

Earlier quoted context omitted.

Another thing which has bothered me in the past was that a set of 2-tuple integers could be mapped by a set of 1-tuple integers, seemingly without any information loss.

The "seemingly" suggests that you remain unconvinced. If you define for example some kind of distance between an arbitrary (x,y) and (0,0), for any (x,y) you can easily count how many points are closer to the origin. There may be ties but it's easy to define a rule to break them. This means you can order all the pairs of integers: (0,0), (1,0), (0,1), (-1,0), (0, -1), (1,1), (1,-1), (-1,-1), (-1,1), (2,0),....

[deleted]

Re: Mathematicians Measure Infinities, Find They’re Equal

#98

Post by jtanderson »

xamuel wrote: post #18

Actual article: https://arxiv.org/pdf/1208.5424.pdf Great results within a very narrow field, which quantamagazine leverages into a clickbaity title.

I was very confused by this. The paper linked by OP (the one I also had in mind when I started the article) is from 2012, but the article is written as if the results are recent (specifically it says from 2016! Admittedly, only five years for such a big result in that field is probably still "recent" to experts; but probably not to the "pop science" readers.

Furthermore, the lack of an actual citation was very disturbing, since if anybody did want to read the actual paper, they had to google for terms that would lead to basically incomprehensible search results.

Re: Mathematicians Measure Infinities, Find They’re Equal

#99

Post by [deleted] »

abnry wrote: post #96
spi wrote: post #92

Earlier quoted context omitted.

Same disclaimer as v64: my background is also mathematics, but not in this field. As he says, what was proved so far is that: a_0 The continuum hypothesis claims that they are all equal; that's been proved by Cohen to be independent of what you call "A" (and is generally called ZFC), by "Forcing" which is not easy, technical, and has won Cohen a Fields medal... And you're right in saying that what is proved in an axi…

Thanks. I think the idea I had in my head was this: If p=t or p!=t is decidable in ZFC+AoC, then it must be that p=t because otherwise that contradicts that the continuum hypothesis is undecidable. So a better way to sum up this result is: "The question of whether p=t or p!=t is decidable in ZFC+AoC. And oh yeah, they happen to be equal." How does that sound?

[deleted]

Re: Mathematicians Measure Infinities, Find They’re Equal

#100

Post by mturmon »

logfromblammo wrote: post #78
kmill wrote: post #63

Earlier quoted context omitted.

That infinitesimal is not a real number. To simplify a little, a real number is something which is the limit of a sequence of rational numbers. Or, given an error bound 1/n, you can write down a rational number within 1/n of the real number. Two real numbers are the same if the difference between their approximations converges to 0 as n gets arbitrarily large. A number with infinitely many zeros after the decimal poi…

Fine, then. Just use the smallest positive nonzero real number, instead. Edit: I really hope I'm not the only one laughing.

"...only one laughing..."

You are illustrating the reason that mathematicians generally disparage the concept of an "infinitesimal", when it's used as proof rather than conceptual aid. (Yes, I know about https://en.wikipedia.org/wiki/Non-standard_analysis)

Not only do you get wrong conclusions, you get tedious, hard-to-adjudicate arguments.

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