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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

#101

Post by kmill »

logfromblammo wrote: post #76
lisper wrote: post #65

Earlier quoted context omitted.

> before the last digit, which is '1' No. There is no last digit. That's the whole point. If there were a last digit your argument would be correct, but there isn't, so it's not.

I don't see the problem with having a first digit and a last digit and an infinite number of digits in between. Edit: Infinitesimal divided by two is infinitesimal, in the same way that infinity multiplied by two is infinity. So 0.000...0001 / 2 = 0.000...0001 . Infinitesimal multiplied by any finite number is infinitesimal. Infinitesimal multiplied by infinity is every number in the interval from infinitesimal to in…

> Don't confuse the limitations on mathematical notation with a limitation on imagination

Good luck proving or calculating anything.

You can define "infinitesimal/2 == infinitesimal", but nothing good will come out of it. A definition is no good unless it lets you do something.

Letting e=infinitesimal, you have e/2==e, so e==2e so 0==2e-e so 0==e. This definition is inconsistent with being able divide by non-zero integers and subtraction.

Re: Mathematicians Measure Infinities, Find They’re Equal

#104

Post by logfromblammo »

kmill wrote: post #101
logfromblammo wrote: post #76

Earlier quoted context omitted.

I don't see the problem with having a first digit and a last digit and an infinite number of digits in between. Edit: Infinitesimal divided by two is infinitesimal, in the same way that infinity multiplied by two is infinity. So 0.000...0001 / 2 = 0.000...0001 . Infinitesimal multiplied by any finite number is infinitesimal. Infinitesimal multiplied by infinity is every number in the interval from infinitesimal to in…

> Don't confuse the limitations on mathematical notation with a limitation on imagination Good luck proving or calculating anything. You can define "infinitesimal/2 == infinitesimal", but nothing good will come out of it. A definition is no good unless it lets you do something. Letting e=infinitesimal, you have e/2==e, so e==2e so 0==2e-e so 0==e. This definition is inconsistent with being able divide by non-zero int…

That's not the definition, that's just what it does.

The definition of infinitesimal is "the smallest-magnitude number that is greater than zero". If you divide a finite number by infinity, infinitesimal is what you get, but don't go thinking that if you multiply it by infinity again that you will get the same number back, because you won't.

The floating point standard does not include a representation for infinitesimal, but an underflow now hints at its existence, instead of just going to zero.

It's probably easier to think of quantities like zero, one, infinity, and infinitesimal as the base vectors in mutually orthogonal dimensions. Their behaviors can be defined separately, such that whatever rules you choose for them can produce different types of math, perhaps useful for different purposes (or none beyond cranking out the dissertation), in the same way that slightly changing the Euclidian parallel lines property can produce elliptic and hyperbolic geometries.

Re: Mathematicians Measure Infinities, Find They’re Equal

#105

Post by throwawayjava »

jtanderson wrote: post #98
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 distur…

The work was first posted to arxiv in 2012. It can take a while until a result is understood by enough experts to confirm its correctness. Imagine if journalists wrote an article claiming P=NP is closed every time someone posted their attempt to arxiv.

Also, this work very recently won a prestigious award.

Also, result published ---> result accepted ---> journalist writes high-quality piece explaining result to lay people can take years.

> the lack of an actual citation was very disturbing

The first linked text in the article is a link to the arxiv paper.

Re: Mathematicians Measure Infinities, Find They’re Equal

#106

Post by adenadel »

zelah wrote: post #103

Please answer me this one question: What is the average number of bits that are necessary to represent an arbitrary natural number? If the average number of bits is finite then I will shut up!

This is equivalent to asking what is the average number of digits necessary to represent an arbitrary natural number. If you assume that this is a finite number you can quickly arrive at a contradiction.

Re: Mathematicians Measure Infinities, Find They’re Equal

#107

Post by jasomill »

gwf wrote: post #69
thriftwy wrote: post #62

BTW: Why then does wikipedia say that the cardinality of the set of all real numbers (denoted c and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted ℵ 0 'aleph-naught')? UPD: I get it, they are actually talking about a third set in the article in a way that's not immediately apparent.

The cardinality of the reals has been known to be strictly greater than the cardinality of the naturals since Cantor. What the Continuum Hypothesis considers is the cardinality of the reals and the cardinality of the power set of the naturals. The power set of another set is the set of unique subsets of the first set. If the first set has cardinality of N, then the power set has cardinality 2^N. Thus, the reals can b…

To be a bit more specific: the cardinality of the reals is equal to the cardinality of the power set of the naturals. This has also been known since the time of Cantor (read: late 19th century, at the dawn of set theory). CH states that no cardinal numbers exist between the cardinalities of the reals and the naturals[1].

Cantor believed CH to be true, but couldn’t prove it, and it was eventually (ca. 1960) proven to be independent of the usual (ZFC) axioms of set theory.

[1] Though CH could of course be stated in infinitely many equivalent ways, by replacing “reals” and “naturals” in this statement with any other pair of sets whose cardinalities are equal to those of the reals and naturals, respectively.

For (uncountably) infinitely many stupid examples, take any non-empty open subset of any connected, finite-dimensional (Hausdorff, second-countable) manifold with at least two points, and any infinite subset of the integers.

Re: Mathematicians Measure Infinities, Find They’re Equal

#108

Post by ColinWright »

davidivadavid wrote: post #102

Is it me or they forgot a step in the diagonalization argument? (adding/subtracting 1)

They say to change every number, and that's enough. Although they don't explicitly say so you can change each number in (nearly) any way you like.

Re: Mathematicians Measure Infinities, Find They’re Equal

#109

Post by gus_massa »

jtanderson wrote: post #98
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 distur…

In Math usually the peer review time is very long. Perhaps one or one and half year (unless you are unlucky and when you contact the editor after a year he realizes that one of the referees "crashed" and must be "reseted").

In Physics people gets mad if the referees+editor take more than two months.

Each branch of science has its own typical time.

Re: Mathematicians Measure Infinities, Find They’re Equal

#110

Post by ColinWright »

samoright wrote: post #81
gwf wrote: post #69

Earlier quoted context omitted.

The cardinality of the reals has been known to be strictly greater than the cardinality of the naturals since Cantor. What the Continuum Hypothesis considers is the cardinality of the reals and the cardinality of the power set of the naturals. The power set of another set is the set of unique subsets of the first set. If the first set has cardinality of N, then the power set has cardinality 2^N. Thus, the reals can b…

Can you simplify for us what exactly have the mathematicians in this article proven? I get that they have proven some infinity A = some infinity B but can you tell us what these A and B are. Also, isn't the cardinality of reals equinumerous with that of the cardinality of the power set of naturals?

Quoting from the article:

> Briefly, p is the minimum size of a collection of infinite sets of the natural numbers that have a “strong finite intersection property” and no “pseudointersection,” which means the subsets overlap each other in a particular way; t is called the “tower number” and is the minimum size of a collection of subsets of the natural numbers that is ordered in a way called “reverse almost inclusion” and has no pseudointersection.

In short, there are two infinities, p and t, that are implicitly defined by some characteristics. It was previously known that there was a relationship between them, but it was not suspected that there were, in fact, equal.

Considerable work is required to understand what these are, and what the result means.

And to answer your explicit question, yes, the reals can be put in 1-1 correspondence with 2^N, the power set of the naturals.

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