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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

#81

Post by samoright »

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…

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?

Re: Mathematicians Measure Infinities, Find They’re Equal

#82

Post by kmill »

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.

What would that be?

If c>0 were smallest, then c/2 would be smaller and still positive!

Re: Mathematicians Measure Infinities, Find They’re Equal

#83

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…

There's not a problem if you can answer this: What is 0.000...1 + 0.000...9? (The 1 and 9 are digits after infinitely many zeros.)

You are not allowed to say "undefined" if these are real numbers, because you are supposed to be able to add any two real numbers.

You are also not allowed to say 0.000...10 since that changes the place values.

Well, you are allowed to say 0.000...10 if you are imagining a real number is a pair (normal real number, natural number). If this is what you are imagining, then there's not a problem if you can answer this: What is the value of 0.000...1 divided by two?

Re: Mathematicians Measure Infinities, Find They’re Equal

#84

Post by clintonc »

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.

Re: Mathematicians Measure Infinities, Find They’re Equal

#85

Post by samoright »

bjl wrote: post #80
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…

Not all uncountable infinities are the same cardinality.

So does this article prove that two uncountable infinities are equal? Can you tell us precisely what two entities have been proven equal in this article?

I understand aleph-0, aleph-1 and 2^aleph-0 and I also understand that if continuum hypothesis is true, then aleph-1 = 2^aleph-0. Is this what these mathematicians have proven, or have they proven something else?

Re: Mathematicians Measure Infinities, Find They’re Equal

#86

Post by samoright »

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.

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?

Re: Mathematicians Measure Infinities, Find They’re Equal

#87

Post by lisper »

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…

You can do that, but then the "last digit" doesn't behave the way you intuitively expect it to. For example, 0.0...1 is exactly equal to 0 for the same reason that 0.999... is exactly equal to 1. So you can't add them up to get a non-zero number.

The reason that adding numbers with a finite number of zeros before the first non-zero digit works to give you any number is that that carries go off to the left. But if there are an infinite number of zeros before the first non-zero digit then there will remain an infinite number of zeros after every addition. The carries can only "overcome" a finite number of zeros.

Re: Mathematicians Measure Infinities, Find They’re Equal

#88

Post by bjl »

samoright wrote: post #85
bjl wrote: post #80

Earlier quoted context omitted.

Not all uncountable infinities are the same cardinality.

So does this article prove that two uncountable infinities are equal? Can you tell us precisely what two entities have been proven equal in this article? I understand aleph-0, aleph-1 and 2^aleph-0 and I also understand that if continuum hypothesis is true, then aleph-1 = 2^aleph-0. Is this what these mathematicians have proven, or have they proven something else?

The paper proves that the smallest cardinality of a set of integers (such that every finite subset has infinite intersections and has no almost-intersections) is equal to the smallest cardinality of a tower.

Re: Mathematicians Measure Infinities, Find They’re Equal

#89

Post by [deleted] »

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…

[deleted]

Re: Mathematicians Measure Infinities, Find They’re Equal

#90

Post by [deleted] »

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…

[deleted]
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