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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

#141

Post by davidshepherd7 »

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…

There are infinite types of infinity not just two. The result is that they found that two types of infinity that were long assumed to be different turned out to be the same.

This is more interesting than it sounds because these infinities are in between (but possibly equal to one of) the size of the natural numbers and the size of the real numbers. There are hard limits on what can be known about such infinities in the usual mathematical framework (zfc): it's impossible to prove if an infinity which is strictly between the two exists. The hypothesis that no such​ infinity exists is called "the continuum hypothesis".

Writing this on mobile, but I hope I've made sense...

Re: Mathematicians Measure Infinities, Find They’re Equal

#142

Post by zelah »

cjslep wrote: post #20
zelah wrote: post #3

I find Cantor's diagonal argument unconvincing. The claim is that there are more real numbers in the range from zero to one than there are natural numbers. To see that this is false simply realize that you don't actually have to write a decimal point to specify the real numbers in this range. Without the decimal point these real numbers just become natural numbers. Can a rational person believe that there are infinit…

There is one English definition of infinity, but when applied to ordinals ("degrees of freedoms"), you get different mathematical concepts of infinity. Vsauce's explanation is approachable: https://youtu.be/SrU9YDoXE88

Thanks - that was very fun!

Re: Mathematicians Measure Infinities, Find They’re Equal

#143

Post by myren »

kmill wrote: post #48
zelah wrote: post #3

I find Cantor's diagonal argument unconvincing. The claim is that there are more real numbers in the range from zero to one than there are natural numbers. To see that this is false simply realize that you don't actually have to write a decimal point to specify the real numbers in this range. Without the decimal point these real numbers just become natural numbers. Can a rational person believe that there are infinit…

I think I figured it out: you must be one of the aliens predicted by the downward Löwenheim–Skolem theorem! If we could have a model of set theory (a set1 of all set2s, where a set1 is a set in our set theory and a set2 a modeled set, like an interpreter), which is necessarily infinitely large, then the downward Löwenheim–Skolem theorem implies there is a model that is only countably infinite. There is a model of the…

I don´t think this is an alien that applied downwards Löwenheim-Skolem to the reals. Rather I think this is an alien which applied upwards Löwenheim-Skolem to peano arithmetic. I think this because the alien talks about infinite natural numbers, aka non-standard natural numbers. It has used upwards Löwenheim-Skolem to get a model of peano arithmetic that has the cardinality of the continuum, and then there is of course a bijection between our reals and the thing that it calls the natural numbers. I see no sign that it´s using a non-standard model of the reals, but lots of signs that it´s using a non-standard model of the naturals.

Re: Mathematicians Measure Infinities, Find They’re Equal

#144

Post by myren »

zelah wrote: post #3

I find Cantor's diagonal argument unconvincing. The claim is that there are more real numbers in the range from zero to one than there are natural numbers. To see that this is false simply realize that you don't actually have to write a decimal point to specify the real numbers in this range. Without the decimal point these real numbers just become natural numbers. Can a rational person believe that there are infinit…

"If there really are an infinite number of natural numbers then some of them must be of a transfinite number of digits or else you would be including numbers in the list more than once."

Theorem: There are no infinite natural numbers.

Proof: By Induction:

Induction Start: 0 is finite.

Induction Step: If n is finite then n+1 is finite.

The principle of induction then tells us: All natural numbers are finite.

If all natural numbers are finite, then there are no infinite natural numbers.

End of proof.

Re: Mathematicians Measure Infinities, Find They’re Equal

#145

Post by zimpenfish »

gumby wrote: post #116

Photos of humans in an article about math are pretty useless but nevertheless I find it interesting that the article included a photo of the male collaborator, Saharon Shelah, and another male mathematician, but not one of the female collaborator, Maryanthe Malliaris.

There’s a comment from the author below to say she specifically requested no photo.

Re: Mathematicians Measure Infinities, Find They’re Equal

#146

Post by chriswarbo »

lisper wrote: post #140
zelah wrote: post #124

Earlier quoted context omitted.

Why is it that Cantor can do an infinite procedure of diagonalization but I can't? If he can diagonalize then I can too. Is it possible that Cantor's "algorithm" is not really an algorithm? Knuth says an algorithm must be correct and must also terminate. Regardless, I have a lot to ponder.

Cantor is playing by the exact same rules that you are. His burden is: given an integer i, produce in finite time the i'th digit of a real not in your list. (He can't produce the whole thing in finite time because it's infinite, obviously.) He does this by using your algorithm to produce the i'th digit of the i'th row and adds one to it (mod 10).

> given an integer i, produce in finite time the i'th digit of a real not in your list

Yes; to be clear, my point is simply that there are two variants of this task:

- Given an integer i and our list (or the infinite procedure that generates it), then the task is easy since we can diagonalise.

- Given an integer i and no knowledge of our list, the task becomes impossible, since the supposed "real not in your list" is actually independent of the list (by definition); hence we're free to choose any list we like, including waiting until after the supposed counter-example has been generated, and sticking that at the head of our list.

Re: Mathematicians Measure Infinities, Find They’re Equal

#147

Post by logfromblammo »

kmill wrote: post #132
dragonius wrote: post #131

Earlier quoted context omitted.

He's already answered you: If you are going to allow this kind of notation, then 0.000...1 = 0 is equivalent to 0 0.000...9 = 9 x 0 = 0 0.000...1 + 0.000...9 = 0 + 0 = 10 x 0 = 0.000...10 (0.000...10) / 2 = 0/2 = 0 You're getting hung up on notation and missing the concept.

Neither has logfromblammo answered me nor am I hung up on notation. The notation is only incidental. They are claiming that it is a well-defined number system with numbers "having a first digit and a last digit and an infinite number of digits in between." I say show that it works. You are saying the way this works is to disregard the digits after the infinitely many digits. Sure, that would make a consistent system.…

I'm not going to answer you, as I haven't made any claim that I care to defend. I made one little post in support of its parent, and people crawled out of the woodwork to tell me how wrong I am, and apparently try to convince me that infinitesimals are not allowed in serious mathematics, or at least not allowed in the way I was trying to use them.

And now every post I have made in this thread tree is getting downvoted. So I'm out. Y'all can argue about nothing--and nearly-nothing--by yourselves.

Re: Mathematicians Measure Infinities, Find They’re Equal

#148

Post by logfromblammo »

jtc1983 wrote: post #115
mturmon wrote: post #100

Earlier quoted context omitted.

"...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.

I agree with your thought, and that of the majority of mathematicians for many years, that infinitesimals are better construed heuristically than literally. You mention that you know of non-standard analysis and indicate that it's irrelevant. Though I don't know why you think this, I agree with you. I just wanted to plug non-standard analysis as both mathematically interesting and also very useful. One can jettison t…

Thanks for your support, but I never said anything about non-standard analysis. I made one post in support of its parent, and people jumped on it to say how wrong I am, as apparently I accidentally stumbled over a sore spot in mathematics. I don't know why infinitesimals apparently aren't allowed, but the tone around here has convinced me to not care.

And now every post I have made in this thread tree is getting downvoted. So I'm out. Y'all can argue about nothing--and nearly-nothing--by yourselves.

Re: Mathematicians Measure Infinities, Find They’re Equal

#149

Post by burkaman »

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 article links to their journal publication from last year: http://www.ams.org/journals/jams/2016-29-01/S0894-0347-2015-...

Re: Mathematicians Measure Infinities, Find They’re Equal

#150

Post by kd0amg »

chriswarbo wrote: post #146
lisper wrote: post #140

Earlier quoted context omitted.

Cantor is playing by the exact same rules that you are. His burden is: given an integer i, produce in finite time the i'th digit of a real not in your list. (He can't produce the whole thing in finite time because it's infinite, obviously.) He does this by using your algorithm to produce the i'th digit of the i'th row and adds one to it (mod 10).

> given an integer i, produce in finite time the i'th digit of a real not in your list Yes; to be clear, my point is simply that there are two variants of this task: - Given an integer i and our list (or the infinite procedure that generates it), then the task is easy since we can diagonalise. - Given an integer i and no knowledge of our list , the task becomes impossible, since the supposed "real not in your list" i…

What does this prove though? That no specific real is guaranteed to be left out of every list? Was that ever in dispute?
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