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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

#51

Post by jerf »

peteretep wrote: post #34
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…

What a shame you're getting downvoted, simply because you're wrong. > Without the decimal point these real > numbers just become natural numbers. If your argument is true, presumably you could write a simple program that would generate all the real numbers with a single, infinite loop? I wonder how you'd manage to generate 0.1 and 1.0 with your scheme.

"What a shame you're getting downvoted, simply because you're wrong."

It's not simply because of being wrong. Being wrong is a hazard to your karma, yes, even sometimes just asking questions can be a hazard (which I dislike and do what little I can to fight, but it's still obviously true). But there's a much bigger hazard being invoked here.

Re: Mathematicians Measure Infinities, Find They’re Equal

#52

Post by Zaak »

empath75 wrote: post #15
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…

> 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. There are no natural numbers with infinite digits, so this is not correct.

> There are no natural numbers with infinite digits, so this is not correct.

Exactly. If you allow an infinite number of digits you wind up with the p-adic numbers, which are uncountable just like the reals.

Re: Mathematicians Measure Infinities, Find They’re Equal

#53

Post by logfromblammo »

lisper wrote: post #12
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…

> Without the decimal point these real numbers just become natural numbers. No, they don't, because the vast majority of them have an infinite number of digits to the right of the decimal point. That's the key: there are more numbers with an infinite number of non-zero digits (the reals) than there are numbers with a finite number of non-zero digits (the naturals). > The problem with Cantor's argument comes down to t…

Start with zero. Add an infinitesimal epsilon an infinite number of times. Now go back to zero and subtract the same epsilon an infinite number of times. You have now traversed all the real numbers.

The decimal representation of the epsilon has an infinite number of zeroes after the decimal point and before the last digit, which is '1'. So if you were to just chop off the leading zero and decimal point to make an equivalence with natural numbers, the epsilon is as much a representation of the natural number 1 as 0.1 or 0.01 or 0.001 or 0.0001 . Infinite real number equivalents to one natural number.

Re: Mathematicians Measure Infinities, Find They’re Equal

#54

Post by [deleted] »

protonfish wrote: post #44
deckar01 wrote: post #36

I did not know that this was an unsolved problem. It is quite paradoxical. You can map the subset of rational numbers in the range [0, 1) to the set of natural numbers, so there are an infinite number of disjoint subsets of the rational numbers that map to the natural numbers. Intuitively a 2nd dimension of infinity should be bigger, but the count of a set is defined to be one dimensional regardless of the dimensiona…

How? Which real number maps to the natural number 42?

[deleted]

Re: Mathematicians Measure Infinities, Find They’re Equal

#55

Post by kgwgk »

threatofrain wrote: post #41
deckar01 wrote: post #36

I did not know that this was an unsolved problem. It is quite paradoxical. You can map the subset of rational numbers in the range [0, 1) to the set of natural numbers, so there are an infinite number of disjoint subsets of the rational numbers that map to the natural numbers. Intuitively a 2nd dimension of infinity should be bigger, but the count of a set is defined to be one dimensional regardless of the dimensiona…

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),....

Re: Mathematicians Measure Infinities, Find They’re Equal

#56

Post by yorwba »

iamlucaswolf wrote: post #28
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…

Phew, you have some courage, questioning the foundations of modern Mathematics in a place like this. But I can relate to your concerns about Cantor's argument. When I first heard it, it also felt artificial and unconvincing to me. What helped me (as with many proofs and concepts in Math) was an image, a visual metaphor if you like. Imagine a very, very large paper on which you place infinitely many dots in a grid. Th…

That's a very nice intuition, but for the wrong concept. What you have been describing is the difference between a dense set (almost no holes) and a nowhere-dense set (holes everywhere).

It turns out that there is a nowhere-dense set that is still uncountably infinite: https://en.wikipedia.org/wiki/Cantor_set

Re: Mathematicians Measure Infinities, Find They’re Equal

#58

Post by 0xBABAD00C »

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 find that people who question Cantor's (obviously correct) theorems and concepts seem to share commonalities with people questioning Einstein's (obviously correct) theories and concepts. Let's just say if Cantor's last name was Rasmussen and the infinities weren't indexed Alephs, I wouldn't expect people to get their panties in a bunch over abstract math, and care so much about proving him wrong, a fraud, a lunatic, etc (without even a basic understanding of the subject).

It hits the subconscious strings of jealousy and mistrust of the majority towards the more successful almost-the-same-but-not-quite minority so very perfectly, especially when the subject matter put forward by the minority is cryptic or unintuitive at the first glance...

Just a theory :)

Re: Mathematicians Measure Infinities, Find They’re Equal

#59

Post by xamuel »

burkaman wrote: post #49
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.

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."

Re: Mathematicians Measure Infinities, Find They’re Equal

#60

Post by SubiculumCode »

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…

Congratulations. You hijacked a thread.
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