Mathematicians Measure Infinities, Find They’re Equal • Hacker News Board

Re: Mathematicians Measure Infinities, Find They’re Equal

#61

Post by bllguo » Wed, Sep 13, 2017, 4:40 PM UTC

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.

That is extremely disingenuous of you. He's being downvoted for his massive arrogance.

Re: Mathematicians Measure Infinities, Find They’re Equal

#62

Post by thriftwy » Wed, Sep 13, 2017, 4:42 PM UTC

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.

Re: Mathematicians Measure Infinities, Find They’re Equal

#63

Post by kmill » Wed, Sep 13, 2017, 4:42 PM UTC

logfromblammo wrote: post #53

lisper wrote: post #12

Earlier quoted context omitted.

> 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 equ…

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 point is within 1/n of zero no matter the n. Therefore the number is zero.

This is like how 0.9999... is 1. The reason is that 1-0.999... is within 1/n of 0 no matter the n.

Suppose you had an infinitesimal epsilon (outside the real numbers --- this is fine, and people do this). How many times are you planning on adding it to itself? To get any actual real number, you are going to have to add it to itself well more than countably many times, though I'm not sure this makes much sense.

Re: Mathematicians Measure Infinities, Find They’re Equal

#64

Post by yorwba » Wed, Sep 13, 2017, 4:43 PM UTC

logfromblammo wrote: post #53

lisper wrote: post #12

Earlier quoted context omitted.

> 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 equ…

> Now go back to zero and ...

You can't ever finish adding epsilon infinitely often, so everything after that is dead code. There is no "now" to speak of.

The same thing comes into play when you speak about an infinite number of zeroes after the decimal point and before the last digit. There is no last digit. You could have numbers with two "ends", but the so-called real numbers are different. They stretch infintely to the right, without end in sight.

There are also p-adic numbers, which stretch infinitely to the left, but they behave very different from real numbers. https://en.wikipedia.org/wiki/P-adic_number

Re: Mathematicians Measure Infinities, Find They’re Equal

#65

Post by lisper » Wed, Sep 13, 2017, 4:46 PM UTC

logfromblammo wrote: post #53

lisper wrote: post #12

Earlier quoted context omitted.

> 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 equ…

> 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.

Re: Mathematicians Measure Infinities, Find They’re Equal

#66

Post by andrewla » Wed, Sep 13, 2017, 4:46 PM UTC

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.

I agree that the downvoting is unfortunate. I guess the assumption is that the post is simply a troll, which seems to be backed up by some of the down-thread replies. Even so, the original post just seems like a list of common misunderstandings about Cantor's notion of infinity and how it corresponds to the way that we use the word in colloquial use.

In defense of the GP, I think they were simply thinking of numbers in [0,1). If you write them backwards, it almost seems like it would work:

    1    ->  .1
    2    ->  .2
    ...
    9    ->  .9
    10   ->  .01
    11   ->  .11
    12   ->  .21
    ...
    3124 ->  .4213
    ...

Done!

Unfortunately, you are either stuck with the fact that some numbers (even simple rational ones) do not have a finite decimal expansion. In most formal proofs of the diagonal theorem, we use the infinite representation without trailing 0s (using trailing 9's instead) to force uniqueness, which makes it even trickier.

Or you're stuck trying to assume that there are natural numbers with an infinite representation, so the decimal representation of the rational number 1/3 would correspond to an actual natural number.

Re: Mathematicians Measure Infinities, Find They’re Equal

#67

Post by deckar01 » Wed, Sep 13, 2017, 4:46 PM UTC

agf wrote: post #50

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…

No, this is incorrect. You can map the set of rational numbers to the set of natural numbers. The set of reals in that range is shown to be uncountably infinite by Cantor diagonalization. https://www.reddit.com/r/explainlikeimfive/comments/1r5aya/e...

Ah, I was thinking of the rationals. Thanks for the ELI5 link.

Re: Mathematicians Measure Infinities, Find They’re Equal

#68

Post by schoen » Wed, Sep 13, 2017, 4:47 PM UTC

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.

Three things:

First, this research didn't show that all infinities are equal, it just showed that two particular ones are equal. Indeed, in Cantor's formulation, the powerset of any set is larger than the original set, which immediately shows that there is no largest infinite set.

Second, this research wasn't about ℵ₀ and c at all, but about another pair.

Third, the issue about ℵ₀ and c isn't about whether they're equal, but whether there is or isn't a third infinity that's larger than ℵ₀ and smaller than c. It's already known that both answers (yes and no) are consistent with standard mathematical axioms. Edit: this result does have some connection with the continuum hypothesis, but I don't know exactly what that connection is. However, these researchers definitely didn't claim either to prove or disprove the continuum hypothesis.

Re: Mathematicians Measure Infinities, Find They’re Equal

#69

Post by gwf » Wed, Sep 13, 2017, 4:48 PM UTC

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 be considered to be exponentially more dense than the naturals.

Some intuition: while there may be N unique stocks on the NYSE, there could in theory be 2^N unique mutual funds that are based on the different combinations of stocks.

Re: Mathematicians Measure Infinities, Find They’re Equal

#70

Post by joshumax » Wed, Sep 13, 2017, 4:49 PM UTC

arstin wrote: post #57

I only had one one set theory course so can't vouch for the details, but this seemed like a well done article on a subject that could easily have been more eyeglazingly superficial or obscure.

Much of my research on number theory and primality relies on a deep understanding in axiomatic set theory. Based on other news sites I've read that attempt to cover naive set theory, I was pleasantly surprised at the article's fairly decent coverage on the topic.