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Mathematicians Measure Infinities, Find They’re Equal

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Re: Mathematicians Measure Infinities, Find They’re Equal

#31

Post by moyix »

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…

Based on your other replies, I suspect it's not worth engaging, but for anyone else reading, here's a great article by a math journal editor writing about all of the different attempts to disprove Cantor's diagonal argument he received and tracing out some common mistakes he saw:

http://www.logic.univie.ac.at/~ykhomski/ST2013/Hodges.pdf

Re: Mathematicians Measure Infinities, Find They’re Equal

#32

Post by thriftwy »

v64 wrote: post #29
thriftwy wrote: post #23

So they basically found a way to represent any real number as an unique sequence of bits? Doesn't sound too hard when spelled like that. I could probably do that, win some prize (not anymore). Do they have anything else like that?

If you've got some free time, can you look into whether or not the real part of every non-trivial zero of the Riemann zeta function is 1/2?

Can you reword this one as a discrete math primer either?

Re: Mathematicians Measure Infinities, Find They’re Equal

#33

Post by mturmon »

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.

Your answer is buried, which is too bad, because it is the one-sentence rebuttal to the construction above.

Re: Mathematicians Measure Infinities, Find They’re Equal

#34

Post by peteretep »

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.

Re: Mathematicians Measure Infinities, Find They’re Equal

#35

Post by littlestymaar »

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…

The error in your reasoning lies here :

> Without the decimal point these real numbers just become natural numbers

This is wrong because irrational numbers have an infinite number of digit, if you remove the «dot» you end up with a number infinitely long, which is not a natural number. □

Edit: the set of natural numbers is infinite, which means it can contain arbitrarily big numbers, yet it doesn't contain «numbers» with an infinite amount of digits.

Re: Mathematicians Measure Infinities, Find They’re Equal

#36

Post by deckar01 »

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 dimensionality of the members.

Edit: real -> rational

Re: Mathematicians Measure Infinities, Find They’re Equal

#38

Post by v64 »

thriftwy wrote: post #32
v64 wrote: post #29

Earlier quoted context omitted.

If you've got some free time, can you look into whether or not the real part of every non-trivial zero of the Riemann zeta function is 1/2?

Can you reword this one as a discrete math primer either?

Let d(n) be the sum of the divisors of n (for example, d(6) = 12 because 1, 2, 3, and 6 divide 6 and 1+2+3+6 = 12). Let h(n) be the nth harmonic number, that is, h(n) = 1 + 1/2 + 1/3 + ... + 1/n. Let ln(x) be the natural logarithm and let e be, well, e.

Show that the inequality:

d(n) holds for all n >= 1.

Best I can do, hope that works! [1]

[1] http://www.theoremoftheday.org/NumberTheory/RobinLagarias/To...

Re: Mathematicians Measure Infinities, Find They’re Equal

#39

Post by ColinWright »

dogecoinbase wrote: post #27
zelah wrote: post #5

Earlier quoted context omitted.

It is possible. The inverse of m is called q. The function q takes an infinite sequence of coin flips and one by one changes every heads to 1 and every tails to 0. The infinite string of zeros and ones is then prepended with a 1 and interpreted as a transfinite natural number in binary notation. This will not take forever because each change will only take half as long as the previous one. A transfinite natural numbe…

Hundreds of thousands of mathematicians are wrong. No. No natural number has an infinite number of digits, and you are wrong. If you weren't, of course, it would be easy to prove me wrong -- simply name the natural number to which the successor function is applied that results in a "transfinite" number, whatever that is.

Be careful, there are such things as transfinite numbers, and they are well-founded. They let us do wonderful things like transfinite induction, and to prove, for example, that Goodstein's Theorem is true, even though it's unproveable in Peano Arithmetic.

But transfinite numbers are not "natural numbers", they are not in the set N, they don't have infinitely many digits, and in the context of this thread, they are a red herring.

Re: Mathematicians Measure Infinities, Find They’re Equal

#40

Post by zardo »

zelah wrote: post #8
gus_massa wrote: post #6

Earlier quoted context omitted.

[I'll try a non technical argument to convince you. It's also not a complete argument, so you must think about this for a while.] > If you believe n is a natural number then you must also believe that n x 10 is a natural number. One more digit! If you interpret the natural number in this way, the important property is that they have only a finite amount of "interesting" digits. Almost all their digits are zero. You c…

>I's much easier to consider the infinite strings of digits like "0.765653625367523765..." or "0.5265362556..." or "0.000073468763478..." and also the one with repetitions like "0.0006767000000..." or "0.0072257822222222...". This is essentially a copy of the real number, but in this copy "0.2999999999999..." is different from "0.300000000000000..." This trick makes much easier to prove that the diagonal ´+1 in each…

>Transfinite natural numbers must exist otherwise you do not have an infinite set

Transfinite

This word does not mean what you think it means. I'm not entirely sure what you think it means, but it's definitely not what it means to everyone else.

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