Question from a Texan Textbook

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Re: Question from a Texan Textbook

#101  Postby Evolving » Sep 14, 2015 2:27 pm

I wouldn't say it was "trivial". It took Cantor to think of it!

Obviously, once you've seen it, it's natural to think, "I could have thought of that!"

In fact: see my sig.
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Re: Question from a Texan Textbook

#102  Postby scott1328 » Sep 14, 2015 2:34 pm

crank wrote: I had difficulties with some bad arguments I tried to counter

What bad arguments?

crank wrote:with some wrong assumptions on my part, and other issues.

this is true

crank wrote: Chaitin's claim, if I understand it, is that reals are infinite precision,

That wasn't his claim.
crank wrote:but that isn't a mathematical term, so whatever, since uncomputable numbers will have an infinite, non-repeating decimal representation, they in effect contain infinite information,

uncomputable numbers have no decimal representation whatsoever, if they did, they would be computable. Uncomputables contain no information whatsoever.
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Re: Question from a Texan Textbook

#103  Postby Pulsar » Sep 14, 2015 2:38 pm

scott1328 wrote:Here is a diagram of how the mapping works Nicko.
Image

There's a more elegant way, by using Stern-Brocot numbers:

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Re: Question from a Texan Textbook

#104  Postby crank » Sep 14, 2015 5:33 pm

scott1328 wrote:
crank wrote: I had difficulties with some bad arguments I tried to counter

What bad arguments?

crank wrote:with some wrong assumptions on my part, and other issues.

this is true

crank wrote: Chaitin's claim, if I understand it, is that reals are infinite precision,

That wasn't his claim.
crank wrote:but that isn't a mathematical term, so whatever, since uncomputable numbers will have an infinite, non-repeating decimal representation, they in effect contain infinite information,

uncomputable numbers have no decimal representation whatsoever, if they did, they would be computable. Uncomputables contain no information whatsoever.

You got one right. Uncomputables have decimal representation, you can't right one out, but you can begin to generate one, in principal, the easiest would be a random number generator [not the computer algorithmic ones, true random number generators].

Chaitin:
But when you work on a computer, the last thing on earth you're ever going to see is a real number, because a real number has an infinite number of digits of precision,
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Re: Question from a Texan Textbook

#105  Postby scott1328 » Sep 14, 2015 6:00 pm

crank wrote:
scott1328 wrote:
crank wrote: I had difficulties with some bad arguments I tried to counter

What bad arguments?

crank wrote:with some wrong assumptions on my part, and other issues.

this is true

crank wrote: Chaitin's claim, if I understand it, is that reals are infinite precision,

That wasn't his claim.
crank wrote:but that isn't a mathematical term, so whatever, since uncomputable numbers will have an infinite, non-repeating decimal representation, they in effect contain infinite information,

uncomputable numbers have no decimal representation whatsoever, if they did, they would be computable. Uncomputables contain no information whatsoever.

You got one right. Uncomputables have decimal representation, you can't right one out, but you can begin to generate one, in principal, the easiest would be a random number generator [not the computer algorithmic ones, true random number generators].

Chaitin:
But when you work on a computer, the last thing on earth you're ever going to see is a real number, because a real number has an infinite number of digits of precision,

Chaitin was clearly speaking in the context of computing real numbers. He was not making the claim you have put into his mouth.


But to say you can begin writing an uncomputable number out is false because all you have done with your random number generator is generate a number that equally identifies countably infinite rational number, countably infinite computable numbers, and uncountably infinite non-computable numbers.

In what sense of the word have you identified any number whatsoever with your random number generator?
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Re: Question from a Texan Textbook

#106  Postby crank » Sep 14, 2015 6:26 pm

scott1328 wrote:
Chaitin was clearly speaking in the context of computing real numbers. He was not making the claim you have put into his mouth.


But to say you can begin writing an uncomputable number out is false because all you have done with your random number generator is generate a number that equally identifies countably infinite rational number, countably infinite computable numbers, and uncountably infinite non-computable numbers.

In what sense of the word have you identified any number whatsoever with your random number generator?


Chaitin is clearly claiming what I said he claimed. All you have to do is watch the video and hear him for yourself, obviously even a quote from a Chaitin paper isn't good enough. You are getting unreasonable in the extreme. How does the context in this case in any way change the meaning of what he explicitly says? "a real number has an infinite number of digits of precision," How much more blatant can something get?

I never said I could identify one, from my very first post I said as much. I never said the random number generator would identify one, only that it could start to generate one. Put another way, stipulate an infinite sequence of random digits, that's an uncomputable. We know what .33... means, we also know what an infinite string of digits means, and we know that all reals will be represented by some combination of those digits. That is exactly what Cali's definition is. This is obvious. Your objections are all way off, either strawman or irrelevant, or plain wrong.
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Re: Question from a Texan Textbook

#107  Postby scott1328 » Sep 15, 2015 1:18 am

I have listened to the video and read the paper you quoted. And in a video and paper aimed at popularizing deeply technical ideas, I would expect such language and metaphors to make his ideas accessible. I do not believe he is claiming what you think he is.

And if he is claiming that reals have infinite precision, then he is playing fast and loose with the mathematics.

Chaitin does make a particularly egregious error in the paper you cited

So in fact there are more uncomputable reals than computable reals. From Cantor's theory of infinite sets, we see that the set of uncomputable reals is just as big as the set of all reals, while the set of computable reals is only as big as the set of whole numbers. The set of uncomputable reals is much bigger than the set of computable reals.

#{uncomputable reals} = #{all reals} = ℵ1
#{computable reals} = #{computer programs} = #{whole numbers} = ℵ0
ℵ1 > ℵ0

In the text I've quoted Chaitin asserted that the cardinality of the reals, typically referred to as C, is equivalent to Aleph-1. This assertion, called the Continuum Hypothesis is formally undecidable in standard formulations of mathematical set theory, that he would assert it as true in his popular paper, leads me to doubt the seriousness of this entire paper.
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Re: Question from a Texan Textbook

#108  Postby crank » Sep 15, 2015 10:09 am

scott1328 wrote:I have listened to the video and read the paper you quoted. And in a video and paper aimed at popularizing deeply technical ideas, I would expect such language and metaphors to make his ideas accessible. I do not believe he is claiming what you think he is.

And if he is claiming that reals have infinite precision, then he is playing fast and loose with the mathematics.

Chaitin does make a particularly egregious error in the paper you cited

So in fact there are more uncomputable reals than computable reals. From Cantor's theory of infinite sets, we see that the set of uncomputable reals is just as big as the set of all reals, while the set of computable reals is only as big as the set of whole numbers. The set of uncomputable reals is much bigger than the set of computable reals.

#{uncomputable reals} = #{all reals} = ℵ1
#{computable reals} = #{computer programs} = #{whole numbers} = ℵ0
ℵ1 > ℵ0

In the text I've quoted Chaitin asserted that the cardinality of the reals, typically referred to as C, is equivalent to Aleph-1. This assertion, called the Continuum Hypothesis is formally undecidable in standard formulations of mathematical set theory, that he would assert it as true in his popular paper, leads me to doubt the seriousness of this entire paper.


I think he's claiming what he says he's claiming, I can't read his mind, can you? You directly assert what is contradictory to exactly what he says.

Now, the ℵ1 vs C question, what do you think more likely, that a mathematician of Chaitin's caliber made a mental slip, or that he is unaware of such a profound issue in the area of infinities?

Plus, I went to wiki, they say it quite differently from what you say:

\aleph_1 is the cardinality of the set of all countable ordinal numbers, called ω1 or (sometimes) Ω. This ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore \aleph_1 is distinct from \aleph_0. The definition of \aleph_1 implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between \aleph_0 and \aleph_1. If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus \aleph_1 is the second-smallest infinite cardinal number.


I appears you can use the terminology the way Chaitin did, there are different ways of using the aleph-numbers depending on what assumptions you want to use.

You obviously know a hell of a lot more maths than I do, but you keep making boneheaded errors, and instead of trying to discuss the issues, you mostly have tried to make me look stupid and wrong. I had a perfectly reasonable conversation with Thommo, we came to agreement on almost everything. I said upfront, I didn't know the proper terminology, but what I was saying was plain enough for the most part, and we were able to clear up the confusion. I was wrong about maths treating integers differently than reals, but only a little, the mind surely does treat them differently. I was wrong about using 'will' when the probability was 100%, that was a technical error. Everything else there was no real error in what I said.

The infinite precision thing is a good example. You need infinite information to specify an uncomputable number, that is what he's talking about. You can't express one, no one disputes this and everyone states this. To express any real, you need an infinite string of coefficients, these are seldom explicitly stated, like they are in Cali's definition, but they are there, by implication, by convention, by whatever, but they are there, in order to ensure the repeating fraction or the zeros. This is not claiming you need infinite information, like I've explained repeatedly. That is only true of the uncomputables. You keep objecting to the 'information' ideas, and the 'infinite precision' idea, but give no reasoning, and you are at odds with most physicist and mathematicians. In order to distinguish between two numbers, you can't ever know if they are different without this infinity of digits. You can't even make sense of these things without first having to just think of the number and its infinity of digits as being there, all at once, like on a number line. As any two point get closer and closer together, you have to look closer and closer to see if there is a difference, in other words, go further and further out to the right of the decimal. In effect, take in the whole thing at once to see if the numbers are the same, but this requires all the digits, an infinity of them. You can't get the granularity fine enough, in other words, look at enough digits, to ever guarantee distinguishing two uncomputables. I can't think of any more ways to explain what is obvious to me.
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Re: Question from a Texan Textbook

#109  Postby ED209 » Sep 17, 2015 11:15 pm

Evolving wrote:I wouldn't say it was "trivial". It took Cantor to think of it!

Obviously, once you've seen it, it's natural to think, "I could have thought of that!"

In fact: see my sig.


Sure it's not trivial to discover, but once someone else had thought of it and you see it, it's trivial to grasp. Even memorable - and that's speaking as someone who's forgotten more mathematics than most people ever learn (but only because I learned a bit more than most people do and subsequently forgot all of it :lol: ) - making it a good thing to teach kids. The exam question was inexcusable, and can only be the result of a system that is staggeringly broken.
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Re: Question from a Texan Textbook

#110  Postby VazScep » Sep 18, 2015 5:03 am

The enumerability of the rationals is easy to discover. The hard part is coming up with the idea of one-one correspondences and then convincing others that they are a really big deal. Until then, you might not have any means to even phrase the problem, let alone get others to care about its answer.

That was the much harder task for Cantor and Dedekind, since it amounts to inventing a new field of mathematics, for which they get plenty of credit.
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Re: Question from a Texan Textbook

#111  Postby LjSpike » Apr 21, 2016 4:09 pm

Newmark wrote:But it's factually wrong. There is a one-to-one correspondence between the integers and the rationals, as they are both countable sets of the same cardinality (aleph-0)*. Infinities may be a bit counter-intuitive, as a set may contain as many elements as a proper subset of said set, but anyone who can't be bothered to look this up has no business writing a math book.

For anyone who's interested, wiki on countable sets is a good place to start...

[EDIT] *Or rather, it's the other way around, but the cardinality of the sets should given you a hint even if you don't understand the proof.

I'm way out of my league, and technically we do have infinite numbers of each one, but that is a bit of a cheats way out of it isn't it, just to write down that numbers go on infinitely...
See this is a secondary schoolers approach to your problem:
There is an infinite number of integers, as fractions normally have the denominator and numerator represented by an integer
So we could take the area between 0 and 1.
We have 2 integers here, 0 and 1. However, we have an infinite number of real rational numbers between these integers, as we have 1 over every integer that exists (so 1 over x infinite times over) but we also have 1 being able to be replaced by anything up to x and above 0 and it still being between 0 and 1 or 0 and 1. So we have an infinite number of integers, but we have an infinity*infinity number of rationals between each increment in integers.

So the relationship is 1:infinity*infinity

This is working of simpler sequences solutions / proofs logic. (I've done so many of these sorts of questions and variants on a GCSE level its slightly dull at times now... "When will both trains be back at the station?")

Any sequence is infinite, unless it has a specified reason not to be, so to prove it on an easier level look at a single setup in the first sequence (integers) and look at the equivalent numbers of steps up in the second sequence, and you give yourself the ratio (or relationship) in a 1:x format. Its a fairly universal solution I've found, as both trains will be in the station when the multiple of x is an integer. :coffee:
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Re: Question from a Texan Textbook

#112  Postby scott1328 » Apr 21, 2016 5:40 pm

LjSpike wrote:
Newmark wrote:But it's factually wrong. There is a one-to-one correspondence between the integers and the rationals, as they are both countable sets of the same cardinality (aleph-0)*. Infinities may be a bit counter-intuitive, as a set may contain as many elements as a proper subset of said set, but anyone who can't be bothered to look this up has no business writing a math book.

For anyone who's interested, wiki on countable sets is a good place to start...

[EDIT] *Or rather, it's the other way around, but the cardinality of the sets should given you a hint even if you don't understand the proof.

I'm way out of my league, and technically we do have infinite numbers of each one, but that is a bit of a cheats way out of it isn't it, just to write down that numbers go on infinitely...
See this is a secondary schoolers approach to your problem:
There is an infinite number of integers, as fractions normally have the denominator and numerator represented by an integer
So we could take the area between 0 and 1.
We have 2 integers here, 0 and 1. However, we have an infinite number of real rational numbers between these integers, as we have 1 over every integer that exists (so 1 over x infinite times over) but we also have 1 being able to be replaced by anything up to x and above 0 and it still being between 0 and 1 or 0 and 1. So we have an infinite number of integers, but we have an infinity*infinity number of rationals between each increment in integers.

So the relationship is 1:infinity*infinity

This is working of simpler sequences solutions / proofs logic. (I've done so many of these sorts of questions and variants on a GCSE level its slightly dull at times now... "When will both trains be back at the station?")

Any sequence is infinite, unless it has a specified reason not to be, so to prove it on an easier level look at a single setup in the first sequence (integers) and look at the equivalent numbers of steps up in the second sequence, and you give yourself the ratio (or relationship) in a 1:x format. Its a fairly universal solution I've found, as both trains will be in the station when the multiple of x is an integer. :coffee:

This is merely incoherent wibble. The rationals are countably infinite. This means that there is a one to one correspondence between every natural number and ever rational number. Read the thread.
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Re: Question from a Texan Textbook

#113  Postby VazScep » Apr 21, 2016 7:01 pm

scott1328 wrote:This is merely incoherent wibble.
To be fair, LJSpike admitted it was a GCSE perspective (one you typically obtain by studying maths between 14-16 years old in the UK). And maths is taught pretty shit here IMO.
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Re: Question from a Texan Textbook

#114  Postby crank » Apr 21, 2016 7:02 pm

But it's about the area between 0 and 1. So you can immediately see we're dealing with imaginary numbers. But this is out of my area. Though I would note the need to distinguish between those represented by 'i' and those by 'j', they're as different as mathematics and engineering.
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Re: Question from a Texan Textbook

#115  Postby VazScep » Apr 21, 2016 7:59 pm

I've got a library in a couple of programming languages where, given an enumeration of data, you can apply a binary operation to generate an enumeration of all binary applications of that data. This can be defined in a nice way that makes it an example of a very common structure that occurs in algebra, and there it diverges slightly from what Cantor was doing, but it does allow you to enumerate the rationals by saying:

Code: Select all
liftA2 (%) nats nats


where (%) takes a numerator and denominator and gives you the fraction, and nats is the enumeration of naturals.
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Re: Question from a Texan Textbook

#116  Postby LjSpike » Apr 22, 2016 3:30 pm

VazScep wrote:
scott1328 wrote:This is merely incoherent wibble.
To be fair, LJSpike admitted it was a GCSE perspective (one you typically obtain by studying maths between 14-16 years old in the UK). And maths is taught pretty shit here IMO.

Yup. Anyway I could shorten my description now, which I shall do. After using a lot of arrows on paper to try and find the most efficient description.

Integers go on infinitely, we could just say that rationals go on infinitely too, but then again, were not giving a 'concrete figure' to either of them. So if we take an increment of 1 in the integers chain, we can then express all of the rationals in this chunk as
Y/X (Y over X). X is any number from 1 to infinity. Y is any number from X * the lower end of our 'chunk' of the integer chain, to infinity. So we can say X and Y both have a range of infinite size.
So to find every combination of them we find all of the possible values of X, which we know to be infinity. We then find all the possible values of Y for all the possible values of X, as we know Y can have a range of infinite size, but can also have a range of say 1, we can do X * Y, which is Infinity * (Infinity/2) as halfway between 1, and infinity is a half of infinity, which is still infinity (as infinity is infinitely large) so we could say that its infinity * infinity or infinity^2

So for a range of 1 in integers, we have a range of infinity^2 (infinity squared) of rationals.
If we were to choose a range of 2 in integers, we can simply extrapolate the pattern found, to be 2*infinity^2 in rationals. So for a range of infinite integers, we multiply infinity^2 by infinity, giving us infinity*infinity*infinity or infinity^3

So for the total amount of integers to rationals we can say the ratio is infinity:infinity^3
Simplifying this ratio down would divide it by infinity, as that is a common factor of both of them. So we then get 1 : infinity^2 (as anything divided by itself = 1, and dividing a^x by a results in a^(x-1))

I would personally be happy to leave it at that, but as we (theoretically) can't have a number bigger than infinity, we can handle this now at the end (just like rounding in a trig question, or such, you leave it till the end to gain the best accuracy).
So we can say, infinity^2, is still infinity as nothing can be larger than infinity, giving us the answer of 1:infinity.
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Re: Question from a Texan Textbook

#117  Postby VazScep » Apr 22, 2016 4:28 pm

No. As scott1328 says, you're incoherently wibbling, and should wait a bit until you've learned some real maths, which, sadly, doesn't happen these days until you get to a university.
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Re: Question from a Texan Textbook

#118  Postby scott1328 » Apr 22, 2016 4:39 pm

LjSpike wrote:
VazScep wrote:
scott1328 wrote:This is merely incoherent wibble.
To be fair, LJSpike admitted it was a GCSE perspective (one you typically obtain by studying maths between 14-16 years old in the UK). And maths is taught pretty shit here IMO.

Yup. Anyway I could shorten my description now, which I shall do. After using a lot of arrows on paper to try and find the most efficient description.

Integers go on infinitely, we could just say that rationals go on infinitely too, but then again, were not giving a 'concrete figure' to either of them. So if we take an increment of 1 in the integers chain, we can then express all of the rationals in this chunk as
Y/X (Y over X). X is any number from 1 to infinity. Y is any number from X * the lower end of our 'chunk' of the integer chain, to infinity. So we can say X and Y both have a range of infinite size.
So to find every combination of them we find all of the possible values of X, which we know to be infinity. We then find all the possible values of Y for all the possible values of X, as we know Y can have a range of infinite size, but can also have a range of say 1, we can do X * Y, which is Infinity * (Infinity/2) as halfway between 1, and infinity is a half of infinity, which is still infinity (as infinity is infinitely large) so we could say that its infinity * infinity or infinity^2
https://en.wikipedia.org/wiki/Countable_set
https://en.wikipedia.org/wiki/Countable_set

So for a range of 1 in integers, we have a range of infinity^2 (infinity squared) of rationals.
If we were to choose a range of 2 in integers, we can simply extrapolate the pattern found, to be 2*infinity^2 in rationals. So for a range of infinite integers, we multiply infinity^2 by infinity, giving us infinity*infinity*infinity or infinity^3

So for the total amount of integers to rationals we can say the ratio is infinity:infinity^3
Simplifying this ratio down would divide it by infinity, as that is a common factor of both of them. So we then get 1 : infinity^2 (as anything divided by itself = 1, and dividing a^x by a results in a^(x-1))

I would personally be happy to leave it at that, but as we (theoretically) can't have a number bigger than infinity, we can handle this now at the end (just like rounding in a trig question, or such, you leave it till the end to gain the best accuracy).
So we can say, infinity^2, is still infinity as nothing can be larger than infinity, giving us the answer of 1:infinity.


Read this Wiki article, it explains how the rationals can be put into a one-to-one correspondence with the natural numbers.

http://en.wikipedia.org/wiki/Countable_set
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Re: Question from a Texan Textbook

#119  Postby LjSpike » Apr 23, 2016 5:17 pm

scott1328 wrote:
LjSpike wrote:
VazScep wrote:
scott1328 wrote:This is merely incoherent wibble.
To be fair, LJSpike admitted it was a GCSE perspective (one you typically obtain by studying maths between 14-16 years old in the UK). And maths is taught pretty shit here IMO.

Yup. Anyway I could shorten my description now, which I shall do. After using a lot of arrows on paper to try and find the most efficient description.

Integers go on infinitely, we could just say that rationals go on infinitely too, but then again, were not giving a 'concrete figure' to either of them. So if we take an increment of 1 in the integers chain, we can then express all of the rationals in this chunk as
Y/X (Y over X). X is any number from 1 to infinity. Y is any number from X * the lower end of our 'chunk' of the integer chain, to infinity. So we can say X and Y both have a range of infinite size.
So to find every combination of them we find all of the possible values of X, which we know to be infinity. We then find all the possible values of Y for all the possible values of X, as we know Y can have a range of infinite size, but can also have a range of say 1, we can do X * Y, which is Infinity * (Infinity/2) as halfway between 1, and infinity is a half of infinity, which is still infinity (as infinity is infinitely large) so we could say that its infinity * infinity or infinity^2
https://en.wikipedia.org/wiki/Countable_set
https://en.wikipedia.org/wiki/Countable_set

So for a range of 1 in integers, we have a range of infinity^2 (infinity squared) of rationals.
If we were to choose a range of 2 in integers, we can simply extrapolate the pattern found, to be 2*infinity^2 in rationals. So for a range of infinite integers, we multiply infinity^2 by infinity, giving us infinity*infinity*infinity or infinity^3

So for the total amount of integers to rationals we can say the ratio is infinity:infinity^3
Simplifying this ratio down would divide it by infinity, as that is a common factor of both of them. So we then get 1 : infinity^2 (as anything divided by itself = 1, and dividing a^x by a results in a^(x-1))

I would personally be happy to leave it at that, but as we (theoretically) can't have a number bigger than infinity, we can handle this now at the end (just like rounding in a trig question, or such, you leave it till the end to gain the best accuracy).
So we can say, infinity^2, is still infinity as nothing can be larger than infinity, giving us the answer of 1:infinity.


Read this Wiki article, it explains how the rationals can be put into a one-to-one correspondence with the natural numbers.

http://en.wikipedia.org/wiki/Countable_set


I looked into the puzzle further, and it isn't a one-to-one correspondence. It is infinity:infinity, but they are different infinities, thus you can't simplify them down to 1:1.
Another puzzle including different infinities (and a rather nice one I like myself) is the Hotel Paradox: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
(On youtube somewhere TedEd has a brilliant video explaining it).

Again my solution, this time displayed via inequalities (compressed to a single equation without paragraphs of description)
Integers:Rationals = n:(n*x=<y=<x)/1=<x=<infinity)
n = Integer
x = unknown (denominator of rational)
y = unknown (numerator of rational)


This solution handles it without resorting both sides to infinity, presenting an at least very hard to express solution (if not impossible to express) as both infinities are different infinities, not equal infinities. (My best guess at displaying it would be)
inf1:inf2 where inf1 < inf2 for integers:rationals
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Re: Question from a Texan Textbook

#120  Postby VazScep » Apr 23, 2016 6:04 pm

Wibbly.
Here we go again. First, we discover recursion.
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