Posted:

**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.

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.