Number Theory question?

I started off this question, full of confidence in giving an answer to your question. After spending an inordinate amount of time deleting, rewriting, rethinking, deleting (again!), I reluctantly came to realise that I cannot give a satisfactory answer.

However, (at the risk of being shot down ignominiously in flames), I will offer this

First, I assume that we agree that the set of real numbers consists of zero-dimensional points on the number line.

Consider the function

y: R -> k

where

y(x) = k

and k is some constant.

Now, for each point x in the (open) interval ]a, b[, since the two lines y = 0 and y = k are parallel we can rightly say that

the 'length' of the line representing the interval ]a, b[ is equal to the length of the line between the points

y(a) and y(b)

That is, the lengths of the two lines are both |b - a|

But what about y = mx for some real number m?

Generally, for m<>0 the length of the line between y(a) and y(b) is

√((b - a)² + (mb - ma)²) > |b - a|

But y = mx is a bijective function.

Let m increase (negatively or positively) as much as you like.

There is still one-to-one correspondence between the intervals ]a, b[ and ]y(a), y(b)[ even though the lengths of the lines differ.

Surely this implies that there is no effect at all on the number density by 'squeezing' the set of reals into any arbitrary (open) interval ]a, b[

Or am I being naïve?

I think that the density of the numbers between 0 and 1 is the same as the density of all of the real numbers.

Look in here for further information- http://en.wikipedia.org/wiki/Cardinality_of_the_co...

You can't do that because you propose to put the infinite into the finite.