[pherq]

The existence of a bijection between two sets is what "same size" means in set theory. Yes, there are non-prime integers, but you can establish a bijection between the two, so their cardinalities are equal (both have a cardinality of aleph zero).

The reals, on the other hand, cannot be placed in a bijection with the natural numbers, and there are therefore "more" reals than naturals (i.e. there is an injection from the naturals to the reals, but not from the reals to the naturals -- any function from reals to naturals must have some pair x ≠ y with f(x) = f(y)).

[taneq]

I didn't go far into set theory but, informally, it seems to me that for any given natural number N, there will be N natural numbers less than or equal to N (obviously) and P prime numbers less than or equal to N, and P < N. Doesn't taking the limit as N goes to infinity show that there are fewer primes than there are natural numbers? Where am I going wrong?

[chimeracoder]

It might be easier to think of it this way: Are there more natural numbers than there are even natural numbers?

The answer is no, as illustrated by the Hotel paradox[0], in which we have a infinitely many rooms and want to accommodate a (possibly infinite) number of guests.. To summarize: For any finite number of guests, you can always find an even-numbered room to correspond to that guest (assume the guests are numbered sequentially, then double their number and put them in the room that has that number.). This creates a one-to-one correspondence, which means that the sets are the same size.

You might say, 'well, that only works because we're dealing with a finite number of guests. But we're talking about infinity'. There are a few different ways of answering that question. In my opinion, the easiest way to look at it is to remember that there is no such number as 'infinity' - when we say 'infinity', we're really trying to express the concept of growing without bound. So, the above strategy (double the person's number) works for any arbitrarily large group. At no point does it stop working, even as the group size grows larger and larger, so we can say that the two sets have the same size.

On the other hand, we have no such strategy for putting every irrational number in one-to-one correspondence with the counting numbers. That proof is a little harder, and the analogy with the hotel guests breaks down, unfortunately, so it's a bit tougher to explain.

[0] https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand...

[masterzora]

> Where am I going wrong?

In short, infinities are complicated and intuition doesn't work well.

In slightly longer, you're talking about the difference of the rate of growth of two functions rather than actually about the cardinalities of the sets.

We define two sets as having the same cardinality when we can create a bijection between them. We can list the primes in order from smallest to largest and number them with the natural numbers. So we'll have 2 match up with 0, 3 with 1, 5 with 2, 7 with 3, etc. Every single prime number will correspond with a natural number AND every single natural number will correspond with a prime number, no exceptions. So they must be the same size. They are also the same size as the integers and the rational numbers but the set of real numbers is a bigger infinity.

[Retric]

You're confusing a classification system with size.

Is the set of Real Numbers larger, smaller, or the same size as the set of points in a finite 2d object? Can you setup a bijection in either direction?

[pherq]

Assuming you consider the dimensions/axes/whatever of the 2d space to be indexed by reals (which is conventional), then yes one can construct a bijection:

    - normalise x and y coordinates in the shape into the interval (0, 1)
    - interleave the bits of the normalised x and y coordinates

This gives a single real value in the interval (0, 1), which exists and is unique for every point in the space (so it is an injection), and it covers every real number in that interval (so it is a surjection).

This gives you a bijection between points in (a) 2-dimensional space and a segment of the real line (which, in turn, has a bijection with the whole real line if you want to specify that).

Once again, cardinality in set theory is based on injections and bijections. If there is an injection from X into Y, then Y is at least as big as X. If there is a bijection between them (i.e. injections in both directions), then they are the same size.

(Also, bijections are inherently bidirectional.)

[Retric]

You used a countable infinity to tile an uncountable one. The set of 0 to 1 line segments being a countable infinity. 0 to 1 maps 1, 1 to 2 maps to 2 ect.

[jbclements]

I believe you misread the text you're replying to. The bijection is between the points in the 2d space and the points in the line segment. Both of these are uncountably infinite.