Radical Edward said in post #3 :

well in terms of set theory, there are the same number of odd numbers as there are naturals, since you can create a 1:1 correlation. however you can't do this with primes so far as I am aware

 

has my avatar been corrupted for everyone else too?

 

heh

 

I was reading about that last month or so, when I drifted off topic.

 

Here's the site i was reading: http://mathforum.org/isaac/problems/prime1.html

 

Most notably:

Theorem: There are infinitely many prime numbers.

 

As far as your question goes, I'm not really sure, but I suppose the theorem implies something . I hope that site helps.

 

As for the avatar? blike said it had something to do w/ the forum shift.

:/

I have been avidly learning set theory for the past day and a half, and I was wondering if anyone knows if the set of prime numbers is smaller than the set of natural numbers. I would hypothesise that it is, and I have been trying to think of methods to prove it but I am perplexed... is there a method for this?

 

thanks

sounds a bit like trying to find the ratio of primes to naturals. Like yourself, I`de go with primes being in a lesser quantity.

Man, You guys get some neat questions!

I base my reasoning upon this, and this only: all Prime numbers are ODD, natural numbers can be either (ODD/EVEN), so for a start you`re down to 50%.

I know of at least 1 number that is ODD but is NOT a prime. therefore I`de conclude that there are less primes than naturals.

well in terms of set theory, there are the same number of odd numbers as there are naturals, since you can create a 1:1 correlation. however you can't do this with primes so far as I am aware

 

has my avatar been corrupted for everyone else too?

pass? been so long since ya said anything and all that, but it still looks ok to me

If you can prove that there's not a bijection between the two sets, then you can prove that they don't have the same cardinality.

 

At a guess, I don't think there's a bijection, but it seems like a little bit of a git to prove.

 

And your avatar is corrupted for me.

Euclid proved there are infinitely many primes ... are you

familiar with his proof?

 

Then you can put the primes in 1-1 correspondence with the

natural numbers, with p(n) being the nth prime (so the function

maps 1 <-> 2, 2 <-> 3, 3 <-> 5, and so on.

 

Does that help?

Well, the cardinality (a kindof posh way of saying 'size' but not that exact) of a set can be different even if the set is infinite. For example, the 'size' of the set of natural numbers is different from that of the rationals and reals.

Also I'm not sure whether that bijection is actually a proper one, you'd have to give a definitive proof.

Let us clear some things up.

 

Two sets have te same cardinality iff there is a bijection between them.

 

There is a bijection between N and the set of primes in N, and further there is a bijection between N and any infinite subset of N. There is also a bijection with Q.

 

Sets that biject with N are called countable,

 

The set of Reals is not countable, nor is the power set of N.

I would reason as follows :

 

Since there are infinitely many prime numbers and since they are a subset of N, the set of prime numbers is clearly a countably infinite set and has thus the same cardinality as N.

 

Mandrake