>>13272537Let's call an ordinal number p>1 a prime if for all ordinals x,y<p, we have xy < p.
Then 2, 3, 5, ... are primes.
2 is the first prime, 3 is the 2rd prime.
The infinieth prime, i.e. w'th prime is w.
w is a prime, since for all a,b<w, ab is finite, hence also <w.
What is the next prime after w? Not w+1, since w*w > w.
It's not hard to show that all w is the only prime countable ordinal.
The next prime after w is w_1, the first uncountable ordinal, since the product of two countable ordinals is countable.
Similarly the next one is w_2, then w_3 and so on. These are the alephs, of which there are infinite number, larger than any set.
We can also give a different definition for an ordinal to be a prime. Definition 2: an ordinal p is a prime if for all ordinals a,b, ab=p => a = p or b=p. Again all the usual numbers 2,3,5 are primes. Is w a prime? We can express w as 2*w or 100*w, but still one of the factors is w. A general argument shows w is indeed a prime. So is w+1. However, w*2 is not. So we get much more primes than with our previous definition. But still the infinieth (w-th) prime is w.
