Actually you are both right -- omega and omega + 1 (or omega^2) have the same 'number' of elements (there is a one to one correspondence) but the correspondence will not be (can not be) order-preserving. Ordinals are a natural extension of the natural numbers. If you take the natural numbers and add a new element that's bigger than all of them then you get a new structure (called omega + 1) which has the same size as the normal natural numbers, but has one new element which is order-distinguishable from all our normal numbers. Some ordinals, which correspond to a jump in 'size' (like 2^omega) are special and called cardinal numbers.
1 + omega == omega < omega + 1 < omega^2
http://en.wikipedia.org/wiki/Ordinal_number