Please, I don't understand the concept of ordinal numbers; for example, since they describe infinities, how can I possibly tell which set has ordinal $\omega+1,\omega+2$ or even $\omega.2$ or $\omega^2$?
Thanks in advance.
Please, I don't understand the concept of ordinal numbers; for example, since they describe infinities, how can I possibly tell which set has ordinal $\omega+1,\omega+2$ or even $\omega.2$ or $\omega^2$?
Thanks in advance.
I like to imagine these ordinals embedded in a familiar place: $\Bbb Q$, the set of rational numbers. A good representative of $\omega$ is the sequence:
$$0,\frac12,\frac23,\frac34,\ldots$$
Conveniently enough, all of these numbers are less than $1$, so if you want a set representing the ordinal $\omega+1$:
$$0,\frac12,\frac23,\frac34,\ldots,1$$
Similary, $2\omega$:
$$0,\frac12,\frac23,\frac34,\ldots,1,1+\frac12,1+\frac23,1+\frac34\ldots$$
In this way, we can embed $n\omega$ into the rational numbers for any $n$. Playing this game in the spaces between every pair of natural numbers, we get $\omega^2$.
Can we go further? Yes, indeed. The function $x\mapsto\frac{x}{x+1}$ maps the interval $[0,\infty)$ injectively and monotically into the interval $[0,1)$. Applying that transformation to our $\omega^2$ sequence compresses the whole thing into the unit interval. Copying the whole thing, by adding $1$ to each number, then gives us $2\omega^2$. Placing copies in each interval, we can get $3\omega^2, 4\omega^2,\ldots \omega^3$. Fun, isn't it?
This process can be repeated as many times as you like, to obtain embeddings in the rational numbers of $\omega^n$ for any $n$. As for $\omega^\omega$.... if there's a good way to embed that into $\Bbb Q$, I'd love to see it!