From Wikipedia says "There are infinite ordinals as well: the smallest infinite ordinal is ω, which is the order type of the natural numbers (finite ordinals) and that can even be identified with the set of natural numbers."
This make the following questions:
- If $\omega$ is identified with the set of natural numbers, why it is the smallest ordinal? Should not the set of natural numbers without $\{1\}$ or $\{0\}$be smaller?
Later it says "Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω"
- If so, what comes after all even numbers? Should not that ordinal be smaller?
My intuition is just broken on this.
In ordinal arithmetic they say:
0 < 1 < 2 < 0' < 1' < 2' < ...
0 < 1 < 2 < ... < 0' < 1' < 2'
After relabeling, the former just looks like ω itself, i.e. 3 + ω = ω.
- I wonder, why they think such relabeling should be allowed and represent the same cardinal number? This grossly contradicts my intuition.
For instance, why they claim that the sets {0,1,2,3,...} and {1,2,3,4,...} correspond to the same cardinal number?
- Why there cannot be well-ordered ordinals in both directions, such as {1,2,3,4,...}, {...,-4',-3',-2',-1',0',1',2',3',4',...}?
Why when attaching a set to the right to form a new cardinal number, it always has the smallest element? This is not required by ordering.
