Thread:HMcCoy/@comment-29331244-20160414012121/@comment-24808864-20160414144627

"Infinity" is not a number that appears on the number line, so if you are making a program about the "number line" it would not formally include infinity. Rather, on the number line, infinity is a concept ("at the end").

The things you are reading about using \omega are ordinals. Techincally the set \omega is defined to be {1,2,3,...}. In this sense, and using the Successor operation, the ordinal \omega +1 is just {1,2,3,...,\omega} or \omega \cup {\omega}.

The concept of "size" is different and is called cardinality. As an easy example consider the fact that {1,2,3} and {a,b,c} are the same "size" since we can define a map that sends 1<->a, 2<->b and 3<->c. The cardinal assigned to \omega is \aleph_0 which is why sometimes you will see references to \omega being the smallest infinity.

Beyond that though, there are certainly different sizes of infinity. For example, one can "count" the natural numbers, but one cannot "count" the real numbers.

The graphic you are displaying shows what are called "Large Cardinal Assumptions" so that is using the cardinal concept of number, not ordinal.

By the way, you see "Woodin" in the middle? That is named after Hugh W. Woodin. I met him when I moved to LA in 2000 because he was here for a conference!

I am trying to keep things "short and simple" because the concepts you are asking about are pretty intense! I will make a blog on the wiki so that you and I can discuss it further for your project if you would like!