User blog comment:Scorcher007/About Cofinality, sipmly/@comment-39541634-20200102080903

Here's a simple intuitive explanation for cofinality.

'''The cofinality of an ordinal X is the minimum number of steps needed to "count" up to X when skipping is allowed. The rules are:'''

1. You are allowed jumps of any size.

2. You are not allowed to name X itself (or ordinals greater than X).

3. For every ordinal under X, you must either name it explicitly or skip over it by naming something bigger.

This alone might seem confusing, but it will be clear after you give some examples.

You start with the finite numbers: To reach (say) 42, we can simply say "41". We've named 41 itself, and we've also skipped over all the smaller numbers. So we are done.

The same is true for all other finite numbers. We can just name the number immediately before it, and that's it.

In other words: The cofinality of any finite number is 1.

So far that's very boring. With infinite ordinals, however, things begin to get interesting:

Take ω, for example. Since there's no "largest number", there's no way to reach ω in any finite number of steps! No matter how much we skip, our list will still be infinite. Moreover, it is the simplest kind of infinite list. It could be something like this:

1,7,42,4791828,a googol,...

Or even:

1,2,3,4,...

Either way, it's a single unbroken infinite row of numbers. That's precisely the structure of ω itself, so when we skip-count to ω, we can't do any better then ordinary counting. The minimum number of steps is ω itself. Hence we say: The cofinality of ω is ω.

Now what about ω+1?

This one is simple. We can skip-count to ω+1 by simply naming "ω". By saying "ω" we've skipped over the finite numbers and we also took care of ω itself, so we're done. The cofinality of ω+1 is 1.

Same is true for ω+2 (just name "ω+1"). In fact, it is easy to see that if X=Y+1 when Y is any ordinal, we can skip-count to X by naming "Y". Hence:

The cofinality of every successor ordinal is 1.

So what about non-successor ordinals (which are also known as "limit ordinals")? Well, as it turns out, the cofinality of an ordinal must be a member of a very special list of ordinals. That list begins like this:

1,ω,ω1,ω2,...

Where ω1 is "the first uncountable ordinal". What that means doesn't matter at this point. All you need to know is that ω1 is very very large. In fact, it is bigger then the ordinal associated with any googological notation.

So let us, for the moment, limit our discussion to ordinals less then ω1. If X < ω1, what could be it's cofinality?

Since ordinary counting is permissible in our "skip-counting" game, it is clear that the cofinality of X cannot be greater than X itself''. ''

So if X < ω1, the cofinality of X can only have two values: 1 or ω.

More specifically if X is a successor ordinal (an ordinal of the form Y+1) then the cofinality of X would be 1. If X is a nonsuccessor ordinal (a limit ordinal) then the cofinality of X would be ω. In the latter case, we can skip-count to X with a simple infinite sequence of the form:

X1, X2, X3, ...

And any such sequence is called a fundamental sequence of X. As you probably know, these are immensely useful in googology.

Now, what happens when we do reach ω1? Well, ω1itself has a cofinality of ω1. What this means, basically, is this:

'''ω1 is so huge, that it is impossible to skip-count up to it with an infinite sequence. In fact, no method of skip-counting will get you to ω1 any "faster" then listing every single ordinal ordinals below it!'''

If that doesn't boggle your mind, read the above statement again. It is completely impossible to "visualize" an ω1-length sequence. The good news is that we don't need to visualize it in order to work with it: Just treat it as a black box that contains all the familiar ordinals (and then some) with "ω1" written on it.

So... what's next? Of-course, ω1+1. This has a cofinality of 1, because we can skip-count to it by simply naming "ω1".

Also, for any X<ω1we can skip-count to ω1+X in the same manner we would have counted to X. If we skip-counted to X like this:

X1, X2, X3, ...

Then we can skip-count to ω1+X like this:

ω1+X1, ω1+X2, +X3, ...

In other words: If X<ω1 is a limit ordinal, then the cofinality of ω1+X is ω.

This goes on for a while, until we reach ω1x2. That's the second ordinal with cofinality of ω1. As we continue up the ordinal ladder, ω1 becomes increasingly common as a cofinality. At this point, we can classify the ordinals into three groups:

1. Ordinals with a cofinality of 1 (the successor ordinals)

2. Ordinals with a cofinality of ω (limit ordinals with can skip-counted to with an ordinary sequence)

3. Ordinals with a cofinality of ω1 (limit ordinals that can only be skip-counted to with an ω1-length sequence).

At least, this is true until we reach ω2, which - just like ω1 -has itself as it's cofinality. Just like ω1, it is impossible to skip-count to ω2 in a way that will get you there "faster" then listing every single ordinal below ω2.

From here on, we have 4 possible cofinality values: 1, ω ,ω1 and ω2.

And of-course this never ends. Eventually we'll have ω3, ω4, ... . Each of these is a milestone where a new possible cofinality is added to the mix.

You can even have ordinals as subscripts, but you gotta be careful about it! For example the cofinality of ωω is not ωω as you might expect. It's cofinality is actually just ω, because we can skip-count to ωω like this:

ω1, ω2, ω3, ...

The cofinality of ωω+1, though, is ωω+1. In general, if the index is a successor ordinal, then the entire ordinal would have itself as its cofinality. Otherwise... well, things can get quite complicated. So this may well be a good point to end this explanation.