More ordinals and the uncountable realm

In my last post on this topic, I left us at gamma_0, or φ(1,0,0). Now, can we go deeper?

Well, obviously, yes. We can just keep going through the Veblen function. We can go to φ(1,0,1) or φ(2,0,0) or φ(w,0,0) or even φ(1,0,0,0), the Ackermann ordinal. We can do as many arguments as we like, of course… going to φ(1,0,0,0,0) or φ(1,0,0,0,0,0) and so forth

So what’s the limit of this? Well, it’s φ(1,0,0,0…) with ω 0’s. This is called the small Veblen ordinal (SVO). But we still have more ordinals to go. For instance, we can assume zeros and assign every entry a coordinate, like so:

φ((1,3),(2,0)) encoding φ(1,0,0,2). The 1 is in the third place and the 2 is in the 0th.

This leads to an obvious result:

SVO = φ((1,w)) — a 1 in the w-th position

Now we can continue this by putting larger ordinals in the coordinate:

φ((1,e0)), φ((1,Γ0)), etc

What about this though? What if we define an ordinal which is φ((1,itself))?

Well, that’s called the Large Veblen Ordinal.

There are still larger countable ordinals, but I’d like to veer off to…

Uncountable infinities

So, the thing is, all these infinities are dwarfed by something else.

The magnitude of the set of the real numbers!

Now, there’s a famous proof that there are more reals than integers. Google cantor’s diagonal proof if you haven’t heard of it. But in essence:

There are infinities which are not like ordinals. Ordinals are sort of like the “name” of a smaller set. These other infinities, cardinals, are the size of those sets. In fact, in math, “cardinality” means “the size of a set”. The cardinality of w is the same as that of w^2 which is the same as that of the LVO which is the same as that of the set of integers– all have cardinality Aleph-naught (henceforth N_0). N_0 is the only countable transfinite cardinal–all other transfinite cardinals are uncountable. But there are uncountable ordinals too. N_1, the next transfinite cardinal, is equal to the ordinal ω₁, which is often known as Ω in this context. And, as it will become relevant soon, there is also an N_a and an w_a for any ordinal a.

Ω is by definition larger than all countables. It’s also the first ordinal which has a “fundamental sequence” longer than normal w. In fact, let’s explain that since it’s greatly important.

Formally, this concept–the length of the fundamental sequence is called cofinality. The cofinality of an ordinal, call it α, is in essence, the length of the shortest sequence of ordinals that, for all ordinals less than α, names at least one ordinal of at least the same size, without naming α. The cofinality of the ordinal 63, for instance, is 1; the sequence being {62}. Note that the cofinality is not the sequence itself!

Now, there are a handful of ordinals (and cardinals– cardinals are also ordinals) called regulars. An ordinal is regular if it is equal to its cofinality (we can abbreviate “the cofinality of x” to “cof(x)”). For instance, 0 is regular, because there are no ordinals less than zero. Therefore, you have already named all the ordinals below zero without naming any at all! Therefore, 0 has cofinality 0. Therefore, 0 is regular. Similarly, 1 is regular, since its cofinality-prove it yourself!- is 1. w is regular as like all countable limit ordinals it has cofinality w (N_0 is also regular as it is equal to w). Ω is also regular, as it has no fundamental sequence (of length w). So is the cardinal N_1. So are N_2 and w_2. In fact, for finite n, N_n is regular. But, N_w is not regular. Why? Because it’s the limit of all N_n for finite n, giving it a fundamental sequence of length w! Lots of cardinals of the form N_(a+1), known as successor cardinals, are regular. But there don’t seem to be any regular cardinals of the form N_a for a limit ordinal a which is at least 1. So are there any uncountable regular cardinals like this?

Well, not provably with ZFC, the standard rules of math. These cardinals are called inaccessible cardinals. The smallest one is simply notated I.

And, after accessing the inaccessible, we can end this post.