But since you had such fun with ordinals here (and here and here), I better add that Ketonen and Solovay later gave a proof based on the ε_{0} stuff and the hierarchy of fast-growing functions. (The variation due to Loebl and Nešetřil is nice and short.) We should talk about this sometime! I wish I understood all the connections better. (Stillwell’s Roads to Infinity offers a nice entry point, though he does like to gloss over details.)

Thanks! Now I feel dumb, because I should have been able to figure that out. But that’s very helpful and reassuring.

]]>We can always have , by choosing to be as large as possible. (We can choose a maximal since the set of possible values is closed; but if then we can increase by one.)

The Veblen normal form for any strongly critical ordinal $\latex \alpha$ such as the Small Veblen Ordinal is .

]]>Got it! Thanks. Fixed.

By the way, Wikipedia says that in Veblen normal form we have

but I’m afraid we might only have

when reaches or exceeds the small Veblen ordinal, which is a fixed point of all the functions

What’s the Veblen normal form of the small Veblen ordinal? I feel I only understand Veblen normal form for ordinals below this.

]]>The paragraphs after that refer to as (and to as )

]]>If I said

someone might point out that I haven’t defined an infinite sum of ordinals, or even pointed to a definition.

Of course the definition of this infinite sum is that it’s the limit of

but that seems more complicated than saying is the limit of

]]>That might be more concise, but perhaps more confusing to the uninitiated. (I think it’s less clear that the ordinal defined by that expression is closed under omega exponentiation, for example.)

]]>Well, maybe it would be easier to say that ?

]]>Royce wrote:

The 1 should be replaced by 0 I think.

You’re right!

perhaps you could add some detail there?

Hmm, I don’t want to turn this road trip into a textbook… and the advantage of blog articles is that people can ask questions… but I’ll mention that is the first ordinal bigger than

Next time I’ll talk about some of this stuff a bit more formally.

]]>