User blog:TheKing44/Ordinal Definable System of Fundamental Sequences

For the purposes of this post, I will be working in the set theory ZF unless otherwise stated.

I will define an exceedingly strong system of fundamental sequences. In particular it will be the strongest system of fundamental sequences definable in first order set theory. This is useful since strong systems of fundamental sequences are needed for using the fast-growing hierarchy.

Let $$\alpha$$ be a countable limit ordinal such that there exists an ordinal definable bijection between $$\alpha$$ and $$\mathbb N$$.The class of ordinal definable sets is well orderable. Moreover, we can *define* a specific well ordering. First, we well ordinal the set of formulas with ordinal parameters. We use shortlex order over the formulas, with ordinal parameters following the symbols of first order set theory in the language. Now, we compare two ordinal definable sets by comparing there earliest defining formulas. This well orders the ordinal definable sets.

Now, let $$f_\alpha$$ be the earliest ordinal definable bijection between $$\alpha$$ and $$\mathbb N$$ according to the well ordering defined above. $$f_\alpha$$ can be interpreted as a sequence of ordinals. We remove from this sequence any ordinal which follows a larger ordinal, to create $$s_\alpha$$. $$s_\alpha$$ is increasing since we removed all the ordinals that followed a larger ordinal. Moreover, its supremum equals $$\alpha$$, since otherwise the supremum+1 was removed from the sequence, implying that an ordinal larger than the supremum+1 was in the sequence, which is contradictory since than $$s$$ would contain an element larger than its supremum. These $$f$$s make up our system of fundamental sequences. I call it the ordinal definable system of fundamental sequences (odsfs).

Is this a system of fundamental sequences? Well, if $$\alpha$$ is in it, then so will any smaller limit ordinal $$\beta$$. That's because we can treat $$f_\alpha$$ as a sequence, remove all the ordinals that are not less than $$\beta$$, and use that to define a bijection between $$\beta$$ and $$\mathbb N$$. Since $$f_\alpha$$ is ordinal definable, and the only parameter we added is an ordinal, this bijection will be ordinal definable.

So, how powerful is this system? Well, if V=OD (a statement which is independent of ZF, or even ZFC), then it includes every countable ordinal! In ZFC, it is well known that a system of fundamental sequences including every countable limit ordinal exists, but usually a specific example is not given (or is impossible to give). Odsfs on the other hand is a specific example!

If we do not assume V=OD, we still have something pretty special. Metatheortically, since ordinal definability is definable in first-order set theory, the system of $$f$$s is definable in first-order set theory. However, no other system of fundamental sequences definable in first order set theory could include a limit ordinal $$\alpha$$ that odsfs does not, since then $$\alpha$$'s fundamental sequence in that system would be ordinal definable, contradicting the fact that is not in odsfs. So, odsfs is the most powerful system of fundamental sequences definable in first-order set theory (and therefore the most powerful explicit example you could work with in ZF or ZFC)!

P.S. The supremum of the domain of odsfs should probably be called $$\omega_1^{\text{HOD}}$$ since [HOD] thinks it is $$\omega_1$$. You can't add it to odsfs because even though it is definable, none of its fundamental sequences are.