User blog comment:Edwin Shade/Questions About Madore's Psi Function/@comment-5529393-20180110030002

I've always assumed "Madore's OCF" refers to the OCF/OCFs described on Wikipedia that Madore has edited. There is a main one with just $$\Omega$$, and then there is a brief description of an extension that goes up to $$\Omega_\omega$$. Still, extending to $$\Omega_{\Omega_{\Omega_\cdots}}$$ seems fairly straightforward, and I will assume this extension when answering your questions.

Question 1:

Well, generally we take $$\Omega = \Omega_1$$, so those two are the same. As for the difference between $$\Omega_1$$ and $$\Omega_2$$, the latter is much bigger, so for its $$\psi$$ value we have to go further to define it. In particular, the values of $$\psi_1$$ will be less than $$\Omega_2$$, so we will be able to apply $$\psi_1$$ in our inductive definitions of $$C(\alpha)$$ for $$\alpha$$ between $$\Omega_1$$ and $$\Omega_2$$.

Here's how I generally think about it: Through the $$\psi_0$$ function, we can use larger and larger multiples of $$\Omega$$ to define larger and larger countable ordinals. For the most well-known OCF, this stops at $$\varepsilon_{\Omega+1}$$, because that's the limit of ordinals we can express in the notation. But certainly having larger multiples of $$\Omega$$ would lead to larger countable ordinals. So, we have transferred the problem of defining large countable ordinals to defining large multiples of $$\Omega$$. Now, we can certainly go beyond $$\varepsilon_{\Omega+1}$$ by using the Veblen notation of the Extended Veblen notation, but we can do better by using another OCF; since we have already used $$\psi_0$$ to collapse ordinals of cardinality $$\Omega$$ into large countable ordinals, we will instead use $$\psi_1$$ to collapse ordinals of cardinality $$\Omega_2$$ into ordinals of cardinality $$\Omega$$. The way it works is much the same:

$$\psi_1(0) = \varepsilon_{\Omega+1}$$

$$\psi_1(1) = \varepsilon_{\Omega+2}$$

$$\psi_1(\Omega_2) = \zeta_{\Omega+1}$$

and so on, analagously to how $$\psi_1$$ turns ordinals of cardinality $$\Omega$$ into large countable ordinals. So, now we have a two-step process where we turn ordinals of cardinality $$\Omega_2$$ into relatively large ordinals of cardinality $$\Omega$$, and then we turn them into really large countable ordinals.

For $$\psi(\Omega_2)$$ we have a fundamental sequence of

$$\psi(\Omega), \psi(\psi_1(\Omega)), \psi(\psi_1(\psi_1(\Omega))), \ldots$$.

For $$\psi(\Omega_5)$$ we have a fundamental sequence of

$$\psi(\Omega_4), \psi(\psi_4(\Omega_4)), \psi(\psi_4(\psi_4(\Omega_4))), \ldots$$ ,

and that's how it generally works for successor cardinals. For a limit cardinal $$\Omega_\alpha$$, the process basically recurs down to $$\alpha$$. So the fundamental sequence for $$\psi(\Omega_\omega)$$ is $$\psi(\Omega_n)$$, and the fundamental sequence for $$\psi(\Omega_\Omega)$$ is

$$\psi(\Omega), \psi(\Omega_{\psi(\Omega)}), \psi(\Omega_{\psi(\Omega_{\psi(\Omega)})}), \ldots$$.

Here's the fundamental sequence for $$\psi(\Omega_{\Omega_2})$$

$$\psi(\Omega_\Omega), \psi(\Omega_{\psi_1(\Omega_\Omega)}), \psi(\Omega_{\psi_1(\Omega_{\psi_1(\Omega_\Omega)})}), \ldots$$.

Question 2:

No, because the left hand side is the limit of an increasing sequence of ordinals of cardinality $$\Omega_\alpha$$, where $$\alpha$$ is always a countable ordinal. The limit will then be some cardinal $$\Omega_\beta$$ for $$\beta$$ a countable ordinal, since the supremum of a countable collection of countable ordinals is always a countable ordinal. On the other hand, if we extend Madore's ordinal collapsing function in the obvious manner, $$\psi_\Omega(0)$$ will be $$\varepsilon_{\Omega_\Omega + 1}$$, which is greater than $$\Omega_\Omega$$ and therefore greater than $$\Omega_\beta$$.

Question 3:

Quick correction: I is not the first fixed point of $$\gamma \mapsto \Omega_\gamma$$; it is the smallest weakly inaccessible ordinal, which is larger than the first fixed point of $$\gamma \mapsto \Omega_\gamma$$, or the first fixed point of the function enumerating the fixed points of $$\gamma \mapsto \Omega_\gamma$$, and so on as far as you care to diagonalize. This makes I useful for collapsing down to "constructible" ordinals in the same way that $$\Omega$$ is useful for collapsing down to countable ordinals; as we get stronger and stronger ordinal collapsing functions, we will get larger and larger countable ordinals, but we know that $$\Omega$$ will forever be larger than any of them, and the same is true with I versus the fixed points of $$\gamma \mapsto \Omega_\gamma$$ and their diagonalizations.

Okay, once we get to inaccessibles we need to deviate a bit from they way Pohlers/Fefferman/Madore defined their collapsing functions. Yes, the way we currently define it $$\psi_\alpha$$ will get you an ordinal of cardinality $$\Omega_\alpha$$; in particular $$\psi_I$$ should get you an ordinal of cardinality I, so it won't be good for defining fixed points of the $$\Omega$$ function. The solution is to use $$\psi_\alpha$$ to define ordinals of cardinality _less_ than $$\alpha$$. In particular, we will use the following definition for our ordinal collapsing function with inaccessibles:

$$C_0 (\alpha, \beta) = \beta \cup \lbrace 0, I \rbrace$$

$$C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, \psi_\pi(\eta) | \gamma, \delta, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace $$

$$C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta) $$

$$\psi_\pi (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \rbrace \cup \lbrace \pi \rbrace) $$

Let me try to translate this a bit. Our $$C(\alpha)$$ is now a $$C(\alpha,\beta)$$, and we now include all ordinals less than $$\beta$$ in our initial set. Then for $$\psi_\pi(\alpha)$$, we look for the smallest ordinal $$\beta$$ such that the portion of $$C(\alpha,\beta)$$ that is below $$\pi$$ is precisely the set $$\beta$$. Now, we know that $$C(\alpha,\beta$$ starts out containing all the ordinals below $$\beta$$ (and I), and then we close the set under a bunch of different operations. What the definition of $$\psi_\pi(\alpha)$$ is saying is, for the right value of $$\beta$$, those closure operations won't generate any new ordinals greater than or equal to $$\beta$$, but less than $$\pi$$.

Let's take a simple example: $$\psi_{\Omega_1}(0)$$. $$C_0(0,\beta)$$ will consist of the ordinals below $$\beta$$, along with I (and 0 if we happened to pick $$\beta$$ to be 0). Then, at each step we close that set under the operations of +, the Veblen function, the function $$\gamma \mapsto \Omega_\gamma$$, and $$\psi_\pi(\eta)$$ where $$\eta < \alpha$$ - but that last part is impossible since $$\alpha = 0$$. So it's just those three operations. Now, the function $$\gamma \mapsto \Omega_\gamma$$ turns out not to matter here, since \pi = \Omega_1, so applying $$\Omega$$ to anything bigger than 0 will get you an ordinal greater than or equal to $$\pi$$, which we aren't worried about. So we just want to find a $$\beta$$ that is closed under addition and the Veblen function; these are the Gamma numbers, and the smallest one is $$\Gamma_0$$. So $$\psi_{\Omega_1}(0) = \Gamma_0$$.

So you can see the similarity with the previous definition of $$\psi$$, except instead of using $$\psi_\alpha$$ to create ordinals of cardinality $$\Omega_\alpha$$, we will instead be using $$\psi_{\Omega_{\alpha+1}}$$. By shifting the indices up one, this allows us new functions $$\psi_\pi$$ when $$\pi$$ is a limit cardinal. But, we will generally not be using limit cardinals for $$\pi$$; the only ones we will use are the weakly inaccessible cardinals. For this particular notation, the only weakly inaccessible cardinal we have is I.

Now $$\psi_I$$ uses the exact same definition as all the other $$\psi_\pi$$, which is nice. However, the result is substantially different. For $$\psi_{\Omega_{\alpha+1}}$$, the $$\gamma \mapsto \Omega_\gamma$$ function allowed you to get $$\Omega_\alpha$$, and that was it; from there you used to addition, Veblen, and $$\psi$$ functions to reach the limit point. But for $$\psi_I$$, there's no largest $$\Omega_\gamma$$ below I, so it's the $$\gamma \mapsto \Omega_\gamma$$ function that is our strongest function to obtain our limit point. $$\var_I(0)$$ has to be closed under the $$\gamma \mapsto \Omega_\gamma$$ function, so the minimum it could be would be the smallest fixed point of the function, and that in fact works. $$\var_I(1)$$ also has to be closed under the $$\gamma \mapsto \Omega_\gamma$$, and also contains $$\var_I(0)$$, so it will be the second smallest fixed point. $$\var_I(I)$$ will then be the smallest ordinal $$\alpha$$ such that $$var_I(\alpha) = \alpha$$, analogously with $$\Omega$$. And the pattern continues, as far as we go with multiples of I.

Question 4:

The traditional definition of a weakly Mahlo cardinal is that it is a cardinal M such that any closed and unbounded set of ordinals below M must contain a regular cardinal.

A set S of ordinals in M is closed if, whenever T is a subset of S and sup T = a < M, then a is in S as well. That is, S is closed under limits/supremums.

A set S of ordinals in M is unbounded if it has no proper upper bound, i.e. there is no a < M for which a is greater than any element of S.

An ordinal/cardinal $$\alpha$$ is regular if there does not exist an unbounded subset of $$\alpha$$ of order type less than $$\alpha$$. The successor cardinals are regular, as are the weakly inaccessible cardinals (since they are defined as precisely the cardinals that are both limit and regular).

If you looking for a "fixed point" definition of a weakly Mahlo cardinal, here is one: A cardinal M is weakly Mahlo if every normal function from M to M has a regular fixed point.

A normal function is a function that is continuous and increasing.

A continuous function f is a function such that for any subset S of the domain, $$f(\sup_{s \in S} s) = sup_{s \in S} f(s)$$.

An increasing function f satisfies $$a > b \implies f(a) > f(b)$$.

This is actually a very similar definition, since the range of a normal function from a limit ordinal to itself is a continuous and unbounded set, and every continuous and unbounded set is the range of a normal function. So I don't know that the second definition is more "easily graspable". To see in part why Mahlo cardinals are so ridiculously large, look at the Wikipedia page for Mahlo cardinals and read the proof that all Mahlo cardinals are hyper-inaccessible; the same method can be used to show that all Mahlo cardinals are hyper-hyper-inaccessible, hyper^omega-inaccessible, hyper^M-inaccessible (where M is the Mahlo cardinal in question) and so on.

It's really this incredible size that is useful to us in ordinal collapsing functions. We want to build a hierarchy of inaccessibles in a similar fashion to the extended Veblen hierarchy; so I(a0) is the a0'th weakly inaccessible, I(1,a0) is the a0'th 1-weakly inacessible (a 1-weakly inaccessible is an inaccessible limit of inaccessibles), I(2,a0) is the a0'th 2-weakly inaccessible, and I(a1,a0) is the a0'th a1-weakly inaccessible. Then, we can define a (1,0)-weakly inaccessible to be a cardinal kappa that is kappa-weakly inaccessible, and I(1,0,a0) is the a0'th (1,0)-weakly inaccessible. And so on, up through arbitrarily many variables. Well, we would like to extend this in the same way that our older ordinal collapsing functions extended the Veblen hierarchy. So, if we define $$\chi(\alpha,\beta)$$ as defined in my Ordinal Notations V blog, we get that

$$\chi(\ldots + M^4 \alpha_4 + M^3 \alpha_3 + M^2 \alpha_2 + M^1 \alpha_1 +\alpha_0, \beta) = I(...,\alpha_4,\alpha_3,\alpha_2,\alpha_1,\alpha_0,\beta)$$

so that M works as a diagonalizer for our inaccessible hierarchy.

So knowing exactly what Mahlo cardinals are is not essential to understanding the corresponding OCFs - the key point is that they satisfy every level of inaccessible as far as we care to go, so they are very large and far apart from each other.