User blog:P進大好きbot/Relation between an OCF and an Ordinal Notation

Googologists here often comfound with an OCF and an ordinal notation, even if they are highly familiar with analysis. For example, I got surprised when rpakr stated "an OCF is an ordinal notation" yesterday.

So I roughly explain what an ordinal collapsing function and an ordinal notation system are.

= OCF =

An OCF is a shorthand of an ordinal collapsing function, and is a notion in set theory. It is a sort of a function which reduces some "degree" of an ordinal. So "a function \(\psi\) is an OCF" is not a mathematical statement, but is an explanation of a role of \(\psi\) in the context.

Purpose
We need a large number!

In order to define a large number, we use recursive countable ordinals with a recursive system of fundamental sequences through functions which send recursive countable ordinals to natural numbers, e.g. FGH and HH. Therefore we need a large recursive countable ordinal.

In order to define a large recursive countable ordinal, we use cardinals through functions which send cardinals to recursive countable ordinals, e.g. several \(\psi\)-functions and \(\vartheta\)-functions. So these are OCFs. They reduce the cofinality (or the cardinality) of cardinals. Therefore we need a large cardinal.

In order to define a large cardinal, we use weakly (hyper) inaccessible cardinals through functions generalising OCFs above sending weakly (hyper) inaccessible cardinals to cardinals, e.g. several extended \(\psi\)-functions. They reduce the degree of the weak (hyper) inaccessibility. Therefore we need a large weakly (hyper) inaccessible cardinal, and also a larger cardinal far beyond them, e.g. a weakly Mahlo cardinal.

In order to define a large... Ok. I stop it.

Construction
To begin with, choose There are essentially two ways to define a new ordinal \(\psi_{\kappa}(\alpha)\) for ordinals \(\kappa\) and \(\alpha\) by transfinite induction on \(\alpha\).
 * finitely many (possibly multi-variable) functions \(F_0,F_1,\ldots,F_m\) which sends ordinals to ordinals, e.g. the constant map \(\alpha \mapsto 0\), the addition \((\alpha,\beta) \mapso \alpha + \beta\), and Veblen function \((\alpha,\beta) \mapsto \varphi_{\alpha}(\beta)\), and
 * a set \(C_{\kappa}^0\) of ordinals.


 * 1) Define a set \(C_{\kappa}(\alpha))\) of ordinals presentable by a combination of ordinals in \(C_{\kappa}^0\), \(F_0,F_1,\ldots,F_m\), and \(\psi_{\pi}(\beta)\) for some \(\beta < \alpha\) and \(\pi\). Then define \(\psi_{\kappa}(\alpha)\) as the smallest ordinal which does not belong to \(C_{\kappa}(\alpha)\).
 * 2) Define a set \(C_{\kappa}(\alpha))\) in a similar way above, except that combinations of \(C_{\kappa}^0), \(F_0,F_1,\ldots,F_m\), and \(\psi_{\pi}(\beta)\) are allowed only when the expressions are of "normal form". So you need to define the notion of normal form for expressions of ordinals. What is an "expression" of an ordinal? You can formally define an expression as a sequence of constants and functions in set theory.

An OCF defined without using expressions is easy for readers to understand. On the other hand, we need to make implicit efforts to define it, because we are not allowed to define an OCF in a way like "Replace the last appearence of \(\omega\) in the input of \(\psi_{\kappa}\) by blah-blah".

An OCF defined by using expressions is not so easy for readers to understand. Moreover, we need to define the notion of normal form in a way that every ordinal (with a fixed bound if necessary) admits precisely one normal form expression. But after that we are allowed to define an OCF in a way like "Replace the last appearence of \(K\) in the input of \(\psi_{\kappa}\) by blah-blah".

Defining an OCF by using expression is quite similar to defining an OCF and an ordinal notation simultaneously by using each other. Therefore even if they do not use expressions, they often actually need to define the notion of normal (or standard) form for formal strings in order to define the associated ordinal notation.

= Ordinal Notation =

An "ordinal notation" is a shorthand of an ordinal notation system, and is a notion in arithmetic. It is a well-ordered system \((OT,<)\) of formal strings called ordinal terms consisting of fixed symbols such as \(0, \omega, \Omega, I, M, K, T, F_{0=1}, +, \varphi, \psi\), and so on. Although you can use any symbols, it is prefered to follow traditional customs. So \(0\) should be the smallest ordinal term, and \(I\) should play a role similar to weakly inaccessible cardinals. As I wrote above, an ordinal notation is just a notion in arithmetic, and hence there is no actual relation to set theory here.

Purpose
We need a large number!

A large recursive countable ordinal is not sufficient. We also need a recursive system of fundamental system, and a way to descrive it in arithmetic, in order to define a computable large function. Otherwise, enjoy Busy Beaver function. It is also cute, and far beyond computable large functions.

Construction
To begin with, choose There are essentially two ways to define an ordinal notation \((OT,<)\). One is purely arithmetic, and the other one is partially set theoretic.
 * finitely many function symbols \(f_0,f_1,\ldots,f_m,p\), e.g. a constant term symbol \(0\) (a function symbol of arity \(0\)), the addition symbol \(+\) (a function sysmbol of arith \(2\)), and another function symbol \(\varphi\),
 * finitely many brace symbols, e.g. \(,\{,\},\langle,\rangle\),
 * finitely many separator symbols, e.g. comma, colon, and semicolon, and
 * a recursive set \(T\) of formal strings called terms consisting of brace symbols, separator symbols, and \(f_0,f_1,\ldots,f_m,p\) satisfying syntax theoretic restrictions on the arity, e.g. if \(f_0\) is of arity \(0\) and \(f_3\) is of arity \(1\), \(f_3(f_0)\) and \(f_3(f_3(f_0)\) are allowed but \(f_3(f_3)\) or \(f_0(f_0)\) are not allowed.


 * 1) Define a binary relation \(<\) on \(T\) by a recursively computation using the syntax theoretic structure on terms, and a recursive subset \(OT \subset T\) to which the restriction of \(<\) is a well-order.
 * 2) Define an OCF \(\psi\) by choosing other data \(F_0, F_1, \ldots, F_m\) and using a notion of expressions of normal form. Then denote by \(o\) the map from \(T\) to the class of ordinals associated to the correspondence \((f_0,f_1,\ldots,f_m,p) \mapsto (F_0,F_1,\ldots,F_m,\psi)\), and by \(OT \subset T\) the subset of terms \(a\) such that the canonical expression of \(o(a)\) by \(F_0,F_1,\ldots,F_m,\psi\) is of normal form. Finally, "interprete" the relation \(o(a) \in o(b)\) for terms \(a\) and \(b\) into a recursive relation \(<\) on \(T\). To be mor precise, define a recursive relation \(<\) on \(T\) such that \(o(a) \in o(b)\) is equivalent to \(a < b\) for any \(a,b \in OT\) (not \(T\)).

The benefit of the first approach is that it is purely arithmetic. However, it is not so easy to prove that \((OT,<)\) is actually an ordinal notation and that it presents sufficiently large ordinal. Of course, it does not mean that it were impossible.

For example, see my ordinal notation here. I directly defined an ordinal notation with \(\textrm{PTO}(\textrm{ZFC})\) without using set theory itself. If what Rathjen stated without proof on the provability of the well-foundedness of his ordinal notation with the smallest weakly Mahlo cardinal, then I guess that my ordinal notation goes beyond it.

The benefit of the second approach is that the well-foundedness immediately follows from that of ordinals. However, it is not so easy to define an OCF by using expressions of normal form.

Strength
I need to add an explanation of what the strength of an ordinal notation \((OT,<)\) is. Almost all ordinal notations are constructed by using an OCF. and then we have a canonical map \(o\) as explained above. Be careful. The strength of \((OT,<)\) is not the limit of \(o(a)\)'s, but is the ordinal type of \((OT,<)\). Also, the strength of the segment of \((OT,<)\) below \(a \in OT\) is not necessarily \(o(a)\) or \(\sup_{b < a} o(b)\), but is the ordinal type of \((\{b \in OT \mid b < a\},<)\).

For example, consider the ordinal notation below:

The order \(<\) associated to \((o,\in)\) is just the comparison of the length, and hence the strength of \((OT,<)\) is \(\omega\), but not \(\varepsilon_0 \times \omega\).

In order to ensure the strength of \((OT,<)\) is actually the limit of \(o(a)\)'s, we need to show that \(o\) gives a one-to-one correspondence to the limit. Recall that when we use cardinals, the limit obviously goes beyond \(\omega_1\), and hence is not bijective with a countable set \(OT\).

We often consider a segement. Suppose that \(o(W) = \Omega\) for some \(a \in OT\). By the reason above, the strength of the segment below \(W\) is not \(\Omega\). Then you may expect that it is actually the limit of \(o(b)\)'s with \(b < a\). As I said, it is also incorrect. Look at the table above again. The limit of \(o(b)\)'s with \(b < 000\) is \(\varepsilon_0\), but the ordinal type of the segment is just \(2\).

As a consequence, the limit of values of \(o\) is not necessarily the strength of the segment. At least, only when you proved that \(o\) gives a one-to-one correspondence between the segment below \(a\) and the limit of \(o(b)\)'s with \(b < a\), the strength coincides with the limit.

Remember that we used \(C_{\kappa}^0\) in the construction of an OCF. If \(C_{\kappa}^0\) contains a large ordinal \(\gamma\) as a subset, then values of \(\psi_{\kappa}\) always go beyond \(\gamma\). Then the system loses the presentability of ordinals using \(\psi\)'s. As a result, \(o\) enjoys awfully many skippings below \(\Omega_1\). Then the strength of \((OT,<)\) is seriously reduced, because \(o\) does not give a one-to-one correspondence between a segment and the limit.

Therefore in order to ensure the strength of \(OT,<)\), we need to be very careful to control the growth of an OCF, so that \(o\) spreads ordinal terms into sufficiently large countable ordinals without skippings. Also, this tells us why to define the notion of normal form is so useful to define an ordinal notation. If we have a canonical way to express sufficiently large countable ordinals, then it immediately ensure that \(o\) is surjective onto those.