Extended veblen hierachy

In a previous blogpost I talked about my ideas of extending hierachies (fgh, sgh, hh, veblen hierachy, ...) to produce large ordinals, uncountable ordinals,... I seemed strong but everything was defined loosely. In this blogpost I'll try to clean up my entension of the veblen hierachy, define a recursive ordinal notation and system of fundamental sequences.

Terms

The set of formal strings \( T\) we consider is recusively defined by

\( 0 \in T \)
\( a+b \in T \) if \( a,b \in T\)
\( \Omega_a \in T\) if \(a \in T\)
\( \varphi_{\Omega_c}(a,b) \in T \) if \( a,b,c \in T \).

I consider 2 terms equal if they are identical.

Speical terms

I give some special terms a name:

\( \underline 1 := \varphi_{\Omega_0}(0, 0) \)
\( \underline \omega := \varphi_{\Omega_0}(0, \underline 1) \)
\( \underline{\omega^a} := \varphi_{\Omega_0}(0, a) \)

Fundamental sequences and cofiniality

I'll recurisively define a partial function \( \bullet[\bullet] : T\times T \to T \) and \( \DeclareMathOperator\cf{cf} \cf : T \to \{\Omega_0, \Omega_{c+\underline 1} : c\in T \} \) recusively. The fundamental sequences are inspired by fundamental sequences for the veblen function.

For \( a,a',b,b',b' ', c, c', n \in T \) :

I call a term \( a \) a limit if \( a \) isn't 0 and isn't of the form \( a' +\underline1 \)
constants \( \cf(0) := 0, \cf(\underline 1) := \underline 1, \underline 1[n] := 0 \)
addition
1. \( \cf(a+0) := \cf(a), (a+0)[n] := a[n] \)
2. \( \cf(a+b) := \cf(b), (a+b)[n] := b[n] \) if \( b\) isn't of the form \( b' + b' ' \) or 0
Cardinals
1. \( \cf(\Omega_a) := \cf(a), \Omega_a[n] := \Omega_{a[n]} \) if \(a\) limit
2. \( \cf(\Omega_a) := \Omega_a, \Omega_a[n] := n \) otherwise
Base case
1. \( \cf(\underline {\omega^{a+\underline1}}) :=\underline \omega, \underline {\omega^{a+\underline1}} [0] := 0, \underline {\omega^{a+\underline1}} [n+\underline 1] := \underline {\omega^{a+\underline1}} [n] + \underline {\omega^a} \)
2. \( \cf(\underline {\omega^a}) = \cf(a), \underline {\omega^a}[n] = \underline {\omega^{a[n]}} \) if \(a \) limit
3. \( \cf( \varphi_{\Omega_c}(0, a) ) = \cf(\underline{\omega^a}), \varphi_{\Omega_c}(0, a)[n] := \underline{\omega^{a[n]}} \) if \( c \neq 0\)
Succesor case
1. \( \cf(\varphi_{\Omega_c}(a+\underline1, 0)) := \omega, \cf(\varphi_{\Omega_c}(a+\underline1, b+\underline1)) := \underline\omega \)
  \( \varphi_{\Omega_c}(a+\underline1, 0)[0] := 0, \quad \varphi_{\Omega_c}(a+\underline1, 0)[n+\underline1] := \varphi_{\Omega_c}(a, \varphi_{\Omega_c}(a+\underline1, 0)[n]) \)
  \( \varphi_{\Omega_c}(a+\underline1, b+\underline1)[0] := \varphi_{\Omega_c}(a+\underline1, b)+\underline1 \)
  \( \varphi_{\Omega_c}(a+\underline1, b+\underline1)[n+\underline1] := \varphi_{\Omega_c}(a, \varphi_{\Omega_c}(a+\underline1, b+\underline1)[n]) \)
2. \( \cf(\varphi_{\Omega_c}(a+\underline1, b)) := \cf(b) \text{ if \( b \) limit} \)
  \( \varphi_{\Omega_c}(a+\underline1, b)[n] :=\varphi_{\Omega_c}(a+\underline1, b[n]) \text{ if \( b \) limit} \)
If \( \text{ if \( a \) limit} , \cf(a)<\Omega_c \):
1. \( \cf( \varphi_{\Omega_c}(a, b) ) = \cf(b) \text{ if \(b \) limit}\)
2. \( \cf( \varphi_{\Omega_c}(a, b) ) = \cf(a) \) otherwise
3. \( \varphi_{\Omega_c}(a, 0)[n] = \varphi_{\Omega_c}(a[n], 0) \)
4. \( \varphi_{\Omega_c}(a, b+\underline 1)[n] = \varphi_{\Omega_c}(a[n], \varphi_{\Omega_c}(a, b)+\underline 1) \)
5. \( \varphi_{\Omega_c}(a, b)[n] = \varphi_{\Omega_c}(a, b[n]) \text{ if \(b \) limit} \)
If \( \text{ if \( a \) limit} , \cf(a)=\Omega_c \):
1. \( \cf( \varphi_{\Omega_c}(a, b)) = \cf(b) \text{ if \(b \) limit} \)
2. \( \cf( \varphi_{\Omega_c}(a, b)) = \underline \omega \) otherwise
3. \( \varphi_{\Omega_c}(a, 0)[0] = 0 \)
  \( \varphi_{\Omega_c}(a, 0)[n+\underline 1] = \varphi_{\Omega_c}(a[\varphi_{\Omega_c}(a, 0)[n]+\underline 1], 0)\)
4. \( \varphi_{\Omega_c}(a, b+\underline 1)[0] = \varphi_{\Omega_c}(a, b)+\underline 1 \)
  \( \varphi_{\Omega_c}(a, b+\underline 1)[n+\underline 1] = \varphi_{\Omega_c}(a[\varphi_{\Omega_c}(a, b+\underline 1)[n]+\underline 1], 0) \)
5. \( \varphi_{\Omega_c}(a, b)[n] = \varphi_{\Omega_c}(a, b[n]) \text{ if \(b \) limit} \)
If \( \text{ if \( a \) limit}, \Omega_{c'+\underline 1} \cf(a)>\Omega_c \): \(\varphi_{\Omega_c}(a, b)[n] := \varphi_{\Omega_c}( \varphi_{\cf(a)}(a, \Omega_{c'}+\underline 1, b)[n]

Note that the fundamental sequence of \( \varphi_{\Omega_c}(a, b), a \text{limit} \) depends on the order of \( \cf(a), \Omega_c \), defined in the next section.

Comparison algorithm

WIP We recursively define a function \(\DeclareMathOperator\cmp{cmp} \cmp : T\times T \to \{-1, 0, 1\} \) and use it to define the order \( a ~R~ b \iff \cmp(a,b)~R~0, R \in <,\le, >, \ge, \cong \)

\( \cmp( \Omega_{c'} , \Omega_c) = \cmp(c', c) \)
\( \cmp(\varphi_{\Omega_{c'}}(a',n'), \varphi_{\Omega_c}(a, n) \) :
1. If \( \Omega_c < \Omega_{c'} \) return \( -\cmp(\varphi_{\Omega_c}(a,n), \varphi_{\Omega_{c'}}(a', n') \)
2. If \( \Omega_{c'} < \Omega_c \) "lift" a', replace each occurence of \( \Omega_{c'+\delta} \) with \( \Omega_{c+\delta} \)
3. If "behaves good" compare like veblen function:
  1. if \( \alpha' > \alpha \) return \( -\cmp(\varphi_{\Omega_c}(a,n), \varphi_{\Omega_{c}}(a', n') \)
  2. if \( \alpha' < \alpha \) return \( \cmp(n', \varphi_{\Omega_c}(a, n) \)
  3. return \( \cmp(n', n) \)
4. return -1 "behaves bad"
the \phi are less then \Omega
the \phi are additively primitve
+
\( \cmp(a,b) = - \cmp(b,a) \) use this for the missing cases.

Explanation

Informally, terms of the form \( \varphi_{\Omega_c}(\alpha, n) \) should correspond to a function \( \varphi_\Omega(\alpha, n) \) form uncountable cardinals \( \Omega \), ordinal \( \alpha \), and \( n<\Omega \). \( \varphi_\Omega(\alpha, \bullet) \) should correspond to the veblen function for ordinals less then \( \Omega \) diagonalised by \(\Omega\). It should satisfy:

\[ \varphi_\Omega(0,n) = \omega^n \\ \DeclareMathOperator\Enum{Enum} \varphi_\Omega(\alpha+1, \bullet) = \Enum\{n : \varphi_\Omega(\alpha, n) = n \} \\ \varphi_\Omega(\alpha, \bullet) = \Enum\{n : \varphi_\Omega(\alpha[m], n) = n \forall m<\cf(\alpha) \} \quad \cf(\alpha)<\Omega \\ \varphi_\Omega(\alpha, \bullet) = \Enum\{n : \varphi_\Omega(\alpha[n], 0) = n \} \quad \cf(\alpha)= \Omega \\ \varphi_\Omega(\alpha, n) = \varphi_\Omega(\varphi_{\cf(\alpha)}(\alpha, \cf({\alpha})^- + 1 ), n) \quad \cf(\alpha)>\Omega \]

With \( \cf \) the cofinality and \( \Omega^- \) the predecesor cardinal. For an undefined system of fundamental sequences.

Notes

If we make \(\varphi(\alpha, n) = n + \omega^\alpha \) for small \(\alpha\) we don't need \( + \) in the notation
The notation reaches the omega fix point
The notation can be extended with inaccesibles
the comparison algrotihm needs to be finished
the notation corresponds closely with theta OFCs.