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

1. \( 0 \in T \)
2. \( \Omega_c \in T \DeclareMathOperator\if{~if~}\if c \in T\)
3. \( \varphi_{\Omega_c}(a,b)  \in T \if a,b,c \in T \)
4. \( a' + \varphi_{\Omega_c}(a,b)  \in T \if a', a,b,c \in T \).
5. \( a' +\Omega_c  \in T \if a', c \in T \).

I write \( a \equiv b \if a,b \in T \) are identical. \( a = b \) is reserved for the ordering of \( T \) defined later.

Childeren & decendants

I define a functions \( \DeclareMathOperator\children{children} \children : T \to \mathcal P(T), \DeclareMathOperator\desc{descendants} \desc: T \to \mathcal P(T) \) for each term to the set of it's children or set of decendants respectively.

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

1. \( \children(0) = \varnothing \)
2. \( \children(\Omega_a) := \{ a \} \)
3. \( \children(\varphi_{\Omega_c}(a,b)) = \{\Omega_c, a, b \} \)
4. \( \children(a'+\varphi_{\Omega_c}(a,b)) = \{a', \Omega_c, a, b \} \)
5. \( \children(a'+{\Omega_c}) = \{a', \Omega_c \} \)
6. \( \desc(a) = \text{ the minemal set \( S \ni a  \) closed under } b \in S \Rightarrow \children(b) \subseteq S \)

Speical terms

I give some special terms a name:

1. \( \underline 1 := \varphi_{\Omega_0}(0, 0) \)
2. \( \underline \omega := \varphi_{\Omega_0}(0, \underline 1) \)

Fundamental sequences and cofiniality

Terminology

I define 4 groups of terms:

1. zero, only \(0\in T\)
2. succesor terms: of the form \(\varphi_{\Omega_c}(0,0), a' +  \varphi_{\Omega_c}(0,0) \DeclareMathOperator\for{~for~} \for a',c \in T\)
3. limit: non zero, non succesor terms.

I'll recurisively define a partial function \( \bullet[\bullet] : T\times T \to T \) and \( \DeclareMathOperator\cf{cf} \cf : T \to \{0,\underline \omega, \underline1,\Omega_c : c\in T \text{ non-limit} \} \) recusively. The fundamental sequences are inspired by fundamental sequences for the veblen function.

For \( a, b, c, c', n \in T \) :

1. addition
   1. \( \cf(a + b) := \cf(b) \)
   2. \( (a+ b)[n] := a \if b[n] \equiv 0\)
   3. \( (a+ b)[n] := a + b' \if  b[n] \equiv 0+b' \for b'\in T\)
   4. \( (a+ b)[n] := a + b[n] \) otherwise
2. 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
3. Base case
   1. \( \cf(0) := 0 \)
   2. \(\cf(\varphi_{\Omega_c}(0,0)) := \underline1,  \varphi_{\Omega_c}(0, b)[0] := 0 \if b\) non-limit.
   3. \( \cf( \varphi_{\Omega_c}(0, b)) := \underline \omega, \varphi_{\Omega_c}(0, b)[n]:=\varphi_{\Omega_c}(0, b)[n[0]] + \varphi_{\Omega_c}(0, b[0]) \if b, n\) succesor
   4. \( \cf( \varphi_{\Omega_c}(0, b)) := \cf(b),\varphi_{\Omega_c}(0, b)[n]:=\varphi_{\Omega_c}(0, b[n])  \if b\) limit
4. If \(a\) succesor:
   1. \( \cf(\varphi_{\Omega_c}(a, b)) := \cf(b) \text{ if \( b \) limit} \)
   2. \( \cf(\varphi_{\Omega_c}(a, b)) := \underline \omega \) otherwise
   3. \( \varphi_{\Omega_c}(a, b)[0] := 0 \if b \equiv 0\)
   4. \( \varphi_{\Omega_c}(a, b)[0] := \varphi_{\Omega_c}(a, b[0])+\underline 1 \if b \) succesor
   5. \( \varphi_{\Omega_c}(a, b)[n] := \varphi_{\Omega_c}(a[0], \varphi_{\Omega_c}(a, b)[n[0]]) \if b\) non-limit, \( n \) succesor
   6. \( \varphi_{\Omega_c}(a, b)[n] := \varphi_{\Omega_c}(a, b[n]) \text{ if \( b \) limit}\)
5.  \( \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. \( \varphi_{\Omega_c}(a, b)[n] :=\varphi_{\Omega_c}(a[n], \varphi_{\Omega_c}(a, b[0])+\underline 1) \if b\) succesor
   5. \(\varphi_{\Omega_c}(a, b)[n] := \varphi_{\Omega_c}(a, b[n]) \text{ if \(b \) limit} \)
6. \( \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. \(  \varphi_{\Omega_c}(a, 0)[0] := 0 \)
   4. \( \varphi_{\Omega_c}(a, b)[0] := \varphi_{\Omega_c}(a, b[0])+\underline 1 \if b \) succesor
   5. \(\varphi_{\Omega_c}(a, b)[n] := \varphi_{\Omega_c}(a[\varphi_{\Omega_c}(a, b)[n[0]]], 0) \if b\) not a limit, \( n \) succesor
   6. \(\varphi_{\Omega_c}(a, b)[n] := \varphi_{\Omega_c}(a, b[n]) \text{ if \(b \) limit} \)
7.  \( \text{ if \( a \) limit, \(c'\) succesor },  \Omega_{c'} \equiv \cf(a)>\Omega_c \): \(\varphi_{\Omega_c}(a, b)[n] := \varphi_{\Omega_c}( \varphi_{\cf(a)}(a, \Omega_{c'[0]}+\underline 1))[n] \)

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

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

Lifting

We will simulatniously define a partial function \( \DeclareMathOperator\lift{lift} \lift : T \times T \times T \to T : (a, c' , c) \to \lift(a,c',c)  \). Intuitively \( \varphi_{\Omega_{c'}}(a, n) =  \varphi_{\Omega_c}(\lift(a,c',c), n) \) for \( c' < c\), we lift \(a \) to a term which causes the same amount of diagonalisation for a larger \(\Omega\).

For \( a, b, c', c, d \in T \)

1. \( \lift( 0, c' ,c) := 0\)
2. \( \lift( \Omega_a, c', c) := l \)
   1. \( l  := \Omega_a  \text{ if } a<c' \)
   2. Search \( \delta \in T \) such that \( a = c'+\delta \):
      1. \( a \equiv a_1 + \cdots + a_n \) with \( a_i \text{ additively principal }, 1\le i \le n \) 
      2. If \( \exists i : a_i > c' \) then \( \delta := a \)
      3. Else choose \( i \) minemal such that \( a_i = c' \) , \( \delta := a_{i+1} + \cdots + a_n \)
   3. \( l := \Omega_{c+\delta} \)
3. \( \lift( a+b, c' ,c) := \lift( a, c' ,c) + \lift( b, c' ,c)\) if \(b\) additively principal
4. \( \lift( \varphi_{\Omega_d}(a,b), c' ,c) := \varphi_{\lift(\Omega_d,c',c)}(\lift(a,c',c),\lift(b,c',c)) \)

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

1. \( \cmp( 0, a )  := 0 \if a \equiv 0, -1 \) otherwise
2. \( \cmp( \Omega_{c'} , \Omega_c) := \cmp(c', c) \)
3. \( \cmp( a'_0+\cdots a'_{n'}, a_0 + \cdots + a_n ) := r \if a'_{i'}, a_i \in T \) not of the form \( a' + \varphi_{\Omega_c}(a,b), a'+\Omega_c \for  0 \le i' \le n', 0 \le i \le n \) and \(n + n' > 0 \)
   1. Make the sequence decending
      1. \( \if \exists i>0: a'_0 < a'_i, r := \cmp( a'_1+\cdots a'_{n'}, a_0 + \cdots + a_n ) \)
      2.  \( \if \exists i>0: a_0 < a_i, r := \cmp( a'_0+\cdots a'_{n'}, a_1 + \cdots + a_n ) \)
   2.  otherwise compare lexografically:
      1. \( \if \cmp( a'_0 , a_0) \neq 0, r:=\cmp( a'_0 , a_0) \)
      2. \( r:= \cmp( a'_1+\cdots a'_{n'}, a_1 + \cdots + a_n ) \) otherwise
4. \( \cmp(\varphi_{\Omega_{c'}}(a',b'), \Omega_c) := \cmp(b', \Omega_c )\)
5. \( \cmp(\varphi_{\Omega_{c'}}(a',b'), \varphi_{\Omega_c}(a, b)) := r \)
   1. If \( \Omega_{c'} <  \Omega_c \), \( r := \cmp( \varphi_{\Omega_c}(\lift(a',c',c),b'), \varphi_{\Omega_c}(a, b)) \)
   2. If \( \Omega_{c'} = \Omega_c \),
      1. If \( \exists d \in \desc(a') : d < \Omega_c , d \ge \varphi_{\Omega_c}(a, b) \)
         1. \( r:= 0 \if a' = \varphi_{\Omega_c}(a, b), b'=0\)
         2. \( r:= 1 \) otherwise
      2. compare like veblen function:
         1. \( r:= \cmp(b', \varphi_{\Omega_c}(a, b)) \if a' < a \)
         2. \( r := \cmp(n', n) \if a' = a \)
6. \( \cmp(a,b) := - \cmp(b,a) \) if none of the other cases apply.

Number

We're almost ready to define the hardy hierachy indexed by terms \( H : T \times \mathbb N \to N : (a, n) \to H_a(n) \) and a large number \( \ell \), we just need to define the function \( N : \mathbb N \to T \) to convert from natural numbers to terms.

1. \( N(0) := 0 \in T\)
2. \( N(a) := n(a-1) + \underline 1 \if 0 < a \in \mathbb N \)
3. \( H_0(n) :=n \)
4. \( H_a(n) :=H_{a[N(n)]}(n+1) \if a \neq 0\)
5. \( H(n) := H_{\varphi_{\Omega_0 }\left( \left. \Omega_{ \Omega_{\ddots_0}} \right\} n, 0\right)}(n) \)
6. \( \ell := H^9(9) \)

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

• The notation reaches the omega fix point
• The notation can be extended with inaccesibles
• the notation corresponds closely with theta OFCs.