FANDOM


Fundamental Sequences

This blog will map out much of the standard definitions on Fundamental Sequences for Ordinals. It will attempt to simplify the rule-set where possible. The alternative rule-set presented here will be used in my other blog for the J Function and in particular in the Sandpit \(J_8\) blog.

I have created a new blog which updates the material below and provides the detailed rule-set on Fundamental Sequences used by my Beta Function blogs.


Basics (Cantor's Normal Form)

Let \(\gamma\) and \(\delta\) be two arbitrary transfinite ordinals, \(\lambda\) is an arbitrary limit ordinal, and \(n\) is a finite integer. Then:

\((\gamma + 1)[n] = \gamma\)

\((\gamma + \lambda)[n] = \gamma + \lambda[n]\) when \(\gamma >= \lambda\)

\(\lambda.(\delta + 1)[n] = \lambda.\delta + \lambda[n]\)

\(\gamma.\lambda[n] = \gamma.(\lambda[n])\) when \(\gamma >= \lambda\)

\(\lambda^{\delta + 1}[n] = \lambda^{\delta}.(\lambda[n])\)

and

\(\gamma^{\lambda}[n] = \gamma^{\lambda[n]}\)

I have written another blog to further extend Normal Form to provide detailed definitions for ordinals of arbitrary complexity.


Some Common Transfinite Ordinals

\(\omega[n] = n\)

\(\epsilon_0[n] = \omega\uparrow\uparrow n\)

\(\epsilon_1[n] = \epsilon_0\uparrow\uparrow n\)

\(\epsilon_{j+1}[n] = \epsilon_j\uparrow\uparrow n\)

and

\(\epsilon_{\omega}[n] = \epsilon_{\omega[n]} = \epsilon_n\)


Veblen Hierarchy

Continuing into Veblen Hierarchy and the \(\varphi\) function. Lets start with these equations which are equivalent to those in the Common Transfinite Ordinal section.

\(\varphi(1)[n] = \omega[n] = n\)

\(\varphi(1,0)[n] = \epsilon_0[n] = \varphi(n) = \omega\uparrow\uparrow n\)

\(\varphi(1,1)[n] = \epsilon_1[n] = \varphi(1,0)\uparrow\uparrow n\)

\(\varphi(1,j + 1)[n] = \epsilon_{j + 1}[n] = \varphi(1,j)\uparrow\uparrow n\)

and

\(\varphi(1,\omega)[n] = \varphi(1,\omega[n]) = \varphi(1,n)\)

The following extends the Veblen function definition for completeness:

\(\varphi() = 0\)

\(\varphi(0) = 1\)

\(\varphi(1) = \omega\)

and

\(\varphi(n) = \varphi^n(1) = \omega\uparrow\uparrow n = \omega^{\varphi(n-1)}\)


Rule-set (The Aristo Sequence)

The following rule-set has been given a name of the "Aristo Sequence", so that it is clearly distinguishable from other rule-set definitions. Before we start, some notational conventions that will be used are:

\(k^2(n,p_*) = k(n,k(n,p))\)

\(k^2(n_*,p) = k(k(n,p),p)\)

and

\(k(a_{[2]},b_{[3]}) = k(a_1,a_2,b_1,b_2,b_3)\)

The rule-set starts with this arbitrary Veblen function:

\(\varphi(\alpha_{[x]},\beta,0_{[y]})[n]\)

where \(x >= 0\) and \(y >= 1\), \(\alpha\) and \(\beta\) can be any ordinal, but \(\beta >= 1\)

then

  • \(\beta\) is a limit ordinal \(= \varphi(\alpha_{[x]},\beta[n],0_{[y]})\)
  • else
    • \(y > 0\)
      • \(x = 0\) and \(\beta = 1\) then \(= \varphi(1,0_{[y]}) = \varphi^{\omega[n]}(1_*,0_{[y-1]})\)
      • else \(= \varphi^{\omega}(\alpha_{[x]},\beta-1,0_*,0_{[y-1]}) = \varphi^{\omega[n]}(\alpha_{[x]},\beta-1,0_*,0_{[y-1]})\)
    • else

with this additional rule thrown in for completeness:

\(\varphi^{\omega}(m,\varphi(n,p)+1_*)\)

\(= \varphi(m+1,\varphi(n,p)+1)\) when \(m+1 < n\)

\(= \varphi(n,p+1)\) when \(m+1 = n\)

The additional rule can be illustrated as follows:

\(\varphi^{\omega}(1,\varphi(2,0)+1_*) = \varphi(2,1)\)

\(\varphi^{\omega}(1,\varphi(3,0)+1_*) = \varphi(2,\varphi(3,0)+1)\)

\(\varphi^{\omega}(2,\varphi(3,0)+1_*) = \varphi(3,1)\)


Calculated Example

What is the fundamental sequence for \(\zeta_0[2]\) ?

\(\zeta_0[2] = \varphi(2,0)[2] = \varphi^{\omega[2]}(1,0_*) = \varphi^2(1,0_*) = \varphi(1,\varphi(1,0))\)

or

\(\zeta_0[2] = \varphi(1,\varphi(1,0)) = \epsilon_{\epsilon_0}\)

I have created another blog to calculate \(f_{\zeta_0}(2)\) in detail.


\(\Gamma_0\)

Calculating \(\Gamma_0\) we get:

\(\varphi(1,0,0) = \Gamma_0\)

then

\(\Gamma_0[2] = \varphi(1,0,0)[2] = \varphi^{\omega[2]}(1_*,0) = \varphi^2(1_*,0) = \varphi(\varphi(1,0),0)\)


Small Veblen Ordinal (SVO)

SVO is defined as follows:

\(SVO = \varphi(1,0_{[\omega]})\)

Diagonalising SVO for n=2 produces this result:

\(SVO[2] = \varphi(1,0_{[\omega]})[2] = \varphi(1,0_{[2]}) = \varphi(1,0,0) = \Gamma_0\)


Appreciate any comments on this blog.

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