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Fundamental Sequences (used by The Beta Function)

This blog will cover the standard definitions on Fundamental Sequences for Ordinals. It will also provide a precise rule-set for the Fundamental Sequences used by in my Beta Function blogs.

This blog is a complete update of my previous blog on Fundamental Sequences. Please keep this in mind if you refer to that blog.


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

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


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 (used by The Beta Function)

The following rule-set is used by my Beta Function blogs and is intended to be 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 definition of an arbitrary Veblen function:

\(\varphi(\lambda_{[b]})\)

We can unpack the arbitrary Veblen function if we let:

\(b = a + 1 + z\)

Then

\(\varphi(\lambda_{[b]}) = \varphi(\lambda_{[a + 1 + z]})\)

And

\(\varphi(\lambda_{[b]}) = \varphi(\lambda_{[a + 1 + z]}) = \varphi(\delta_{[a]},\gamma,0_{[z]})\)

Where

\(\delta_1 > 0\) if \(a > 0\)

We can continue to unpack the arbitrary Veblen function if we let:

\(a = x + 1 + y\)

Then

\(\varphi(\delta_{[a]},\gamma,0_{[z]}) = \varphi(\delta_{[x + 1 + y]},\gamma,0_{[z]})\)

And

\(\varphi(\delta_{[a]},\gamma,0_{[z]}) = \varphi(\delta_{[x + 1 + y]},\gamma,0_{[z]}) = \varphi(\alpha_{[x]},\beta,0_{[y]},\gamma,0_{[z]})\)

Where

\(\alpha_1 > 0\) if \(x > 0\)

We can now begin the rule-set:

\(\gamma\) \(z\) \(a\) or \(x\) \(\beta\) \(\varphi(\lambda_{[b]})[n] = \varphi(\delta_{[a]},\gamma,0_{[z]})[n]\)
limit any any any \(= \varphi(\delta_{[a]},\gamma[n],0_{[z]})\)
\(= 1\) \(> 0\) \(a = 0\) any \(= \varphi(\gamma,0_{[z]})[n] = \varphi(1,0_{[z]})[n]\)

\(= \varphi^{\omega[n]}(1_*,0_{[z-1]})\)

successor \(> 0\) \(a > 0\) any \(= \varphi^{\omega[n]}(\delta_{[a]},\gamma-1,0_*,0_{[z-1]})\)
\(= 0\) any limit \(= \varphi(\alpha_{[x]},\beta,0_{[y]},\gamma)[n]\)

\(= \varphi^{\omega[n]}(\alpha_{[x]},\beta[n],0_{[y]},\varphi(\alpha_{[x]},\beta,0_{[y]},\gamma-1) + 1_*)\)

\(= 0\) \(x = 0\) \(= 1\) \(= \varphi(\beta,0_{[y]},\gamma)[n] = \varphi(1,0_{[y]},\gamma)[n]\)

\(= \varphi^{\omega[n]}(1,0_{[y-1]},\varphi(1,0_{[y]},\gamma-1) + 1_*)\)

\(= 0\) \(x > 0\) successor \(= \varphi^{\omega[n]}(\alpha_{[x]},\beta-1,0_{[y]},\varphi(\alpha_{[x]},\beta,0_{[y]},\gamma-1) + 1_*)\)


Calculated Example for \(\zeta_0[n]\)

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

\(\zeta_0[3] = \varphi(2,0)[3] = \varphi^{\omega[3]}(1,0_*) = \varphi^3(1,0_*) = \varphi(1,\varphi(1,\varphi(1,0)))\)


Calculated Example for \(\Gamma_0[n]\)

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

\(\Gamma_0[3] = \varphi(1,0,0)[3] = \varphi^{\omega[3]}(1_*,0) = \varphi^3(1_*,0) = \varphi(\varphi(\varphi(1,0),0),0)\)


Calculated Example for Small Veblen Ordinal \(SVO[n]\)

SVO is defined as follows:

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

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

\(SVO[3] = \varphi(1,0_{[\omega]})[3] = \varphi(1,0_{[\omega[3]]}) = \varphi(1,0_{[3]}) = \varphi(1,0,0,0)\)

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