FANDOM


Rule 2B will be corrected

This blog will be used to correct this rule which appears in my blog on fundamental sequences. Rule 2B is one of the rules in the "Aristo Sequence". These notational conventions are used by this sequence:

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

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

and

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

Rule 2b: if \(a_x\) is not a limit ordinal and \(y=0\) then:

\(\varphi(a_{[x]},0_{[y]})[n] = \varphi(a_{[x]})[n] = \varphi(a_{[x-1]},a_x-1)\uparrow\uparrow\omega[n] = \varphi(a_{[x-1]},a_x-1)\uparrow\uparrow n\)


Correction

Rule 2B will be corrected using this definition

\(\gamma\uparrow\uparrow\omega = \varphi(1,\gamma+1) = \epsilon_{\gamma+1}\)

and

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

when n is a finite integer.


Sequence Table

The following table is used to outline the ascending sequence of ordinals beyond \(\zeta_0\):

ascending ordinals
Start \(\varphi(2,0)\)
\(\varphi(2,0)+1\)
... \(\varphi(2,0).2\)
... \(\varphi(2,0)^2\)
... \(\varphi(2,0)\uparrow\uparrow 2\)
... \(\varphi(2,0)\uparrow\uparrow\omega\)
= \(\varphi(1,\varphi(2,0)+1)\)
... \(\varphi(1,\varphi(2,0).2)\)
... \(\varphi(1,\varphi(2,0)^2)\)
... \(\varphi(1,\varphi(2,0)\uparrow\uparrow 2)\)
... \(\varphi(1,\varphi(2,0)\uparrow\uparrow\omega)\)
= \(\varphi^2(1,\varphi(2,0)+1_*)\)
= \(\varphi(1,\varphi(1,\varphi(2,0)+1_*))\)
... \(\varphi(1,\varphi(1,\varphi(2,0)+1)+1)\)
... \(\varphi(1,\varphi(1,\varphi(2,0)+1)+2)\)
... \(\varphi(1,\varphi(1,\varphi(2,0)+1).2)\)
... \(\varphi(1,\varphi(1,\varphi(2,0)+1)^2)\)
... \(\varphi(1,\varphi(1,\varphi(2,0)+1)\uparrow\uparrow 2)\)
... \(\varphi(1,\varphi(1,\varphi(2,0)+1)\uparrow\uparrow\omega)\)
= \(\varphi(1,\varphi(1,\varphi(2,0)+2))\)
... \(\varphi(1,\varphi(1,\varphi(2,0).2))\)
... \(\varphi(1,\varphi(1,\varphi(2,0)^2))\)
... \(\varphi(1,\varphi(1,\varphi(2,0)\uparrow\uparrow 2))\)
... \(\varphi(1,\varphi(1,\varphi(2,0)\uparrow\uparrow\omega))\)
= \(\varphi(1,\varphi(1,\varphi(1,\varphi(2,0)+1)))\)
= \(\varphi^3(1,\varphi(2,0)+1_*)\)
... \(\varphi^{\omega}(1,\varphi(2,0)+1_*)\)
= \(\varphi(2,1)\)
... \(\varphi(2,n)\)
... \(\varphi^{\omega}(1,\varphi(2,n)+1_*)\)
= \(\varphi(2,n+1)\)
... \(\varphi(2,\varphi(2,0))\)
= \(\varphi^2(2,0_*)\)
... \(\varphi^{\omega}(2,0_*)\)
= \(\varphi(3,0)\)

The rule inferred from the above is:

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

The sequence continues for values m >= 3.

ascending ordinals
Start \(\varphi(3,0)\)
... \(\varphi(3,0)\uparrow\uparrow\omega\)
= \(\varphi(1,\varphi(3,0)+1)\)
... \(\varphi^{\omega}(1,\varphi(3,0)+1_*)\)
= \(\varphi(2,\varphi(3,0)+1)\)
... \(\varphi^{\omega}(2,\varphi(3,0)+1_*)\)
= \(\varphi(3,1)\)
... \(\varphi(3,n)\)
... \(\varphi(m,0)\)
... \(\varphi(\varphi(1,0),0)\)
... \(\varphi^{\omega}(\varphi(1,0)_*,0)\)
= \(\varphi(1,0,0)\)

The rule inferred from the above is:

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

A general rule is then proposed as:

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

when n is a finite integer.


Further Observation

The rule inferred from the above is:

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

With a further observation that a general equality is:

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

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

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

This rule helps to clarify the following equivalence:

\(\varphi(1,\varphi(2,0) + 1)\uparrow\uparrow\omega = \varphi(1,\varphi(2,0) + 2)\)

The following result is not equivalent and is incorrect because \(m+1 = n\) or \(1+1 = 2\)

\(\varphi(1,\varphi(2,0) + 1)\uparrow\uparrow\omega = \varphi(1,\varphi(1,\varphi(2,0) + 1) + 1)\) is incorrect.

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