The M Function
- 1 The M Function
- 2 Full Definition of the \(M()\) Function
- 3 Evaluating \(M(x(0),n)\)
- 4 Comparing \(x()\) functions to ordinals
- 5 Some other calculations of the \(x()\) function
- 6 Further References
I have created the M Function that extends recursion ideas from my previous blog on Generalised Recursion.
Basic Definition for this function is:
\(M(n) = n + 1\) and \(M(c + 1,n) = M(c,R(\mathbb{C},\mathbb{C},\mathbb{C}))\)
It is a faster growing version of my \(Z()\) function from the same blog. We can compare the two as follows:
\(Z(1,1) = Z(R(1,1,\mathbb{C})) = Z(R(1,1,2)) = 1 + 2 = 3\)
\(M(1,1) = M(R(\mathbb{C},\mathbb{C},\mathbb{C})) = M(R(2,2,2)) = 2.2 + 1 = 5\)
\(Z(1,2) = Z(R(2,2,3)) = 2.3 + 1 = 7\)
\(M(1,2) = M(R(3,3,3)) = \displaystyle \sum_{i=0}^3 3^i…
Generalised Recursion
- 1 Generalised Recursion
- 2 The Generalised Recursion Function
- 3 Scope of the Generalised Recursion Function
- 4 Some Interesting Functions
- 5 The \(Z()\) Function
- 6 Evaluating the \(Z()\) Function
- 7 Evaluating \(Z(2,1)\)
- 8 Comparing \(Z(1,n)\) to \(f_{\omega}(n)\)
- 9 Comparing \(Z(n,n)\) to \(f_{\omega^2}(n)\)
- 10 Some other calculations of the \(Z()\) function
- 11 Further References
Following on from my previous blog on Functional Notation, I have started to generalise the notation for recursion of functions.
Starting with
\(F(a_{[x]},b|1,d_{[y]}) = F(a_{[x]},b,d_{[y]})\)
\(F(a_{[x]},b|c + 1,d_{[y]}) = F(a_{[x]},\mathbb{C},d_{[y]})\)
We can repeat the use of the \(|\) symbol and define the following:
\(F(a_{[x]},b||1,d_{[y]}) = F(a_{[x]},b,d_{[y]})\)
\(F(a_{[x]},b||c + 1,d…
My Functional Notation
- 1 My Functional Function
- 2 Cardinal Number of Functional Parameters
- 3 Ordinal Position of a Functional Parameter
- 4 Leading Zero Rule
- 5 The Decrement Function
- 6 Functional Recursion
- 7 The Quantum Function
- 8 Further References
I have developed Functional Notation that simplifies the presentation of functions and their rules. Especially when the functions are used recursively as is common for Googological functions.
\(F(a_{[x]}) = F(a_1,a_2,...,a_x)\)
This notation compresses the representation of functions with many parameters. Especially when the cardinal number of parameters may be a Googologically large number.
Examples:
\(F(1_{[2]}) = F(1,1)\)
\(F(a,0_{[4]}) = F(a,0,0,0,0)\)
\(F([x]a) = F(a,0_{[x]})\)
This notation further simplifies the representation of a fu…
The a^n(n) Sequence
- 1 The \(\alpha^n(n)\) Sequence
- 2 The Sequence
- 3 The Calculations
- 4 Further References
Here is the sequence you get when you diagonalise over the the Alpha Function.
\(\alpha^0(0) = 0\) This is a trivial result.
\(\alpha^1(1) = \alpha(1) = 1\)
\(\alpha^2(2) = \alpha(10) = Q(t_{1}^{Q(Q^{2}(Q(t_{0}(t_{0}^{Q(Q(1,t_{0}(0)))}(1_*,0)),t_{1}(0))),t_{1}(0))}(0),2)\)
\(> f_{svo}(2)\)
\(\alpha^3(3) = \alpha^2(33)\)
\(= \alpha(Q(t_{2}(Q^{Q(Q^{Q^{t_{0}(0)}(t_{0}(0))}(1,t_{0}(0)_*))}(Q(Q^{[1]}(Q^{Q(Q(t_{0}(0)),t_{0}(0))}(1,t_{0}(1)_*),t_{0}(1)))_*,t_{0}(1)),0),2))\)
\(> \alpha(Q(t_{2}(t_{0}(1),0),n))\) for any reasonable value of n
\(> \alpha(f_{LVO}(n))\) for any reasonable value of n
\(> f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\) for any reasonable value of n
Yo…
Q(1,0,1) is bigger than TREE(3)
- 1 Q(1,0,1) is bigger than TREE(3)
- 2 Calculating Q(1,0,1)
- 3 Comparing Q(1,0,1)
- 4 my Alpha Function
- 5 Finding Q(1,0,1)
- 6 Further References
This is a worked example of my Quantum Function to compare it to a big number like TREE(3).
\(Q(1,0,1) = Q(t_{Q(1,1)}(0),1)\)
\(= Q(t_{3}(0),1)\)
\(> Q(t_{2}(0),2)\)
\(> f_{LVO}(3)\) where \(LVO \geq \vartheta(\Omega^\Omega)\)
\(> TREE(3)\) or \(\geq f_{\vartheta(\Omega^\omega\omega)}(3)\)
The Quantum Function is computable, and follows simple rules beginning with:
\(Q(n) = n + 1\)
So it is easy to construct incrementally larger number numbers using the rules, for example:
\(Q(Q(1,0,1)) = Q(1,0,1) + 1\)
\(Q(Q(Q(1,0,1))) = Q(1,0,1) + 2\)
\(Q^{3}(Q(1,0,1)) = Q(Q(Q(Q(1,0,1)))) = Q(1,0,1) + 3\)
\(Q^{4}(Q(1,0,1)) = Q(1,0,1) + 4\)
\(Q…