User blog:B1mb0w/Notation Extended

Notation Extended
This blog extends the notation I use that is explained here. You will see this notation in use on my blogs. This blog will present the exact definition I intend for these notations. Any deviance from the notation here is most likely due to an error in one of my other blogs.

They notations are parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*\), and the decremented function \(C\).

Parameter Subscript Brackets
\(M(a,0_{[2]}) = M(a,0,0)\)

\(M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)\)

Leading Zeros Assumption
\(M(0_{[x]},0_{[2]},b_{[3]},1) = M(0_{[x + 2]},b_1,b_2,b_3,1) = M(b_1,b_2,b_3,1)\)

Recursion Parameter Subscript \(*\)
\(M^2(a) = M^2(a_*) = M(M(a))\) and \(M(a,b_*) = M(a,b)\)

\(M^2(a,b_*) = M(a,M(a,b))\)

\(M^2(a_*,b) = M(M(a,b),b)\)

Decremented Function \(C\)
For any function:

\(M(a_{[b]},c + 1,0_{[d]})\)

then function \(C\) is defined as:

\(C = M(a_{[b]},c,0_{[d]})\)

This becomes clearer when it is assumed the scope of \(C\) is limited ónly to the right hand side of an equation for which function \(M\) appears on the left hand side.

Here is an example:

\(M(a_{[b]},c + 1,0_{[d]}) = C^2\)

is equivalent to:

\(M(a_{[b]},c + 1,0_{[d]}) = M(a_{[b]},c,0_{[d]})^2\)

Defining the Veblen function
Here is an example of this notation being used to define the Veblen function.

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

\(\varphi(c + 1) = O^C\)

\(\varphi(1,c + 1) = C\uparrow\uparrow O\)

\(\varphi(1,0_{[y + 1]},c + 1) = \varphi^O(1,0_{[y]},(C + 1)_*)\)

\(\varphi(a_{[x]},b + 1,c + 1) = \varphi^O(a_{[x]},b,(C + 1)_*)\)

\(\varphi(1,0_{[z + 1]}) = \varphi^O(1_*,0_{[z]})\)

\(\varphi(a_{[x]},c + 1,0_{[z + 1]}) = \varphi^O(a_{[x]},c,0_*,0_{[z]})\)

and

\(\varphi^{c + 1}(a_{[x]},b_*,d_{[z]}) = \varphi^c(a_{[x]},(\varphi(a_{[x]},b,d_{[z]}) + 1)_*,d_{[z]})\)

Further References
Further references to relevant blogs can be found here: User:B1mb0w