User blog:Ikosarakt1/Symbols function

I found a function that seems to be computable, but it is probably grows faster than any other computable.

I define s(n) by following rules:

We have basically decimal digits, indexes, # symbol means rest of expression, parenthesis, commas, the triple point (one symbol) and all elementary functions. Then we construct the expression, which can lead to a large number, using at most n symbols, and replace all variables to 10. We need to sort out all such possible expressions and form number from each one. The largest number is the value of s(n).

As n becomes larger, more notations might be defined, so s(n) stands above all recursive notations.

Some values:

\(s(0) = 0\), no possible expressions.

\(s(1) = 10\), one possible expression is single n

\(s(2) = 10^{10}\), I use standard notation and 2 symbols.

\(s(3) = 10^{10^{10}}\)

\(s(4) = 10^{10^{10^{10}}}\)

\(s(5) = 10^{10^{10^{10^{10}}}}\)

Then I define new 6 symbol notation: \({}^nn = n^{...^{n}}\) (there are 6 symbols, two n's before equality sign, third =, and n, ... and n for 4,5 and 6 symbol respectively. At this point, I am not sure that it is lead to the largest number defined using 6 symbols, but though it is lower bound, thus:

\(s(6) >= {}^{10}10 = 10 \uparrow \uparrow 10 \)

\(s(7) >= {}^{^{10}10}10 = 10 \uparrow \uparrow \uparrow 3 \)

\(s(8) >= {}^{^{^{10}10}10}10 = 10 \uparrow \uparrow \uparrow 3 \)

To explain what it is really powerful, I counted number of symbols which need to define multi-dimensional array notation does not surpass 335 symbols, thus:

\s(335) > \lbrace 10,10 (10) 2 \rbrace \)

Just imagine how big might be s(1000). s(1000000) surpass any number which ever defined by notation, s(googol) probably around Rayo's number.

What about s(s(5)). We need pentalogue symbols to define this number. Nextly comes things like s(s(s(1))), s(s(s(s(...(s(s(1)...) (with s(s(1)) symbols), and so on.