FANDOM


So, I'm working on a new number - one which will hopefully be large.

We will assume syntax is clear from the context, context, yet more context and that an amateur googologist will not need to have everything explicitly defined but instead referenced (i.e. mentioning Godel encoding).

Firstly, we will define a function to Godel encode a sentence in FOST + Magnarth: \(\texttt{en}(\cdot)\). We will define the alphabet \(A\), for the encoding, as so:

\(A = \{\langle=\rangle,\langle\in\rangle,\langle\forall\rangle,\langle\wedge\rangle,\langle\neg\rangle,\langle(\rangle,\langle)\rangle,\langle\texttt{mg}\rangle,\langle,\rangle,\langle\{\rangle,\langle\}\rangle\} \cup \bigcup_{n<\omega} {\langle n \rangle, \langle x_n \rangle}\)

We define \(\texttt{mg}(x_a, x_b, x_c)\) (where all sets are \(\in V_\omega\), \(a<\omega\), \(b<\omega\) and, \(c<\omega\)); \(x_a\) shall be a Godel coded sentence in the language of FOST + Magnarth (unless the rank is 0 where the Magnarth predicate is not included); \(x_b\) shall be the rank, a Godel coded finite tuple, which shall be described moreover later, in the form: \((a, b, c, d, ...)\), or \((a)\), or \((a, b)\), or etc. before Godel encoding; \(x_c\) shall be a Godel coded finite tuple of pairs, assigning each variable defined in the Godel coded sentence \(x_a\) a set, in the form: \(((x_0, \{\}),(x_1, \{\{\},\{\}\}), ...)\), or \(((x_0, \{\{\},\{\{\}\}\}))\), or \(((x_0,\{\{\{\}\},\{\}\}),(x_1, \{\{\}\}))\), or etc. before Godel encoding.

\(x_b\): When the tuple is one element, that element \(i\), in the tuple \((i)\), shall decide the maximum rank of any Magnarth predicate \(\texttt{mg}\) defined in Godel coded FOST + Magnarth sentences; if \(i = 0\), no Godel coded FOST sentence can feature the Magnarth predicate; if \(i > 0\), no Godel coded sentence FOST + Magnarth sentence can feature the Magnarth predicate with rank \(> i\).

When the tuple is more than one element, those elements \(i\), \(j\), \(k\) etc, in the tuple \((i, j, k, ...)\), shall decide the maximum rank of any Magnarth predicate \(\texttt{mg}\), also; If the tuple is \((1, 0)\), the rank of any Magnarth predicate \(\texttt{mg}\) defined in Godel coded FOST + Magnarth sentences can be any finite one-element tuple rank; For any \(n+1\)-tuple of the form \((1, 0, 0, 0, ...)\) (with a finite set of zeroes after a one) the rank for any Magnarth predicate is at most any finite \(n\)-tuple; If we have an \(n\)-tuple of the form \((\alpha, 0, 0, 0, ...)\), the rank of any Magnarth predicate can be at most any finite n-tuple with a first element being \(<\alpha\).

We define \(\alpha(n)\) to be the largest natural number defined in the language of FOST + Magnarth in at most n symbols; We define First Magnarth to be \(\alpha^{1000}(10^{100})\), where function iteration is employed.


(to be continued, hidden comments)

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.