User:Emlightened/Sandbox

Smaller of the two Bigeddons, Little Bigeddon is defined by adding an extra sort and a binary truth predicate to the language of set theory so we have \(\mathcal L = \{\in,T\}\).

The left (call it Code) variable on the truth predicate is the encoding of the predicate, and the right (call it Rank) variable is the rank. \(T(a,\beta)\) can be represented as \(T_\beta(a)\). The predicates are ranked so that we can call truth predicates on truth predicates etc. Specifically, \(T_\beta\) will only decide formulas which use the truth predicates \(T_\alpha\) for \(\alpha<\beta\), but does allow quantification, such as \(\forall \alpha<\beta (T_\alpha(\ulcorner\phi(\alpha)\urcorner)\) for some \(\phi\), for instance.

The Godel coding for (parameterless) formulae encodes each formula as some set \(\in V_\omega\) as follows (where \(i<\omega\) and \(j<\omega\)): \(\ulcorner x_i = x_j \urcorner = \langle 0, i, j \rangle\) \(\ulcorner x_i \in x_j \urcorner = \langle 1, i, j \rangle\) \(\ulcorner T(x_i, x_j) \urcorner = \langle 2, i, j \rangle\) \(\ulcorner \varphi \wedge \psi \urcorner = \langle 3, \ulcorner \varphi \urcorner, \ulcorner \psi \urcorner \rangle\) \(\ulcorner \lnot\varphi \urcorner = \langle 4, \ulcorner \varphi \urcorner \rangle\) \(\ulcorner \forall x_i\varphi \urcorner = \langle 5, i, \ulcorner \varphi \urcorner \rangle\) \(\ulcorner \forall_R x_i\varphi \urcorner = \langle 5, i, \ulcorner \varphi \urcorner \rangle\) (We implicitly sort some of the variables into two different types, as we only want to allow certain types of quantification over the Rank variables. \(\forall_R\) is reserved for precisely the Rank variables.)

We also inductively specify the (used) free variables, the bound variables, and the Rank variables (\(\texttt{fr}(\cdot)\), \(\texttt{bd}(\cdot)\), \(\texttt{rk}(\cdot)\), respectively) of an encoded (parameterless) formula as follows: \(\texttt{fr}(\ulcorner x_i = x_j \urcorner) = \texttt{fr}(\ulcorner x_i \in x_j \urcorner) = \texttt{fr}(\ulcorner T(x_i, x_j) \urcorner) = \{i, j\}\) \(\texttt{fr}(\ulcorner \varphi \wedge \psi \urcorner) = \texttt{fr}(\ulcorner \varphi \urcorner) \cup \texttt{fr}(\ulcorner \psi \urcorner)\) \(\texttt{fr}(\ulcorner \not \varphi \urcorner) = \texttt{fr}(\ulcorner \varphi \urcorner)\) \(\texttt{fr}(\ulcorner \forall x_i\varphi \urcorner) = \texttt{fr}(\ulcorner \forall_R x_i\varphi \urcorner) = \texttt{fr}(\ulcorner \varphi \urcorner) \setminus \{i\}\) \(\texttt{bd}(\ulcorner x_i = x_j \urcorner) = \texttt{bd}(\ulcorner x_i \in x_j \urcorner) = \texttt{bd}(\ulcorner T(x_i, x_j) \urcorner) = \emptyset\) \(\texttt{bd}(\ulcorner \varphi \wedge \psi \urcorner) = \texttt{bd}(\ulcorner \varphi \urcorner) \cup \texttt{bd}(\ulcorner \psi \urcorner)\) \(\texttt{bd}(\ulcorner \not \varphi \urcorner) = \texttt{bd}(\ulcorner \varphi \urcorner)\) \(\texttt{bd}(\ulcorner \forall x_i\varphi \urcorner) = \texttt{bd}(\ulcorner \forall_R x_i\varphi \urcorner) = \texttt{bd}(\ulcorner \varphi \urcorner) \cup \{i\}\) \(\texttt{rk}(\ulcorner x_i = x_j \urcorner) = \texttt{rk}(\ulcorner x_i \in x_j \urcorner) = \emptyset\) \(\texttt{rk}(\ulcorner T(x_i, x_j) \urcorner) = \{j\}\) \(\texttt{rk}(\ulcorner \varphi \wedge \psi \urcorner) = \texttt{rk}(\ulcorner \varphi \urcorner) \cup \texttt{rk}(\ulcorner \psi \urcorner)\) \(\texttt{rk}(\ulcorner \not \varphi \urcorner) = \texttt{rk}(\ulcorner \forall x_i\varphi \urcorner) =  \texttt{fr}(\ulcorner \forall_R x_i\varphi \urcorner) = \texttt{rk}(\ulcorner \varphi \urcorner)\)

Now we define the set of valid (parameterless) formulae \(\texttt{form}\) by induction as follows:

\(\texttt{form} = \cup_{n<\omega} form_n\) \(form_0 = \{\langle a, i, j \rangle: a<3 \wedge i<\omega \wedge j<\omega\}\) \(form_{n+1} = form_n \cup \{\ulcorner \varphi \wedge \psi \urcorner, \ulcorner \lnot\varphi \urcorner : \varphi, \psi \in form_n\} \cup \{\ulcorner \forall x_i\varphi \urcorner: i\in \texttt{fr}(\ulcorner\varphi\urcorner)\setminus \texttt{bd}(\ulcorner\varphi\urcorner) \wedge \ulcorner\varphi\urcorner \in form_n\cdots\)