The new biggest defined number?


Anyway I believe that this is the biggest number yet to grace this website. It is not defined in the language \(\mathcal L = \{\in\}\), like Rayo's number, but in a larger language, like BIG FOOT.

Recently, a discussion happened wherein LittlePeng9 came to the conclusion that the language FOOT is as expressive as \(\{\in,T\}\), where \(T\) is a truth predicate for parametered formulae in the language \(\{\in\}\). To see this, note that such a predicate can define the class of correct cardinals FOOT uses, and that a proper class of correct cardinals can be used to define such a truth predicate, by using the least correct cardinal above all of the given parameters.

An alternate but equivalent approach, incidentally, is to extend the language of FOOT by introducing correct for correct cardinals cardinals, correct for correct for correct cardinals cardinals cardinals, ad transfinitum.

Anyway, for now, here's my number:

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

We assume that basic notations need not be explicitly defined, and that use of set builder notation and induction is clear from context. Define the following:

\(a = \langle b,c\rangle \leftrightarrow a=\{\{b,b\},\{b,c\}\}\)

\(a = \langle b,c,d\rangle \leftrightarrow a=\langle \langle b,c\rangle ,d \rangle \)

\(a = \langle b,c,d,e\rangle \leftrightarrow a=\langle \langle b,c,d \rangle ,e \rangle \)

\(On(a) \leftrightarrow \forall b\in a(\cup b\in a \wedge\forall c\in b(\cup c \in b))\)

The left (call it Code) variable on the truth predicate is the encoding of the predicate, the middle (call it Rank) variable is the rank, and the last (call it Param) is the function that assigns parameters to the free variable indicies. \(T(a,\beta,c)\) can be represented as \(T_\beta(a,c)\). 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 \(\forall \alpha<\beta (T_\alpha(\ulcorner\phi(\alpha)\urcorner,c)\) for some \(\phi\), for instance.

The following definition of Godel coding is technically redundant as we could assign tuples directly, however has been included for clarity. The Godel coding for (parameterless) formulae encodes each formula as some set \(\in V_\omega\) as follows (where \(i<\omega\), \(j<\omega\) and \(k<\omega\)):

\(\ulcorner x_i = x_j \urcorner = \langle 0, i, j \rangle\)

\(\ulcorner x_i \in x_j \urcorner = \langle 1, i, j \rangle\)

\(\ulcorner T(x_i, x_j,x_k) \urcorner = \langle 2, i, j,k \rangle\)

\(\ulcorner \varphi \wedge \psi \urcorner = \langle 3, \ulcorner \varphi \urcorner, \ulcorner \psi \urcorner \rangle\)

\(\ulcorner \lnot\varphi \urcorner = \langle 4, \ulcorner \varphi \urcorner \rangle\)

\(\ulcorner \forall x_i\varphi \urcorner = \langle 5, i, \ulcorner \varphi \urcorner \rangle\)

\(\ulcorner \forall_R x_i\varphi \urcorner = \langle 6, i, \ulcorner \varphi \urcorner \rangle\)

(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:

\(\texttt{fr}(\ulcorner x_i = x_j \urcorner) = \texttt{fr}(\ulcorner x_i \in x_j \urcorner)= \{i, j\}\)

\(\texttt{fr}(\ulcorner T(x_i, x_j, x_k) \urcorner) = \{i, j, k\}\)

\(\texttt{fr}(\ulcorner \varphi \wedge \psi \urcorner) = \texttt{fr}(\ulcorner \varphi \urcorner) \cup \texttt{fr}(\ulcorner \psi \urcorner)\)

\(\texttt{fr}(\ulcorner \lnot \varphi \urcorner) = \texttt{fr}(\ulcorner \varphi \urcorner)\)

\(\texttt{fr}(\ulcorner \forall x_i\varphi \urcorner) = \texttt{fr}(\ulcorner \forall_R x_i\varphi \urcorner) = \texttt{fr}(\ulcorner \varphi \urcorner) \setminus \{i\}\)

\(\texttt{bd}(\ulcorner x_i = x_j \urcorner) = \texttt{bd}(\ulcorner x_i \in x_j \urcorner) = \texttt{bd}(\ulcorner T(x_i, x_j, x_k) \urcorner) = \emptyset\)

\(\texttt{bd}(\ulcorner \varphi \wedge \psi \urcorner) = \texttt{bd}(\ulcorner \varphi \urcorner) \cup \texttt{bd}(\ulcorner \psi \urcorner)\)

\(\texttt{bd}(\ulcorner \lnot \varphi \urcorner) = \texttt{bd}(\ulcorner \varphi \urcorner)\)

\(\texttt{bd}(\ulcorner \forall x_i\varphi \urcorner) = \texttt{bd}(\ulcorner \forall_R x_i\varphi \urcorner) = \texttt{bd}(\ulcorner \varphi \urcorner) \cup \{i\}\)

\(\texttt{rk}(\ulcorner x_i = x_j \urcorner) = \texttt{rk}(\ulcorner x_i \in x_j \urcorner) = \emptyset\)

\(\texttt{rk}(\ulcorner T(x_i, x_j, x_k) \urcorner) = \{j\}\)

\(\texttt{rk}(\ulcorner \varphi \wedge \psi \urcorner) = \texttt{rk}(\ulcorner \varphi \urcorner) \cup \texttt{rk}(\ulcorner \psi \urcorner)\)

\(\texttt{rk}(\ulcorner \lnot \varphi \urcorner) = \texttt{rk}(\ulcorner \forall x_i\varphi \urcorner) =  \texttt{rk}(\ulcorner \forall_R x_i\varphi \urcorner) = \texttt{rk}(\ulcorner \varphi \urcorner)\)

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

\(\texttt{form} = \cup_{n<\omega} form_n\)

\(form_0 = \{\langle 0, i, j \rangle: \wedge i+j<\omega\} \cup \{\langle 1, i, j \rangle: \wedge i+j<\omega\} \cup \{\langle 2, i, j, k \rangle: \wedge i+j+k<\omega\}\)

\(form_{n+1} = form_n \cup \{\ulcorner \varphi \wedge \psi \urcorner, \ulcorner \lnot\varphi \urcorner : \varphi, \psi \in form_n\} \)

\(\cup \{\ulcorner \forall x_i\varphi \urcorner: i\in \texttt{fr}(\ulcorner\varphi\urcorner)\setminus \texttt{rk}(\ulcorner\varphi\urcorner) \wedge \ulcorner\varphi\urcorner \in form_n\}\)

\(\cup  \{\ulcorner \forall_R x_i\varphi \urcorner: i\in \texttt{fr}(\ulcorner\varphi\urcorner)\cap \texttt{rk}(\ulcorner\varphi\urcorner) \wedge \ulcorner\varphi\urcorner \in form_n\}\)

Lastly, we need to define axioms for \(T\):

\( T(a,b,c)\leftrightarrow[a\in \texttt{form} \wedge On(b)\)

\( \wedge \forall d\in c(\exists e\exists f(d=\langle e,f\rangle)\wedge\forall e [\exists!f(d=\langle e,f\rangle)\)

\( \wedge\forall f(d=\langle e,f\rangle \rightarrow [e\in \texttt{fr}(a)\wedge(e\in\texttt{rk}(a)\rightarrow f\in b)])])\)

\(\wedge (\exists d\in\omega\exists e\in\omega[a=\ulcorner d=e\urcorner\wedge c(d)=c(e)]\)

\(\vee\exists d\in\omega\exists e\in\omega[a=\ulcorner d\in e\urcorner\wedge c(d)\in c(e)]\)

\(\vee\exists d\in\omega\exists e\in\omega\exists f\in\omega[a=\ulcorner T(d,e,f)\urcorner\wedge T(c(d),c(e),c(f))]\)

\(\vee\exists d\in \texttt{form}\exists e\in \texttt{form}[a=\ulcorner d\wedge e\urcorner\wedge T(d,b,c)\wedge T(e,b,c)]\)

\(\vee\exists d\in\texttt{form}[a=\ulcorner \lnot d\urcorner\wedge \lnot T(d,b,c)]\)

\(\vee\exists d\in\omega\exists e\in \texttt{form}[a=\ulcorner \forall x_de\urcorner\wedge\forall f(T(e,b,c\cup\{\langle d,f\rangle\}))]\)

\(\vee\exists d\in\omega\exists e\in \texttt{form}[a=\ulcorner \forall_Rx_de\urcorner\wedge\forall f\in b(T(e,b,c\cup\{\langle d,f\rangle\}))])]\)

Aaaaaaand we're almost done.

Little Biggeddon = The largest number \(k\) such that there is some unary formula \(\varphi\) in the language \(\mathcal L = \{\in,T\}\) of quantifier rank \(\leq 12\uparrow\uparrow 12\) such that \(\exists! a(\varphi(a))\wedge \varphi(k)\).

Aaaaaaand we're done.

Anyway, if you spot any mistakes or have any questions, please comment them.

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