BIG FOOT

BIG FOOT is a counterpart of Rayo's number based on an extended version of the language of first-order set theory. As a result, it was considered to be among the largest named numbers. However, BIG FOOT is ill-defined by the reason explained a later section. It was defined in October 2014 by an author under the pen name "Wojowu" or "LittlePeng9", and was given its name by Sbiis Saibian.

Its definition is almost identical to Rayo's number, another well-known large number which diagonalizes over first-order formulas in the von Neumann universe (which is the universe of discourse for first-order set theory). BIG FOOT extends first-order set theory by making use of a unique domain of discourse called the oodleverse, using a language called first-order oodle theory (FOOT), and was supposed to generalize nth-order set theory of arbitrarily large n.

Letting \(\text{FOOT}(n)\) denote the largest natural number uniquely definable in the language of FOOT in at most \(n\) symbols, we define BIG FOOT as \(\text{FOOT}^{10}(10^{100})\), where \(\text{FOOT}^{a}(n)\) is \(\text{FOOT}(n)\) iterated \(a\) times (recursion). BIG FOOT is thus equal to:

FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(10100))))))))))

Definition of FOOT
The language of first-order oodle theory is defined as the language of set theory augmented with the symbols \([\) and \(]\). The universe of discourse consists of oodles, which are subject to the Tarskian definition of truth for a set theory. We call \(\in\)-transitive oodles oodinals, and consider \(\in\) as the ordering relation amongst them (so that we can speak of "larger" and "smaller" oodinals).

The FOOT function is the oodle-theory analogue to Rayo's function, where oodle-theory is an extension of set-theory.

Because all the structures involved are elements of the universe of discourse, FOOT has turned out to be equivalent in strength to FOST with a single adjoined.

Ill-definedness
The main problem in the definition of BIG FOOT is the lack of the precise clarification of axioms other than the axioms of extensionality and power set. In order to define an uncomputable large number, we need to fix under what axioms we define it, but Wojowu just explained that he assumed that oodles satisfy all the properties which are considered "natural" properties. In mathematics, we traditionally omit the axioms only when we work in ZFC set theory. On the other hand, since the definition of FOOT uses the existence of several oodinals which does not follow from the axioms of \(\textrm{ZFC}\) set theory, it is reasonable to guess that Wojowu assumed stronger axioms.

Unfortunately, such a set theory contradicts. Namely, for any set theory \(T\) extending \(\textrm{ZFC}\) set theory, if FOOT is well-defined in \(T\), then \(T\) is inconsistent. The following proof is originally posted by the Googology Wiki user p進大好きbot :

Suppose that the transfinite ordinal \(\alpha_0\) in the original definition is formalised in \(T\) by a defining formula \(\varphi(\alpha)\) with a free occurrence of a variable term \(\alpha\). Then the existence of \(\alpha_0\) satisfying \(\varphi(\alpha_0)\) ensures that the existence of \(\beta < \alpha_0\) satisfying \(\varphi(\beta)\) by the definition of \(\alpha_0\). By the minimality of \(\alpha_0\), it implies \(\alpha_0 = \beta < \alpha_0\), which contradicts the well-foundedness of \(\alpha_0\).

Even if we ignore the problem above, we have another problem. Wojowu required that for any formula \(\varphi(\alpha)\) with a free occurrence of a variable term \(\alpha\), for any \(A \in V_{\textrm{Ord}}\), then \(\varphi(A)\) is true in \(V\) if and only if it is in \(V_{\textrm{Ord}}\). It implies that for any closed formula \(\varphi\), \(V_{\textrm{Ord}} \models \varphi\) is equivalent to \(\varphi\). Let \(\varphi\) denote the existence of \((\alpha_n)_{n < \omega}\), which is true under \(T\) by the assumption. Then \(V_{\textrm{Ord}}\) satisfies \(\varphi\), and hence \((\alpha_n^{V_{\textrm{Ord}}})_{n < \omega}\) is well-defined. On the other hand, \(\alpha_0^{V_{\textrm{Ord}}}\) satisfies the same property as \(\alpha_0\). Indeed, for any parameter-free formula \(\phi\), \(\phi^{V_{\textrm{Ord}}}\) is equivalent to \(\phi\) under \(T\). Since many formulae, e.g. \(\beta \in \alpha\), "\(\alpha\) is oodle", "\(n\) is the Goedel number of a formula", and so on, is absolute with respect to the inclusion \(V_{\textrm{Ord}} \hookrightarrow V\), it implies \(\alpha_n^{V_{\textrm{Ord}}} = \alpha_n\) for any \(n < \omega\). We obtain \(\textrm{Ord}^{V_{\textrm{Ord}}} = \textrm{Ord}\), which contradicts the smallness of \(\textrm{Ord}\).

As a conclusion, FOOT and BIG FOOT are ill-defined. In particular, salad numbers containing them are ill-defined.

Alternative formulation
As is shown above, the original definition of BIG FOOT contradicts reasonable set theories. The first problem is the ill-definedness of \(\alpha_0\), whose reflection property universally quantifies (Goedel numbers of) formulae \(\varphi(\alpha)\) with a free occurrence of a variable term \(\alpha\). On the other hand, if we consider the theory \(T_0\) given by adding a constant term symbol \(a_0\) to the language of the first order oodle theory and the schema \(a_0 \in \textrm{On} \land ((\exists \beta \in \textrm{On}, \Phi(\beta)) \to (\exists \beta < a_0, \Phi(\beta)))\) on formulae \(\Phi(\alpha)\) with a free occurrence of a variable term \(\alpha\) to the original unspecified axiom, a variant of \(\alpha_0\) is definable as \(a_0\) in \(T_0\). Here \(\textrm{On}\) denotes the class of oodinals. Although \(a_0\) does not formalise the original \(\alpha_0\), it can play a similar role. Similarly, a variant \(a_n\) of \(\alpha_n\) is definable in the theory \(T_n\) constructed in a similar way for any meta-theoretic natural number \(n\).

Here, a natural number in the least common extension \(T_{\infty}\) of the tower \((T_n)_{n \in \mathbb{N}}\) of the first order oodle theory is not necessarily equal to a meta-theoretic natural number \(n\), i.e. the natural number whose defining formula is syntax theoretically given as "the \(n\)-th successor of \(0\)". Therefore constructing \(a_n\) for each meta-theoretic natural number \(n\) does not gives a well-defined sequence \((a_n)_{n \in \mathbb{N}}\) in \(T_{\infty}\). In order to construct a variant of \(\textrm{Ord}\), we need further arguments.

One may define a variant of BIG FOOT in such a direction, but the resulting number will be completely different from BIG FOOT.