Little Bigeddon is a googologism based on an extension of the language of set theory. It was defined on the 5th January 2017 by user Emlightened.[1] User LittlePeng9, the creator of BIG FOOT, wrote "... I'd say this is a large number worth losing to ..." on the original blog post.[2] Little Bigeddon was supposed to be larger than BIG FOOT. Unfortunately, BIG FOOT is ill-defined, and hence this comparison does not make sense.

Disregarding naive extensions, Little Bigeddon was generally considered the largest named number. However, Oblivion, Utter Oblivion, and any Oblivion-based functions could have been considered larger than Little Bigeddon, but it was questionable if they are sufficiently well-defined and sufficiently compliant to the basic rules of googology, to take that title. At last, it has turned out that the definition of Little Bigeddon also includes many errors.

Definition of Little Bigeddon

To the language of set theory, we add an extra sort of variables, which is called the rank variables, which can be quantified by a designated rank quantifier \(\forall_R\), and a trinary predicate \(T\), which is the transfinitely iterated truth predicate. We then define the Little Bigeddon as 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 \(\exists\lnot a(\varphi(a))\land\varphi(k)\).

Issue

The definition contains many errors pointed out by p進大好きbot. For example, there is no \(d \in c\) satisfying the condition \(\forall e \exists ! f(d = \langle e, f \rangle)\) in the second line of the definition of \(T\), i.e. \(T\) is always false. The Goedel codes \(\ulcorner d = e \urcorner\) and \(\ulcorner d \in e \urcorner\) in the fourth and fifth lines do not make sense, and hence seem to be typos of \(\ulcorner x_d = x_e \urcorner\) and \(\ulcorner x_d \in x_e \urcorner\). The evaluation \(c(d)\) in the fouth, fifth, and sixth lines is undefined.

Therefore, strictly speaking, Little Bigeddon is ill-defined unless such massive errors are removed.

Sources

  1. Emlightened. Little BigeddonLarge Ordinums
  2. Emlightened. Little Bigeddon

See also

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