## FANDOM

10,824 Pages

You wanted more.

I forgot for a while.

Wojowu helped.

This. Is Big Bigeddon.

Although Sasquatch fits better.

And I'm not going to be quite as rigorous as my last post. That said, we work in the language $$(\in, \bar\in, <)$$, where equality is a defined symbol. $$\in$$, $$\bar\in$$ and $$<$$ are binary predicates, we also define the unary functions $$F$$ and $$R$$ from these.

Fix a Godel numbering of formulae:

$$\ulcorner t_1 \in t_2 \urcorner = \langle 0, \llcorner t_1 \lrcorner, \llcorner t_2 \lrcorner \rangle$$

$$\ulcorner \varphi \wedge \psi \urcorner = \langle 1, \ulcorner \varphi \urcorner, \ulcorner \psi \urcorner \rangle$$

$$\ulcorner \lnot\varphi \urcorner = \langle 2, \ulcorner \varphi \urcorner \rangle$$

$$\ulcorner \forall x_i\varphi \urcorner = \langle 3, \llcorner x_i \lrcorner, \ulcorner \varphi \urcorner \rangle$$

And of terms:

$$\llcorner x_i \lrcorner = \langle 4, i \rangle$$

$$\llcorner a \lrcorner = \langle 5, a \rangle$$

$$\llcorner R(t) \lrcorner = \langle 6, \llcorner t \lrcorner \rangle$$

$$\llcorner F(t') \lrcorner = \langle 7, \llcorner t' \lrcorner \rangle$$

Where $$t'$$ contains no free variables, and $$\llcorner a \lrcorner$$ represents the set $$a$$

Fix $$<$$ to be a wellordering of $$V$$. As there is no guarantee that this is, in general, definable, let it be the standard wellorder of $$HOD$$ in the standard class forcing extension $$V[G]$$ where $$V=HOD$$.

Let $$\llcorner a\lrcorner \bar \in \llcorner b\lrcorner \leftrightarrow a\in b$$ for all sets $$a$$, $$b$$, and adjunction be the binary function $$(a,b)\mapsto a\cup\{b\}$$.

If $$(\bar\in,R,F)\vDash t\text{ is an ordinal}$$, define $$R(t)$$ inductively as $$R(0) = \emptyset$$, $$R(\alpha) = (\text{closure of }R(\beta)\cup\{R(\beta)\}\cup C\text{ under adjunction and }F)^\bar\in$$ where $$C=(5\times V)^\in$$$$\beta$$ is the $$\bar\in$$-maximal element of $$\alpha$$ if it exists, and $$R(\lambda) = \cup^\bar\in\{R(\theta):\theta\bar\in\lambda\}$$ if it does not. Otherwise, $$R(t)=\emptyset$$.

Note that we don't explicitly define $$R$$ in $$V$$, just for $$\bar\in$$. This makes no difference as only the $$\bar\in$$ behaviour of $$R$$ matters, and we can simply let it's $$\in$$ value be $$\llcorner R(\cdot)\lrcorner$$ if necessary..

Define $$F(\ulcorner \phi \urcorner)$$ (for unary $$\phi$$, possibly with parameters) to be $$\{a\}$$ where $$a$$ is the $$<$$-minimal element of $$V$$ such that $$(\bar\in,R,F)\vDash\phi(a)$$ if $$\exists b \varphi(b)$$, where $$\varphi$$ is obtained by repacing the function $$R$$ with a function enumerating the $$\Sigma_n$$-correct cardinals, where $$\phi \in \Sigma_n$$ (n minimal, and $$\emptyset$$ otherwise.

I conjecture that, for fixed $$V$$ and $$\in$$, this produces unique values of $$\bar\in$$ and $$<$$. (In the case of ambiguity of 'standard' in the definition of the $$HOD$$ class ordering, we'll go with the first one that Thomas Jech taught anyone.) Assuming this to be the case (and even if it's not), we define Sasquatch as:

The largest number $$k$$ such that there is some unary formula $$\phi$$ in the language $$\{\bar\in,Q\}$$ (where $$Q(a,b) \leftrightarrow R(a)=b$$) of quantifier rank $$\leq 12\uparrow\uparrow 12$$ such that $$\exists ! a (\phi(a)) \wedge \phi(k)$$.

And bye for another short while. Hopefully I'll stay around long enough to answer questions in the comments.