User blog:TheKing44/Grand Finale Functions

Rayo's function grows faster than any function definable in first-order set theory. However, it would be incorrect that this function settles Googology in first-order set theory, since it is not itself definable in that language. It would only be fair to compare it to other functions in second-order set theory.

In this post, I will define a family of extremely fast growing functions that can be defined in first-order set theory. First, some prerequisites.

Satisfaction
A satisfaction predicate tells you which arguments satisfy a formula. More formally, for a satisfaction predicate $$\#$$, we must have $$\forall x,y,\dots,z. \#('\phi', (x,y,\dots,z)) \iff \phi(x,y,\dots,z)$$ for every formula $$\phi$$.

We can not define a satisfaction predicate in first order set theory that works for every formula in first order set theory. We can however define a satisfaction predicate over a subset of such formulas.

Bounded Quantifier Formulas
We can define a $$\#_\text{Bounded}$$ for formulas that only contain bounded quantifiers. Given a formula $$\phi$$ with arguments $$x,y,\dots,z$$, we first pick some set that contains $$x,y,\dots,z$$ and is transitive. This set will function as our universe. We will call it $$U$$. We will also require $$U$$ to be contain each of its finite subsets.

Then there will be a unique relation $$\#_U$$ on variables assignments with range $$U$$ and formulas such that


 * $$\#(x ``='' y, \overrightarrow v) \iff \overrightarrow v(x) = \overrightarrow v(y)$$
 * $$\#(x ``\in'' y, \overrightarrow v) \iff \overrightarrow v(x) \in \overrightarrow v(y)$$
 * $$\#(``\lnot \phi, \overrightarrow v) \iff \lnot \#(``\phi, \overrightarrow v)$$
 * $$\#(``\phi \lor \psi, \overrightarrow v) \iff \#(``\phi, \overrightarrow v) \lor \#(``\phi'', \overrightarrow v)$$
 * $$\#(``\phi \land \psi, \overrightarrow v) \iff \#(``\phi, \overrightarrow v) \land \#(``\phi'', \overrightarrow v)$$
 * $$\#(``\exists x ``\in y ``. \phi, \overrightarrow v) \iff \exists z \in \overrightarrow v(y). \#(``\phi, \overrightarrow v[x := z])$$
 * $$\#(``\forall x ``\in y ``. \phi, \overrightarrow v) \iff \forall z \in \overrightarrow v(y). \#(``\phi, \overrightarrow v[x := z])$$

(where $$x$$ and $$y$$ represent variable symbols from whatever variable alphabet we are using).

For convenience, we will extend the language with terms for finite sets. This will make working with tuples easier. This is why we required $$U$$ to contain each of its subsets. We also modify $$\#_U$$ to evaluate these terms as needed.

So, $$\#_\text{Bounded}(\phi, (x, y, \dots, z))$$ holds if and only if there exists a $$U$$ and $$\#_U$$ such that $$(\phi, (x, y, \dots, z)) \in \#_U$$.

Lévy hierarchy formulas
The Lévy hierarchy is a way of categorizing formulas in the language of first-order set theory. Each (is equivalent to a formula that) has a fixed number of unbounded quantifiers.

How do we define satisfaction predicates for these? Well, since the quantifiers are unbounded, there can not be a suitable relation $$\#_U$$ since it will be the size of a proper class (which first-order set theory can not talk about). However, we can basically "hard code" the unbounded quantifiers in. For example, here is a satisfaction predicate for $$\Pi^{ZFC}_3$$:


 * $$\exists \psi. \psi \in \Sigma_3 \land ZFC \vdash \phi ``\iff'' \psi \land \forall q. \exists r. \forall s. \#_\text{Bounded}(\overline \psi, (q, r, s, x, y, \dots, z))$$

where $$\overline \psi$$ is $$\psi$$ with the unbounded quantifiers removed (giving it three more free variables).

The Grand Finale Functions
For some satisfaction predicate $$\#$$, that predicate's grand finale function applied to $$n$$ is the greatest natural number $$k$$ such that there exists $$\phi$$ in $$\#$$'s domain with less than $$n$$ symbols such that $$\forall j \in \mathbb N. (\#(\phi, j) \iff j = k)$$ plus 1 (or 0 if no such $$k$$ exists). If $$\#$$ is definable in first-order set theory, so will its grande finale function. However, there is no formula in first-order set theory assigning such predicates to grand finale functions in general.

As a concrete example, let $$f(n)$$ equal (drum roll please):


 * $$\min(\{m : m \in \mathbb N, \forall k \in \mathbb N. k < m \impliedby \exists \phi \in \Sigma_{100}. |\phi| < n \land \forall j \in \mathbb N. ($$
 * $$\exists a. \forall a \prime . \exists b. \forall b \prime. \exists c. \forall c \prime . \exists d. \forall d \prime . \exists e. \forall e \prime . \exists f. \forall f \prime . \exists g. \forall g \prime . \exists h. \forall h \prime . \exists i. \forall i \prime . \exists l. \forall l \prime . \exists o. \forall o \prime . \exists p. \forall p \prime . \exists q. \forall q \prime . \exists r. \forall r \prime . \exists s. \forall s \prime . \exists t. \forall t \prime . \exists u. \forall u \prime . \exists v. \forall v \prime . \exists w. \forall w \prime . \exists x. \forall x \prime . \exists y. \forall y \prime . \existsz. \forall z \prime . \exists \alpha. \forall \alpha \prime . \exists \beta. \forall \beta \prime . \exists \gamma.\forall \gamma \prime . \exists \delta. \forall \delta \prime . \exists \epsilon. \forall \epsilon \prime . \exists\zeta. \forall \zeta \prime . \exists \eta. \forall \eta \prime . \exists \theta. \forall \theta \prime . \exists \iota. \forall \iota \prime . \exists \kappa. \forall \kappa \prime . \exists \lambda. \forall \lambda \prime . \exists \mu. \forall \mu \prime . \exists \nu. \forall \nu \prime . \exists \xi. \forall \xi \prime . \exists \pi. \forall \pi \prime . \exists \rho. \forall \rho \prime . \exists\sigma. \forall \sigma \prime . \exists \tau. \forall \tau \prime . \exists \upsilon. \forall \upsilon \prime . \exists \chi. \forall \chi \prime . \exists \psi. \forall \psi \prime . \exists \omega. \forall \omega \prime . \exists\varepsilon. \forall \varepsilon \prime . \exists \varkappa. \forall \varkappa \prime . \exists \varpi. \forall \varpi \prime . \exists \varrho. \forall \varrho \prime . \exists \varsigma. \forall \varsigma \prime . \exists \vartheta. \forall \vartheta \prime .$$
 * $$\#_\text{Bounded}(\overline \phi, (a, a \prime, b, b \prime, c, c \prime, d, d \prime, e, e \prime, f, f \prime, g, g \prime, h, h \prime, i, i \prime, l, l \prime, o, o \prime, p, p \prime, q, q \prime, r, r\prime, s, s \prime, t, t \prime, u, u \prime, v, v \prime, w, w \prime, x, x \prime, y, y \prime, z, z \prime, \alpha, \alpha \prime, \beta, \beta \prime, \gamma, \gamma \prime, \delta, \delta \prime, \epsilon, \epsilon \prime, \zeta, \zeta \prime, \eta, \eta \prime, \theta, \theta \prime, \iota, \iota \prime, \kappa, \kappa \prime, \lambda, \lambda \prime, \mu, \mu \prime, \nu, \nu \prime, \xi, \xi \prime, \pi, \pi \prime, \rho, \rho \prime, \sigma, \sigma \prime, \tau, \tau \prime, \upsilon, \upsilon \prime, \chi, \chi \prime, \psi, \psi \prime, \omega, \omega \prime, \varepsilon, \varepsilon \prime, \varkappa, \varkappa \prime, \varpi, \varpi \prime, \varrho, \varrho \prime, \varsigma, \varsigma \prime, \vartheta, \vartheta \prime, j)) \iff j = k\})$$

As a concrete example of a number, we will call $$f(10^{100})$$ the finale number.

Growth Rate
Although these functions are definable in first-order set theory, they are more convenient to analyze using second-order set theory, which we will do in this section.

Suppose some function $$f$$ is definable by a binary formula $$\phi$$ in first-order set theory. For any $$n$$, we can define $$k = f(n)$$ via $$\phi(n, k)$$ quite straight forwardly. Assuming we are using von neumann ordinals to represent the natural numbers, this means that $$g(c2^x + |phi| + 1)$$ grows faster than $$f$$ for some constant $$c$$, where $$g$$ is the grande finale function of the satisfaction predicate over the level of the Lévy hierarchy containing $$\phi$$. Of course, this a very loose lower bound, but already shows how quickly the grande finale functions grow. The only condition we placed on $$f$$ is that it was definable by a binary formula for some level of the Lévy hierarchy. (I have not figured out exactly how many, but its clear that moving a couple (constant) number of levels in the Lévy hierarchy allows $$f(x)$$ to beaten by $$g(x)$$.)

In particular, this means that sense every function definable in first order set theory falls in the Lévy hierarchy, there is a grand finale function such that $$g(c2^x + |phi| + 1)$$ beats $$f$$ for any first-order set theory definable function $$f$$. Moreover, finding the $$g$$ to beat it can be done mechanically: simply count the number of quantifiers used in $$f$$'s definition, and then write the corresponding grande finale function. This makes the grande finale functions the most powerful Googolism in first-order set theory. Any function definable in first-order set theory is beaten by a naive extension of the grand finale function defined above. Even if a function incorporates a grand finale function into its definition, a grande finale function higher up in the Lévy hierarchy will still beat that function, and no function definable in first-order set theory can incorporate all the grande finale functions into its definition.

Generalizations
It appears that the grande finale functions could be generalized to first-order and second-order arithmetic, as well as second-order set theory. It most likely generalizes to what ever meta-theories the authors of Little Bigeddon and Sasquatch used.

Conclusion
When the Googolplex was initially defined, its value was determined by the endurance of the person writing. Dr. Kasner, however, was displeased with this definition:

A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr Einstein, simply because he had more endurance.

Mathematics and the Imagination

He averted it by redefining Googolplex to be $$10^{10^100}$$. However, the grande finale functions proof his efforts to be in vain. One can not turn the number of quantifiers in the grande finale function into an argument; each and every quantifier must be included in the definition. This means that Carnera or whatever dexterous person you wish to name can defeat any Googolism by writing a grande finale function that simply includes enough quantifiers to defeat it. Even if the competing Googolist tries to incorporate the resulting grande finale function into their definition in interesting and powerful ways, the result can still be defined by yet a longer grande finale function. He can not claim that their adversary is making a naive extension of their function. The adversary is making a naive extension of their own function, only taking into account the length of the Googolists definition.

However, all good things must come to an end. We have had lots of fun all the way, learning powerful maths, making friends, and hopefully learning something about ourselves. We've had highs and, well, higher highs. Although all numbers are smaller than most numbers, the numbers we defined were still pretty awesome.

All good things must come to an end, and so it is fitting that Googology have not merely a finale, but a grand finale. However, do not consider this the end, but a new beginning. Please, join us next week for part three, return of the hand crampers. But please, first give our performers a hand.

''Well known Googologists come unto the stage and take a bow while the audience gives thunderous applaud and throws flowers. Curtains close and credits roll via projector.''

THE END