User blog:P進大好きbot/Whether Rayo's number is well-defined or not

Hi, everyone. I hope that you all are enojoying daily googology.

First of all, I suspect that Rayo's number is not well-defined, even if one adds mathematically precise descriptions to the original definition. You know that the original definition lacks the declarations of what axioms of second order logic the definition of Rayo's function is based on and of what axioms of first order set theory the definition of the satisfaction relation used in the description of Rayo's function. Here, I do not mind it. Instead, I would like to consider other evidence which implies that Rayo's number is not well-defined. The point is "the impossible reference to the Platonist universe of the coded theory in the base theory". Of course, my thought might be wrong. I appreciate any opinions and corrections.

=Definition=

As a traditional terminology in formal logic, a formula \(\Phi\) of the base theory is said to be a definition if there is a unique variable \(x\) of the formal language of the base theory such that the formula \(\exists ! x(\Phi)\) of the base theory is provable. Therefore the sentence "\(\Phi\) is a definition" is a formula in the meta-theory. So in order to define a large number, one needs to write down (an abbreviation/interpretation by a natural language of) a formula \(\Phi\) of the base theory such that the formula \((\exists ! x(\Phi)) \wedge (\forall x(\Phi \to (x \in \mathbb{N})))\) of the base theory is provable.

=Conventon=

In order to define a natural number, we need a formal theory in order to avoid serious ambiguity and paradoxes such as Berry's one. Of course, we can use a formal theory coded in a given formal theory. In order to distinguish them, I would like to call the formal theory in which we defined a natural number the base theory and the theory defined in it the coded theory.

Rayo's number lies in the base theory by definition. Therefore Rayo's function should lie in the base theory, because so does its value. It implies that the domain of Rayo's function also lies in the base theory.

On the other hand, the set theory given by the formal language FOST and some (not precisely declared) axioms is the coded theory.

=Satisfaction Relation=

Now it is time to consider the function Sat. It has two imputs \([\Phi]\) and \(s\). The first imput \(\Phi\) is a coded formula, i.e. a formula of FOST, which is a term (realised as a natural number or a formal sequence of letters) in the base theory. The second imput \(s\) is a variable assignment, which assings variables in FOST to natural numbers. The natural numbers are terms in the base theory, because Rayo's number, which lies in the base theory, is defined as the minimum of a certain set of them. In order to assign variables in the coded theory to natural numbers in the base theory, one needs to fix a correspondence from natural numbers in the base theory to formulae of the coded theory defining terms in the coded theory.

Since we are working with a set theory as the coded theory, there is a canonical choice. The well-known von Neuman construction yields a purel syntax-theoretical correspondence, as long as we assume the axiom of extensionality, the axiom of the empty set, the axiom of pairing, and the axiom of union for the coded set theory. Say, would you mind if I assumed that the coded theory is the ZFC set theory? Since no axiom is declared, it is the most reasonable choice, I think.

There are three ways to defined the satisfaction relation.


 * For a class \(M\) in the coded theory and a formula \(\Phi\) of the coded theory, the sentence "\(M\) satisfies \(\Phi\)" is precisely the formula \(\Phi^M\) of the coded theory, which is obtained by replacing all quantifiers by those restricted to \(M\).
 * For a class \(M\) in the base theory and a formula \(\Phi\) of the formal language \(L_M\) constructed from \(M\), the sentence "\(M\) satisfies \(\Phi\)" is precisely the formula \(\textrm{Sat}(M,\Phi)\) of the base theory.
 * For a class \(M\) in the coded theory and a formula \(\Phi\) of the coded theory, the sentence "\(M\) satisfies \(\Phi\)" is precisely the formula \(\textrm{Sat}(M,\Phi_M)\) of the coded theory, where \(\Phi_M\) denotes the canonical interpretation of \(\Phi\) as a formula of \(L_M\).
 * For a definable class \(M\) in the coded theory and a formula \(\Phi\) of the coded theory, the sentence "\(M\) satisfies \(\Phi\)" is precisely the formula of the base theory given as the provability of \(\Phi^M\) in the coded theory.

Since Rayo's function uses the satisfaction relation in order to define the set of namable natural numbers, it should be a formula in the base theory so that the axiom of separation is applicable. Therefore the candidates are the second definition and the fourth definition. Since the formal language of the coded theory is fixed as FOST, which does not coincides with \(L_M\) for a class \(M\), the second definition is not the right one. Therefore the definition of the satisfaction relation used in Rayo's function is the fourth one. Moreover, there is no description of a (set or class) model \(M\) of the coded theory in the base theory, the only reasonable choice of \(M\) is the definable class \(V\) of all sets in the coded theory. However, \(\Phi^V\) is equivalent to \(\Phi\) under logical axioms, and hence the provability of \(\Phi^V\) is equivalent to that of \(\Phi\) under the coded theory. You know, the provability itself is too weak to define a incomputable large number, because ZFC set theory is effectively axiomised, So this is not the right interpretation of the definition of Rayo's number.

How can we define the satisfaction relation of a formula \(\Phi\) of the coded theory as a formula of the base theory without using the provability?

OK. Some people believe that a Platonist universe might work here.

=Platonist Universe=

A Platonist universe is an abstract object in human mind which satisfies given axioms of set theory and whose informal existence people may or may not accept. It is not precisely defined in terms of pure mathematics because it is a philosophical notion, but is quite useful for us to avoid highly formal arguments on sets.

Of course, the use of a Platonist universe is not the Almighty Universe, even though it is not precisely defined. I suspect that almost all googologists disagree with any statements such as "I am using the greatest Platonist universe, and hence I can regard \(\infty\) as a natural number! I win!!" or "My universe satisfies \(\perp\), and hence I can prove that my large number is bigger than whatever you will define!".

In order to avoid such non-sense statements, I have a proposal, which I think that is not so harmful.Say, how about considering a Platonist universe \(V\) as a class in the coded theory. What formula does \(V\) satisfy? OK. Any formulae that you choose, as long as there is no contradiction.

But... What does satisfy mean here? We need to avoid using provability also here. So it should be the first definition or the third definition above. However, they are relations given as formulae of the coded theory. In order to define Sat in the base theory, we need the satisfaction relation realised as a formula of the base theory.

Is my proposal that a Platonist universe is a class in the coded theory bad? Should we deal with it as a purely philosophical notion? Even if we do so, it is an object which is not a term or a formula of the base theory or the coded theory. Then the base theory should refer to the truth in a Platonist universe of the coded theory. Therefore there is no way to apply the axiom of separation in order to define the set of namable natural numbers. As a result, the use of a Platonist universe does not work here.

= Alternative Direction? =

Let \(V\) variable of the coded theory such that the axiom of the coded theory is precisely the union of the following two sets of all formulae;


 * A maximal consistent set \(\Sigma\) of formulae of the coded theory in which \(V\) never occurs and which contains the ZFC axiom.
 * The set \(\{\Phi^V \mid \Phi \in \Sigma\}\).

I mean that I give up to use ZFC set theory as the coded theory. Of course, it is not so reasonable, because there is no declaration of the axiom in the original definition. This is not an affirmative interpretation of the definition of Rayo's number any more, but is the definition of a new large number.

The provability of \(\Phi^V\) behaves as the truth. Indeed, since \(\Sigma\) is maximal, for any formula \(\Phi\) of the coded theory in which \(x\) does not occur, precisely one of \(\Phi^x\) and \((\neg(\Phi))^x = \neg(\Phi^x)\) is contained in the latter set of axioms.

I note that such a \(\Sigma\) exists by Zorn's lemma in the base theory, and can be chosen canonically because FOST can be explicitly equipped with a well-order. Therefore using \(V\), we can defined a well-defined Rayo-like large number.

PS:

I do not know why I can not display mathematical formulae in my blog post. Could anyone tell me the right way to write them? Thank you!