User blog:P進大好きbot/List of common mistakes on formal logic appearing in googology

In order to avoid the repetition of the same explanations, I list several common mistakes on formal logic appearing in googology.

= Why is formal logic important in googology? =

Because it helps us to reject Berry's paradox like "My large number is the natural number definable by \(10^{100}\) or less letters! I win!" Of course, formal logic itself is useful for us to argue in a precise manner.

= What does "definition" mean in mathematics? =

A definition of a term such as a large number in formal logic is simply a formula \(F\) with free variable \(x\) such that \(\exists ! x(F)\) is provable under the axiom on which you are working. In particular, you can never define non-trivial term without axioms.

= What is the meta theory? =

The meta theory is the theory in which the formal language of the theory on which you are working is defined. I need more precisse explanation.

Let \(L\) be a formal language, and \(A\) be a set of axioms written in \(L\). For example, \((L,A)\) is the pair of a formal language and the axiom of group theory, or the pair of the formal language \(\textrm{FOST}\) and the ZFC axiom \(A\) written in \(\textrm{FOST}\). Imagine that you are working on the theory \(T\) based on \((L,A)\). Then the meta theory \(MT\) is the theory in which \(L\) and \(A\) are terms. Namely, the theory \(T\) is coded in \(MT\).

The meta theory is usually arithmetic or set theory, because it possesses the presentablity of formal languages. In arithmetic, you can define a formal language by using the pair function and Goedel correspondence. In set theory, you can do in a similar way in arithmetic, or you can use the free monoid generated by a specific set.

Of course, \(MT\) might also be a coded theory in another theory, say, the meta meta theory \(MMT\). Otherwise, it is described as a naive finite collection of symbols without non-trivial equalities in the axiom.

I note that \(MT\) is fixed before you consider \(T\). Therefore when you fix \(T\), the maximum of the length of the tower of theories \(T,MT,MMT,\ldots\) is fixed. In particular, when you want to define a large number in \(T\), you are not allowed to arrange the length of the tower of meta theories saying like "I take the meta theory, the meta meta theory, and the meta times \(\infty\) theory! Using them all, I define a great number! I win!".

= Is every consistent first order theory incomplete? =

No. You need to read the conditions of the incompleteness theorem carefully. For example, \(\textrm{RCF}\) does not satisfy the condition. I am afraid that you are confounding incompleteness theorem with completeness theorem.

Also, there is a consistent first order theory which is not incomplete. Let \(\Sigma\) denote the set of consistent systems of axioms written in a fixed formal language. Then by Zorn's lemma, there is a maximal element \(A\)in \(\Sigma\) with respect to the inclusion relation. If there is a closed formula \(F\) independent of \(A\), \(A \cup \{F\}\) is consistent. It contradicts the maximality of \(A\). Therefore \(A\) is not incomplete.

= Can we deal with natural numbers without axioms? =

No. First, see this. Then you notice that if you could define the notion of natural numbers in an appropriate way (e.g. \(0 \neq 1\)) in first order logic without axioms, then you can define it in the same way in any first order logic such as group theory and category theory. It contradicts the obvious fact that they have a model whose underlying set is a singleton.

= What is a model? =

There are several definitions of the notion of a model. \(L\) be a formal language of first order set theory, and \(A\) be an appropriate set of axioms of set theory such as ZF or ZFC. Imagine that you are working on the set theory \(T\) based on \((L,A)\). I denote by \(MT\) the meta theory of it.


 * For a set \(\Sigma\) of formulae in \(T\), which is a term of \(MT\), the formula \(M \models \Sigma\) with free variable \(M\) is defined as the formula in \(T\) given as the satisfaction property presented by using the formal language \(L_M\), which is a term of \(T\), associated to \(M\).
 * For a set \(\Sigma\) of formulae in \(T\), which is a term of \(MT\), the formula \(X \models \Sigma\) with definable class \(X\) is defined as the formula in \(MT\) which is the synonym of \(X \vdash \Sigma).

I note that for a term \(M\) of \(T\), the correspondence \(F \mapsto F^M\) is defined in \(MT\), but not in \(T\). For example, if you consider a formal language \(\textrm{FOST}\) of first order set theory which is a term on \(T\), then you might think that you can consider \(F^M\) for each formula \(F\) written in \(\textrm{FOST}\). However, you can never define a term \(F^M\) of \(MT\) from a term \(F\) of \(T\). It works only when \(F\) is the code of a formula in \(T\). Therefore a model \(M\) of \(T\) does not form a model of ZFC axiom written in \(\textrm{FOST}\).

You might say "Huh? But \(\textrm{FOST}\) is the language of set theory. So every formula written in \(\textrm{FOST}\) is derived from \(T\)!" However, it is wrong. A formula in the coded theory is not necessarily the code of the base theory, even if both theories are set thoeries. Indeed, if you could do so, since there is a model \(M\) of \(\textrm{ZFC} + \neg \textrm{Con}(\textrm{ZFC})) under the assumption of \(\textrm{Con}(\textrm{ZFC})\) in \(MT\) by incompleteness theorem and completeness theorem, and the sequence of formulae forming the proof of \(\perp\) in \(M\) should be derived from a sequence of formulae in \(T\) and also a sequence of formulae in \(MT\). It contradicts the assumption of \(\textrm{Con}(\textrm{ZFC})\) in \(MT\).

At least, for each formula \(F\) written in \(\textrm{FOST}\), you can define a subset of the set of maps from the set of free variables occurring in \(F\) to \(M\) in a recursive way. It might work as well as \(F^M\).

= Can I use a model in the definition of a large number? =

Partially yes. You can include a model \(M\) in the definition of a large number, as long as either one of the following holds:
 * you are working on set thoery \(T\) and your definition using \(M\) is actually a definition in \(T\) in the sense of this.
 * you are working on \(M\) itself and your definition names a natural number relative to \(M\)

For example, imagine that you are working on set thoery \(T\). The sentence like "I denote by \(n \in \mathbb{N}\) the cardinality of a finite model \(G\) of group thoery" is not a closed formula in \(T\) defining \(n\) as long as the definition of \(G\) is specified in \(T\).

Similarly, even if \(T\) is ZFC set theory, then you can never use a model \(M\) of ZFC axiom by the same reason as long as you are participating in a large number contest with a rule specifying the use of such an \(M\). The sentence like "I denote by \(n \in \mathbb{N}\) (caution: it is not \(\mathbb{N}^M\)) the minimum of the set of natural numbers \(k\) such that there is a formula \(F\) in \(T\) with \(M \models F(k)\)" is not a definition in \(T\) as long as the definition of \(M\) is specified.

Of course, if you are working on a model \(M\) of \(T\) but not on \(T\) itself, then you can use \(M\). It depends on the rule of the googology which you are enjoying. In this case, the sentence like "I denote by \(n \in \mathbb{N}^M\) the minimum of the set of natural numbers greater than any natural number \(k\) satisfying some formula \(P\) in \(T\) with length smaller than or equal to \(10^{100}\)" can be easily interpreted to be a definition, as long as the rule allows you to refer to the meta theory \(MT\).

Indeed, let \(T\) denote the base set theory on which you are working, \(\Sigma\) the set of formulae in \(T\) with length smaller than or equal to \(10^{100}\) with free variable \(x\), and \(F\) the formula with free variable \(y\) given by connecting \(\forall x \in \mathbb{N}, (P \to (x < y))\) for each \(P \in \Sigma\) by \(\wedge\). They are terms of \(MT\), and \(n\) can be defined as the minimum of the subset of \(\mathbb{N}\) defined by \(F^M\) by the separation axiom.

Also, if you are working on \(T\) and if you are allowed to use a model \(M\) to define a natural number \(n \in \mathbb{N}\) (not \(\mathbb{N}^M\)), then you can interpret your definition in a similar way.

Be careful that you can never use \(M\) as a model of set theory based on a coded formal language \(\textrm{FOST}\), which is a term of \(T\). See this.

= Can Rayo's number be defined without axioms? =

No. See this.