User blog:P進大好きbot/Ordinal Notation with the PTO of ZFC

I constructed a recursive ordinal notation corresponding to a "proof theoretic analogue" of the PTO of \(\textrm{ZFC}\) in a previous blog post. Applying a similar strategy, I define an ordinal notation corresponding to the PTO of ZFC.

The ordinal notation does not admit a recursive subset of terms of standard forms to which the restriction of the canonical strict partial order is a recursive well-order, but yields a computable large function under \(\textrm{ZFC}\).

As in the previous blog post on the "proof theoretic analogue" of the PTO of \(\textrm{ZFC}\), I work in \(\textrm{ZFC}\) set theory \(T\) coded in a meta theory \(MT\) with the assumption of \(\textrm{Con}(T)\).

= Rough Sketch =

I will define a map \(\textrm{RWO}\) sending an \(n \in \mathbb{N}\) to (the Goedel number of) a formula \(\textrm{RWO}(n)\) in an internal set theory such that the statement that \(\textrm{RWO}(n)\) is a definition of a recursive well-order on a recursive subset of \(\mathbb{N}\) admits a formal proof. Then \(\textrm{RWO}\) yields a non-recursive "well-founded" strict partial order \(<_{\textrm{RWO}}\) on \(\mathbb{N}\).

The "well-foundedness" is similar to that in the previous blog post, and roughly means that there exists no infinite strictly decreasing sequence with respect to \(<_{\textrm{RWO}}\) consisting of pairs \((n,d)\) of natural numbers presented by the repetition of successors. As I explained in the previous blog post, a natural number is not necessarily presentable in that way.

The main differences from the ordinal notation in the previus blog post are the following:
 * The map \(\textrm{RWO}\) is a \(1\)-ary function, and hence does not have the parameter \(d\) corresponding to a bound of formal proofs of additional data.
 * The well-order defined by the formula \(\textrm{RWO}(n)\) is not assumed to be comparable to other well-orders.

= Review =

I recall several notions introduced in the previous blog post.

I defined a formal language \(L\) using arithmetic and an internal formal theory with \(\textrm{ZFC}\) axiom \(A\) described in \(L\). Variables in \(L\) are written as \(x^L(i)\) for some \(i \in \mathbb{N}\). The relation symbol in \(L\) is written as \(\in^L\). Similarly, formal strings in \(L\), e.g. formulae, formal proofs, and syntax sugars, are written in a similar way.

Also I introdced the following maps: \begin{eqnarray*} \textrm{Prj}_0 \times \textrm{Prj}_1 \colon \mathbb{N} & \stackrel{\cong}{\to} & \mathbb{N} \times \mathbb{N} \ (\textrm{pair function})\\ \textrm{Prm} \colon \mathbb{N} & \to & \mathbb{N} \ (\textrm{enumeration of primes}) \\ v_p \colon \mathbb{N} \setminus \{0\} & \to & \mathbb{N} \ (p \textrm{-adic valuation}) \\ \textrm{Lng} \colon \mathbb{N} \setminus \{0\} & \to & \mathbb{N} \ (\textrm{length as an array}) \\ \textrm{Fml}^L \colon \mathbb{N} & \hookrightarrow & L \ (\textrm{Goedel correspondence of formulae}) \\ \textrm{Prf}^L \colon \mathbb{N} & \hookrightarrow & \mathbb{N} \ (\textrm{enumetation of proofs}) B \colon \mathbb{N} & \to & \mathbb{N} \\ d & \mapsto & 2 \uparrow^4 10^{100}(d+1) \end{eqnarray*}

For a \(d \in \mathbb{N}\), a formal proof \(\textrm{Prf}(N)\) in \(L\) is said to be "of complexity \(\leq d\)" if \(N \leq d\).

= Ordinal Notation =

I define a partial order \(<_{\textrm{RWO}}\) on \(\mathbb{N}\), which is "well-founded" in the sense in Rough Sketch.

Enumeration of Provably Recursive Well-Orders
I denote by \(\textrm{RWO}^L\) the formula in \(L\) which states that \(x^L(0)\) is a recursive well-order on a recursive subset of \(\mathbb{N}^L\).

Let \(n \in \mathbb{N}\). I denote by \(\textrm{RWO}(n) \in \mathbb{N}\) in the smallest natural number satisfying the following: By the argument of the evaluation of the old \(\textrm{RWO}(n,d)\) and \(N\) in the previous blog post, the new \(\textrm{RWO}\) forms a total function on \(\mathbb{N}\).
 * 1) The inequality \(\textrm{RWO}(n) \neq \textrm{RWO}(m)\) holds for any \(\mathbb{N}\) with \(m < n\).
 * 2) There is a formal proof of \(\exists !^L x^L(0), (\textrm{Fml}^L(\textrm{RWO}(n)) \wedge^L \textrm{RWO}^L)\) of complexity \(\leq B^{n+1}(1)\).

By the definition, the image of \(\textrm{RWO}\) coincides with the set of Goedel numbers of definitions of provably recursive well-orders on recursive subsets of \(\mathbb{N}^L\).

Strict Partial Orders
First, I define a binary relation \(<_{\textrm{RWO},d}\) on \(\mathbb{N}\) for each \(d \in \mathbb{N}\).

For an \(n \in \mathbb{N}\), I abbreviate to \(\Phi(n,d)\) the statement that there is a formal proof of \(\exists !^L x^L(0), (\textrm{Fml}^L(\textrm{RWO}(n)) \wedge^L \textrm{RWO}^L)\) of complexity \(\leq d\).

The relation \(n_0 <_{\textrm{RWO},d} n_1\) is true if the following hold: Then \(<_{\textrm{RWO},d}\) forms a recursive vubary relation, which is an irreflexive (not necessarily transitive) asymmetric binary relation under the assumption of \(\textrm{Con}(\textrm{ZFC})\), and \(n_0 <_{\textrm{RWO},d} n_1\) implies \(n_0 <_{\textrm{RWO},d+1} n_1\) by the definition.
 * 1) \(\Phi(n_0,d)\) and \(\Phi(n_1,d)\)
 * 2) There is a formal proof of the existence of an order-preserving injective non-surjective map from the well-ordered set defined by \(\textrm{Fml}^L(\textrm{RWO}(n_0))\) to the one defined by \(\textrm{Fml}^L(\textrm{RWO}(n_1))\) of complexity \(\leq d\).

Next, I define a binary relation \(<_{\textrm{RWO}}\) on \(\mathbb{N}\). It is very simple. The relation \(n_0 <_{\textrm{RWO}} n_1\) is true if \(n_0 <_{\textrm{RWO},d} n_1\) for some \(d \in \mathbb{N}).

Then \(<_{\textrm{RWO}}\) forms a non-recursive binary relation, which is a strict partial order under the assumption of \(\textrm{Con}(\textrm{ZFC})\), and \(n_0 <_{\textrm{RWO}} n_1\) roughly means that there is a formal proof of the existence of an order-preserving injective non-surjective map from the well-ordered set defined by \(\textrm{Fml}^L(\textrm{RWO}(n_0))\) to the one defined by \(\textrm{Fml}^L(\textrm{RWO}(n_1))\).

Predecessor
Let \((n,d) \in \mathbb{N} \times \mathbb{N}\) with \(\Phi(n,d)\). An \(m \in \mathbb{N}\) is called the predecessor of \(n\) with respect to \(<_{\textrm{RWO},d}\) if \(m\) is the smallest natural number satifying the following:
 * 1) \(m \neq n\).
 * 2) \(\Phi(m,d)\).
 * 3) For any \(m' \in \mathbb{N}\) with \(m' <_{\textrm{RWO},d} n\), either one of the following holds:
 * 4) \(m' = m\).
 * 5) \(m' <_{\textrm{RWO},d} m\).

Then by the definition, the predecessor of \(n\) with respect to \(<_{\textrm{RWO},d}\) is unique and computable without the assumption of \(\textrm{Con}(\textrm{ZFC})\) as long as it exists, and the existence of the predeccesor is decidable.

Of course, \(n\) does not necessarily admit the predecessor with respect to \(<_{\textrm{RWO},d}\). If \(n\) does not admit the predecessor with respect to \(<_{\textrm{RWO},d}\), then I say that \(n\) is minimal with respect to \(<_{\textrm{RWO},d}\).

Fundamental Sequence
For an \(n \in \mathbb{N}\), I define a map \begin{eqnarray*} \mathbb{N} & \to & \mathbb{N} \\ s & \mapsto & n[s] \end{eqnarray*} in the following way: I note that this system of fundamental sequences is defined to any \(n\) which does not necessarily correspond to a limit ordinal, and hence is not a fundamental sequence in the usual sense.
 * 1) If \(n\) is minimal with respect to \(<_{\textrm{RWO},s}\), then \(n[s] = n\).
 * 2) Otherwise, \(n[s]\) is the predecessor of \(n\) with respect to \(<_{\textrm{RWO},s}\).

= "Well-Foundedness" =

The "well-foundedness" of \(<_{\textrm{RWO}}\) means that there is no infinite decreasing sequence consisting of meta natural numbers in the sense of the previous blog post with respect to \(<_{\textrm{RWO}}\)

It can be verified in a completely simillar way to the "well-foundedness" in the previous blog post. Therefore I omit the proof.

= FGH =

For an \(n \in \mathbb{N} \times \mathbb{N}\), I define a computable function \(f_n(s)\) in the following recursive way: The domain of \(f_n\) contains any meta natural number \(s\) as long as \(n\) is also a meta natural number, because the \(s\)-th entry of the fundamental sequence of a meta natural number is a meta natural number. Otherwise, the domain of \(f_n\) might be empty. At least, the correspondence \((n,s) \mapsto f_n(s)\) is a well-defined (not necessarily total) computable function on \(\mathbb{N} \times \mathbb{N}\).
 * 1) If \(n[s] = n\), then \(f_n(s) = s+1\).
 * 2) Otherwise, \(f_n(s) = f_{n[B(s)]}^s(s)\).

= Large Number =

I denote by \(f\) the computable partial function in the base theory defined as \(f(n) := f_n(B(n))\). It is not provably total, because the termination of the recursion process is just true on \(\omega\)-models of the base theory.

At least, \(10^{100}\) is a meta natural number, and \(f_n(s)\) and \(n[s]\) send a pair \((n,s)\) of meta natural numbers to meta natural numbers. Therefore \(f^{10^{100}}(10^{100})\) is a well-defined computable large number in \(\textrm{ZFC}\) set theory, as long as I assume the existence of an \(\omega\)-model of the base theory on the meta theoryby the same reason explained in the previous blog post.

= Analysis =

Although I constructed an ordinal notation whose limit is the PTO of \(\textrm{ZFC}\), I only used the provable order relations so that the resulting function \(f(n)\) forms a computable one under \(\textrm{ZFC}\) axiom. Therefore the growth rate of \(f(n)\) might be lower than the PTO of \(\textrm{ZFC}\).

On the other hand, every provably recursive countable ordinal \(\alpha\) into \((\mathbb{N},<_{\textrm{RWO}})\), and hence the recursion along \(<_{\textrm{ZFC}}\) with respect to the predecessor construction is actually stronger than that of \(\alpha\) with a provably recursive system of fundamental systems such that the formula \(\forall \beta \in \alpha \cap \textrm{Lim}, \beta[n] \in \beta\) admits a formal proof of complexity \(\leq (2 \uparrow^4 2)(n + 1)\) for any meta natural number \(n\).

It is because the bound of the complexity of formal proofs in the computation of the \(B(n)\)-th entry of the fundamental sequence grows much faster than \(C(n+1)\) for any positive real constant \(C\).

It implies that the growth rate of \(f(n)\) (restricted to meta natural numbers) is much larger than many big recursive countable ordinals.