User blog:P進大好きbot/Ordinal Notation for a "Proof-Theoretic Analogue" of the PTO of ZFC

Throughout this blog post, I work in \(\textrm{ZFC}\) set thoery. I note that a similar argument is valid for any effectively axiomised theory containing arithmetic.

I construct an ordinal notation, which presents a "proof-theoretic analogue" of PTO of ZFC together with a canonical choice of a system of fundamental sequences.

= Rough Sketch =

I will define a map \(\texrm{Def}\) sending an \((n,d) \in \mathbb{N} \times \mathbb{N}\) to (the Goedel number of) a formula \(\textrm{Def}(n,d)\) in an internal set theory such that if there is a formula \(F\) satifying the following two conditions with respect to the notion of the complexity of a formal proof defined later, then \(\textrm{Def}(n,d)\) also satisfies them: Then \(\textrm{Def}\) yields a "well-founded" strict partial order \(<_{\textrm{Def}}\) on \(\mathbb{N} \times \mathbb{N}\), which admits a canonical system of fundamental sequences and a canonical subset of standard forms.
 * 1) The statement that \(F\) is a definition of a recursive well-order on a recursuve subset of \(\mathbb{N} \times \mathbb{N}\) admits a formal proof of complexity \(< 10^{100}(d+1)\).
 * 2) For any \(m \in n\), either one of the statement that the ordinal type of the well-order defined by \(F\) is smaller than that of \(\textrm{Def}(m,d)\) and the statement that the ordinal type of the well-order defined by \(F\) is greaterer than that of \(\textrm{Def}(m,d)\) admits a formal proof of complexity \(< 10^{100}(d+1)\).

I note that the "well-founded" means that there exists no formula that defines an infinite strictly decreasing sequence with respect to \(<_{\textrm{Def}}\). In particular, the sequence constructed by using the system of fundamental sequences starting from any pair \((n,d)\) of natural numbers presented by the repetition of successors eventually terminates.

The second condition of \(\textrm{Def}(n,d)\) is very important in this ordinal notation. Indeed, the construction would work even if I removed it, and the resulting ordinal notation would correspond to the PTO of \(\textrm{ZFC}\) by definition. Is it happy? No. The resulting order on it would become non-recursive, and hence useless to define a computable function through FGH. Therefore the second condition is necessary for me to define a recursive ordinal notation. The second condition is given in a proof-theoretic way, and that is why I wrote that it corresponds to a "proof-theoretic analogue" of PTO of \(\textrm{ZFC}\).

= Preparation =

For an \(n \in \mathbb{N}\), I denote by \(\textrm{Proj}_0(n) \in \mathbb{N}\) the greatest natural number with \(2^{\textrm{Proj}_0(n)} \mid n\), and by \(\textrm{Proj}_1(n) \in \mathbb{N}\) the natural number \(2^{-1}(2^{- \textrm{Proj}_0(n)}n-1)\). Then the pair function \begin{eqnarray*} \mathbb{N} & \to & \mathbb{N} \times \mathbb{N} \\ n & \mapsto & (\textrm{Proj}_0(n),\textrm{Proj}_1(n)) \end{eqnarray*} is a bijective map, whose inverse is the map \begin{eqnarray*} \mathbb{N} \times \mathbb{N} & \to & \mathbb{N} \\ (n_0,n_1) & \mapsto & 2^{n_0}(2n_1+1). \end{eqnarray*}

For an \(i \in \mathbb{N}\), I denote by \(\textrm{Prime}(i) \in \mathbb{N}\) the smallest prime number greater than \(\textrm{Prime}(j)\) for any \(j \in i\).

For a prime number \(p\), I denote by \(v_p \colon \mathbb{N} \setminus \{0\} \to \mathbb{N}) the normalised additive \(p\)-adic valuation.

The map \begin{eqnarray*} \mathbb{N}^{\oplus \mathbb{N}} & \to & \mathbb{N} \\ (e_i)_{i \in \mathbb{N}} & \mapsto & (\prod_{i \in \mathbb{N}} \textrm{Prime}(i)^{e_i}) - 1. \end{eqnarray*} is a bijective map, whose inverse is the map \begin{eqnarray*} \mathbb{N} & \to & \mathbb{N}^{\oplus \mathbb{N}} \\ n & \mapsto & (v_{textrm{Prime}(i)}(n))_{i \in \mathbb{N}}, \end{eqnarray*} where \(\mathbb{N}^{\oplus \mathbb{N}}\) denotes \(\mathbb{N}^{\mathbb{N}} \cap \mathbb{Z}^{\oplus \mathbb{Z}}\).

For an \(n \in \mathbb{N}\), I denote by \(\textrm{Length}(n) \in \mathbb{N}\) the smallest natural number with \(\forall j \in \mathbb{N} \setminus \textrm{Length}(n), (v_{\textrm{Prime}(j)}(n) = 0) )\).

= Internal Formal Theory =

For an \(i \in \mathbb{N}\), I put \begin{eqnarray*} x^L(i) := (0,i). \end{eqnarray*} Also, I put \begin{eqnarray*} \in^L & := & (1,0) \\ \to^L & := & (2,0) \\ \neg^L & := & (2,1) \\ \forall^L & := & (3,0). \end{eqnarray*} Then the set \(L := \{ x^L(i) \mid i \in \mathbb{N} \} \cup \{(1,0),(2,0),(2,1),(3,0)\}\) forms a formal language of first order set theory such that \(x~L_n\)'s are variable symbols, \(\in^L\) is a $2$-ary relation symbol, \(\to^L\) is the logical connective symbol for the implication, \(\neg^L\) is the logical connective symbol for the negation, and \(\forall^L\) is the symbol for the universal quantifier. I denote by \(A\) the \(\textrm{ZFC}\) axiom realised as a set of formulae in \(L\). I use obvious syntax sugars in first order set theory, e.g.\ \(=^L, \subset^L, \wedge^L, \vee^L, \exists^L, \exists !^L, \mathbb{N}^L, \times^L, \ldots).

For an \(n \in \mathbb{N}\), I denote by \(\textrm{Formula}^L(n)\) the formula in \(L\) given in the following recursive way: \begin{eqnarray*} \textrm{Formula}^L(n) := \left\{ \begin{array}{ll} x^L(\textrm{Proj}_0(2^{-1}n)) \in^L x^L(\textrm{Proj}_1(2^{-1}n)) & (n \in 2 \mathbb{N}) \\ \textrm{Formula}^L(\textrm{Proj}_0(2^{-2}(n-1))) \to^L \textrm{Formula}^L(\textrm{Proj}_1(2^{-2}(n-1))) & (n \in 2^2 \mathbb{N} + 1) \\ \neg^L \textrm{Formula}^L(2^{-3}(n-3)) & (n \in 2^3 \mathbb{N} + 3) \\ \forall^L x^L(\textrm{Proj}_0(2^{-3}(n-7))), \textrm{Formula}^L(\textrm{Proj}_1(2^{-3}(n-7))) & (n \in 2^3 \mathbb{N} + 7) \end{array} \right. \end{eqnarray*}

For an \(n \in \mathbb{N}\), I denote by \(\textrm{Proof}^L(n) \in \mathbb{N}\) the smallest natural number \(N\) satisfying the following:
 * 1) For any \(m \in n\), \(\textrm{Proof}^L(m) \in N) holds.
 * 2) The finite sequence \((\textrm{Formula}^L(v_{\textrm{Prime}(i)}(N)))_{i \in \textrm{Length}(N)}\) is a formal proof under \(A\).

For a \(d \in \mathbb{N}\), A formal proof \(P\) under \(A\) is said to be of complexity \(< d\) if it is presented as \(\textrm{Proof^L(n)}\) for some \(n \in \mathbb{N}\) with \(n < d\). By the definition, there are only finitely many formal proofs under \(A\) of complexity \(< d\).

= The Ordinal Notation =

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 \(\mathbb{N}^L \times \mathbb{N}^L\).

For an \((n,d) \in \mathbb{N} \times \mathbb{N}\), I define \(\textrm{Def}(n,d) \in \mathbb{N}) in the following recursive way:
 * 1) If \(d > 0\) and \(\textrm{Def}(n,d-1) \neq 0\), then \(\textrm{Def}(n,d) = \textrm{Def}(n,d-1)\).
 * 2) Otherwise, if there exists an \(N \in \mathbb{N}\) satisfying the following two conditions, then \(\textrm{Def}(n,d)\) is the smallest natural natural number satisfying them:
 * 3) There is a formal proof of \(\exists ! x^L(0), \textrm{Formula}^L(N) \wedge^L \textrm{RWO}^L\) of complexity \(< 10^{100}(d+1)\).
 * 4) For any \(m \in n) with \(\textrm{Def}(m,d) \neq 0\), either one of the following holds:
 * 5) There is a formal proof of the existence of an order-preserving injective non-surjective map from the well-ordered set defined by \(\textrm{Formua}^L(\textrm{Def}(m,d))\) to the one defined by \(\textrm{Formula}^L(N)\) of complexity \(< 10^{100}(d+1)\)
 * 6) There is a formal proof of the existence of an order-preserving injective non-surjective map from the well-ordered set defined by \(\textrm{Formula}^L(N') \) to the one defined by \(\textrm{Formua}^L(\textrm{Def}(m,d))\) of complexity \(< 10^{100}(d+1)\).
 * 7) Otherwise, \(\textrm{Def}(n,d) = 0\).

For a \(d \in \mathbb{N}\), I denote by \(N(d) \in \mathbb{N}\) the smallest natural number satisfying \(\textrm{Def}(N(d),d) = 0\).

I define a recursive binary relation \(<_{\textrm{Def}}\) on \(\mathbb{N} \times \mathbb{N}\). The relation \((n_0,d_0) <_{\textrm{Def}} (n_1,d_1)\) is true if the following hold: Then \(<_{\textrm{Def}}\) forms a strict partial order, and its restriction to \(\mathbb{N} \times \{d\}\) is a well-order whose ordinal type is \(\omega\) for any \(d \in \mathbb{N}\).
 * 1) If \(n_1 < N(d_1)\), then the following hold:
 * 2) \(n_0 < N(d_0)\).
 * 3) There is a formal proof of the existence of an order-preserving injective non-surjective map from the well-ordered set defined by \(\textrm{Formua}^L(\textrm{Def}(n_0,d_0))\) to the one defined by \(\textrm{Formula}^L(\textrm{Def}(n_1,d_1))\).
 * 4) If \(n_1 \geq N(d_1)\), then the following hold:
 * 5) If \(d_0 = d_1\), then \(n_0 < n_1\).
 * 6) Otherwise \(n_1 > 0\) and \((n_0,d_0) <_{\textrm{Def}} (n_1-1,d_1)\).

= "Well-Foundedness" =

In this section, I need to refer to the mata-theory of the base theory. As I defined \(L\) and \(A\) in the base theory, the base theory is defined in the meta theory by using a formal language \(ML\) and the set \(MA\) of \(\textrm{ZFC}\) axiom described in \(ML\). I note that the meta theory is an arithmetic or another formal theory containing arithmetic, e.g. \(\textrm{ZF}^{-}\) set theory.

A meta natural number is a natural number in the meta theory. Through the von Neumann construction, a meta natural number corresponds to a formula in \(ML\) defining a natural number in the base theory. Namely, the meta natural number \(0\) corresponds to the formula \(\forall x_1, x_1 \notin x_0\) defining \(\ulcorner 0 \urcorner := \emptyset\), and the meta natural number \(n+1\) corresponds to the formula defining \(\ulcorner n+1 \urcorner := \ulcorner n \urcorner \cup \{\ulcorner n \urcorner\}\). Therefore every meta natural number is regarded as a natural number in the base theory in a harmless way.

Is every natural number in the base theory a meta natural number? It is impossible to expless the question as a formula in the base theory or the meta theory, and hence none can assume that every natural number is a meta natural number. Moreover, even if a formula on a natural number \(n\) in the base theory is provable for any meta natural number, the formula quantified by \(\forall n\) is not necessarily provable. So the non-existence of an infinite strictly decreasing chain starting from any meta natural number does not imply the well-foundedness.

Now let \(n\) and \(d\) be meta natural numbers. If \(n \geq N(d)\), then any strictly decreasing sequence with respect to \(<_{\textrm{Def}}\) starting from \((n,d)\) eventually lies in the locus of pairs of meta natural numbers corresponding to well-ordered sets.

If \(n < N(d)\), then the formal proof of the well-foundedness of \(\textrm{Def}(n,d)\) in the base theory actually gives a formal proof of the corresponding formula in the meta theory.

Therefore in any cases, there is no infinite strictly decreasing sequence with respect to \(<_{\textrm{Def}}\) starting from \((n,d)\). This is the precise meaning of the "well-foundedness" introduced in Rough Sketch.

= Standard Form =

An \((n,d) \in \mathbb{N} \times \mathbb{N}\) is said to be of standard form if \(d\) is the smallest natural number such that \(\textrm{Def}(n,d) \neq 0\). Then the set of pairs \((n,d)\) of standard form forms a recursive subset of \(\mathbb{N} \times \mathbb{N}\), and the restriction of \(<_{\textrm{Def}}\) to it is a strict total order, which is an "well-order" in a natural sense.

I denote by \(\textrm{Standard}(n,d) \in \mathbb{N} \times \mathbb{N}\) the unique pair of standard form whose first entry is \(n\).

= Fundamental Sequence =

For an \((n,d) \in \mathbb{N} \times \mathbb{N}\), I define a map \((n,d)[s] \colon \mathbb{N} \to \mathbb{N} \times \mathbb{N} \) in the following way: I note that this system of fundamental sequences is defined to any \((n,d)\) which does not necessarily correspond to a limit ordinal, and hence is not a fundamental sequence in the usual sense. If \((n,d)\) corresponds to a limit ordinal, then the sequence of ordinals corresponding to \((n,d)[s]\) converges to the ordinal corresponding to \((n,d)\). Otherwise, the successor of the ordinal corresponding to \((n,d)[s]\) coincides with the ordinal corresponding to \((n,d)\) or \((n,d+s)\).
 * 1) If \(n < N(d)\), then the following hold:
 * 2) If \((n,d+s)\) is minimal in \(\mathbb{N} \times \{d+s\}\) with respect to the restriction of \(<_{\textrm{Def}}\), then \((n,d)[s] = (n,d+s)\).
 * 3) Otherwise, \((n,d)[s]\) is the unique element of \(\mathbb{N} \times \{d+s\}\) whose successor is \((n,d+s)\) with respect to the restriction of \(<_{\textrm{Def}}\).
 * 4) If \(n \geq N(d)\), then the following hold:
 * 5) If \(n = N(d)\), the the following hold:
 * 6) If \(n = 0\), then \((n,d)[s] = (n,d)\).
 * 7) Otherwise, then \((n,d)[s]\) is the greatest element of \(N(d) \times \{d\}\), whose successor is \((n,d)\), with respect to the restriction of \(<_{\textrm{Def}}\).
 * 8) Otherwise, then \((n,d)[s] = (n-1,d)\).

If \((n,d)\) is of standard form, then the alternative seuquence \(\textrm{Standard}((n,d)[s])\) works as a system of fundamental sequences restricted to the subset of pairs of standard form, because \(\textrm{Standard}((n,d)[s])\) is a pair of standard form which shares the corresponding ordinal with \((n,d)[s]\).

= FGH =

For an \((n,d) \in \mathbb{N} \times \mathbb{N}\), I define a computable function \begin{eqnarray*} f_{(n,d)} \colon \mathbb{N} & \to & \mathbb{N} \\ s & \mapsto & f_{(n,d)}(s) \end{eqnarray*} in the following recursive way:
 * 1) If \((n,d)[s] = (n,d+s)\), then \(f_{(n,d)}(s) = s+1\).
 * 2) Otherwise, \(f_{(n,d)}(s) = f_{(n,d)[s]}^s(s)\).

= Large Number =

I denote by \(f\) the computable partial function in the base theory defined as \(f(n) := f_{(n,n)}(n)\). It is not provably total, because the termination of the recursion process is just provable for meta natural numbers. At least, \(10^{100}\) is a meta natural number, and \(f\) sends a meta natural number to a meta natural number. Therefore \(f^{10^{100}}(10^{100})\) is a well-defined computable large number in \(\textrm{ZFC}\( set thoery.