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

I construct a completely well-defined ordinal notation, which presents a "proof-theoretic analogue" of the PTO of \(\textrm{ZFC}\) together with a canonical choice of a system of fundamental sequences, and define a computable large number.

I would like you to analyse this new ordinal notation. I expect that it presents very large recursive ordinals.

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

= Rough Sketch =

I will define a map \(\textrm{RWO}\) sending an \((n,d) \in \mathbb{N} \times \mathbb{N}\) to (the Goedel number of) a formula \(\textrm{RWO}(n,d)\) in an internal set theory such that if there is a formula \(F\) satisfying the following two conditions with respect to the notion of the complexity of a formal proof defined later, then \(\textrm{RWO}(n,d)\) also satisfies them: Then \(\textrm{RWO}\) yields a "well-founded" strict partial order \(<_{\textrm{RWO}}\) 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 recursive subset of \(\mathbb{N} \times \mathbb{N}\) admits a formal proof of complexity \(< 10^{100} \uparrow \uparrow \uparrow (d+1) \).
 * 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 \(\textrm{RWO}(m,d)\) and the statement that the ordinal type of the well-order defined by \(F\) is greater than that of \(\textrm{RWO}(m,d)\) admits a formal proof of complexity \(< 10^{100} \uparrow \uparrow \uparrow (d+1)\).

The "well-founded" roughly means that there exists no infinite strictly decreasing sequence with respect to \(<_{\textrm{RWO}}\) starting from any pair \((n,d)\) of natural numbers presented by the repetition of successors eventually terminates. A reader might think that every natural number is presentable in that way, but it is not true. I will explain it later.

Unfortunately, the order \(<_{\textrm{RWO}}\) itself is not recursive, and hence I could not use it directly in order to define computable large functions and computable large numbers. Instead, I define a generator of \(<_{\textrm{RWO}}\) given as a recursive binary relation \(triangleleft_{\textrm{RWO}}\), which allows me to construct computable ones through FGH.

The second condition of \(\textrm{RWO}(n,d)\) 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 would not admit a generator similar to \(<_{\textrm{RWO}}\), and hence I could not define computable large functions and computable large numbers. 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 the PTO of \(\textrm{ZFC}\).

= Preparation =

For an \(n \in \mathbb{N} \setminus \{0\}\), 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} \setminus \{0\} & \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} \setminus \{0\} \\ (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} \setminus \{0\} \\ (e_i)_{i \in \mathbb{N}} & \mapsto & \prod_{i \in \mathbb{N}} \textrm{Prime}(i)^{e_i}. \end{eqnarray*} is a bijective map, whose inverse is the map \begin{eqnarray*} \mathbb{N} \setminus \{0\} & \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 \(\mathbb{N}^{\mathbb{N}} \cap \mathbb{Z}^{\oplus \mathbb{N}}\).

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

= 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 \(L := \{ x^L(i) \mid i \in \mathbb{N} \} \cup \{\in^L, \to^L, \neg^L, \forall^L \}\) forms a formal language of first order set theory such that \(x^L(i)\)'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 might use obvious syntax sugars in first order set theory, e.g. \(\notin^L, =^L, \neq^L, \subset^L, \wedge^L, \vee^L, \exists^L, \exists !^L, \cup^L, \times^L, (x,y)^L, \{x\}^L, \mathbb{N}^L, \omega^L, \ldots\).

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(\frac{n}{2})) \in^L x^L(\textrm{Proj}_1(\frac{n}{2})) & (n \in 2 \mathbb{N}) \\ \textrm{Formula}^L(\textrm{Proj}_0(\frac{n-1}{4})) \to^L \textrm{Formula}^L(\textrm{Proj}_1(\frac{n-1}{4})) & (n \in 4 \mathbb{N} + 1) \\ \neg^L \textrm{Formula}^L(\frac{n-3}{8}) & (n \in 8 \mathbb{N} + 3) \\ \forall^L x^L(\textrm{Proj}_0(\frac{n-7}{8})), \textrm{Formula}^L(\textrm{Proj}_1(\frac{n-7}{8})) & (n \in 8 \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 positive integer \(N\) 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 \(n \in d\). 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 \(\mathbb{N}^L \times^L \mathbb{N}^L\).

For an \((n,d) \in \mathbb{N} \times \mathbb{N}\), I define \(\textrm{RWO}(n,d) \in \mathbb{N}\) in the following recursive way:
 * 1) If \(d > 0\) and \(\textrm{RWO}(n,d-1) \neq 0\), then \(\textrm{RWO}(n,d) = \textrm{RWO}(n,d-1)\).
 * 2) Otherwise, if there exists an \(N \in \mathbb{N}\) satisfying the following two conditions, then \(\textrm{RWO}(n,d)\) 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 \(< 10^{100} \uparrow \uparrow \uparrow (d+1)\).
 * 4) For any \(m \in n\) with \(\textrm{RWO}(m,d) \neq 0\), 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 \(\textrm{Formula}^L(\textrm{RWO}(m,d))\) to the one defined by \(\textrm{Formula}^L(N)\) of complexity \(< 10^{100} \uparrow \uparrow \uparrow (d+1)\)
 * 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 \(\textrm{Formula}^L(\textrm{RWO}(m,d))\) of complexity \(< 10^{100} \uparrow \uparrow \uparrow (d+1)\).
 * 7) Otherwise, \(\textrm{RWO}(n,d) = 0\).

For a \(d \in \mathbb{N}\), I denote by \(N(d) \in \mathbb{N}\) the smallest natural number with \(\textrm{RWO}(N(d),d) = 0\), which actually exists because there are only finitely many formal proofs of complexity \(<d\).

= Partial Orders =

I define a recursive binary relation \(<_{\textrm{RWO},d}\) on \(\mathbb{N} \times \{d\}\) for each \(d \in \mathbb{N}\). The relation \((n_0,d) <_{\textrm{RWO}} (n_1,d)\) is true if the following hold: Then \(<_{\textrm{RWO},d}\) forms a strict well-order of ordinal type \(\omega\).
 * 1) If \(n_1 < N(d)\), then the following hold:
 * 2) \(n_0 < N(d)\).
 * 3) There is a formal proof of the existence of an order-preserving injective non-surjective map from the well-ordered set defined by \(\textrm{Formula}^L(\textrm{RWO}(n_0,d))\) to the one defined by \(\textrm{Formula}^L(\textrm{RWO}(n_1,d))\) of complexity \(< 10^{100} \uparrow \uparrow \uparrow (d+1)\).
 * 4) If \(n_1 \geq N(d_1)\), then \(n_0 < n_1\).

Also, I define a recursive binary relation \(\triangleleft_{\textrm{RWO}}\) on \(\mathbb{N} \times \mathbb{N}\). The relation \((n_0,d_0) \triangleleft_{\textrm{RWO}} (n_1,d_1)\) is true if either one of the following hold: Then \(\triangleleft_{\textrm{RWO}}\) generates a non-recursive strict partial order \(<_{\textrm{RWO}}\), and the restrictions of \(\triangleleft_{\textrm{RWO}}\) and \(<_{\textrm{RWO}}\) to \(\mathbb{N} \times \{d\}\) coincide with \(<_{\textrm{RWO},d}\) for any \(d \in \mathbb{N}\).
 * 1) If \(d_0 < d_1\), then \(n_0 < N(d_0)\) and either one of the following hold:
 * 2) \(n_0 = n_1\).
 * 3) \((n_0,d_1) <_{\textrm{RWO},d_1} (n_1,d_1)\).
 * 4) If \(d_0 = d_1\), then \((n_0,d_0) <_{\textrm{RWO},d_1} (n_1,d_1)\).
 * 5) If \(d_0 > d_1\), then \(n_1 < N(d_1)\) and \((n_0,d_0) <_{\textrm{RWO},d_0} (n_1,d_0)\).

I emphasise that I do not use the non-recursive strict partial order \(<_{\textrm{RWO}}\) for the definition of large functions and large numbers, because I would like to define computable ones. I just use it to ensure that the resulting computable functions actually halt.

= "Well-Foundedness" =

In this section, I need to refer to the meta 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 \(L^{\textrm{M}}\) and \(\textrm{ZFC}\) axiom \(A^{\textrm{M}}\) realised as a set of formulae in \(L^{\textrm{M}}\). 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 \(L^{\textrm{M}}\) 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 express 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 \(<_{\textrm{RWO}}\) 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 \(\textrm{RWO}(n,d)\) 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 \(<_{\textrm{RWO}}\) starting from \((n,d)\). This is the precise meaning of the "well-foundedness" introduced in Rough Sketch.

= Examples =

The formula \(\forall^L x^L(1), x^L(1) \notin^L x^L(0)\) in \(L\) defines the emptyset \(0^L\), which is itself the unique recursive well-order on the recursive subset \(0^L \subset \mathbb{N}^L \times^L \mathbb{N}^L\). The corresponding Goedel number is computed in the following way: \begin{eqnarray*} & & \forall^L x^L(1), x^L(1) \notin^L x^L(0) \\ & = & \forall^L x^L(1), \neg^L \textrm{Formula}^L(2 \times 2^1(2 \times 0 + 1)) \\ & = & \forall^L x^L(1), \neg^L \textrm{Formula}^L(4) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(8 \times 4 + 3) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(35)) \\ & = & \textrm{Formula}^L(8 \times 2^1(2 \times 35 + 1) + 7) \\ & = & \textrm{Formula}^L(1135) \end{eqnarray*} Since this formula belongs to \(A\), the sequence \((\textrm{Formula}^L(1135))\) is a formal proof under \(A\). By \(\textrm{Length}(2^{1135}) = 1\) and \(v_{\textrm{Prime}(0)}(2^{1135}) = v_2(2^{1135}) = 1135\), the equality \((\textrm{Formula}^L(v_{\textrm{Prime}^L(i)}(2^{1135}))_{i \in \textrm{Length}(2^{1135})} = (\textrm{Formula}^L(1135))\) holds.

I guess that \((1135,1)\) is the unique minimal element of \(\mathbb{N} \times \{1\}\). It would be incorrect if there were a "shorter" formula in \(L\) which defines a recursive well-order on a recursive subset of \(\mathbb{N}^L \times^L \mathbb{N}^L\) whose ordinal such that the statement that its ordinal type is \(\emptyset^L\) is independent of \(A\). Although I do not check all possible formulae, the non-existence of such a formula is not so doubtful. If it is correct, then I obtain the following: \begin{eqnarray*} \textrm{RWO}(n,0) & = & 0 \\ \textrm{RWO}(0,1) & = & 1135 \end{eqnarray*}

Although I need only to consider a recursive well-order on a recursive subset of \(\mathbb{N}^L \times^L \mathbb{N}^L\), it might be good to demonstrate the computation of the Goedel number of the formula \(\forall^L x^L(1), (x^L(1) \in^L x^L(0) \leftrightarrow^L \forall x^L(2), x^L(2) \notin^L x^L(1))\) in \(L\) defining the next easiest example \(1^L\). The corresponding Goedel number is computed in the following way: \begin{eqnarray*} & & \forall^L x^L(1), x^L(1) \in^L x^L(0) \leftrightarrow^L \forall x^L(2), x^L(2) \notin^L x^L(1) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(2 \times 2^1(2 \times 0 + 1)) \leftrightarrow^L \forall x^L(2), \neg^L \textrm{Formula}^L(2 \times 2^2(2 \times 1 + 1)) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(4) \leftrightarrow^L \forall x^L(2), \neg^L \textrm{Formula}^L(24) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(4) \leftrightarrow^L \forall x^L(2), \textrm{Formula}^L(8 \times 24 + 3) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(4) \leftrightarrow^L \forall x^L(2), \textrm{Formula}^L(195) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(4) \leftrightarrow^L \textrm{Formula}^L(8 \times 2^2(2 \times 195 + 1) + 7) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(4) \leftrightarrow^L \textrm{Formula}^L(12519) \\ & = & \forall^L x^L(1), (\textrm{Formula}^L(4) \to^L \textrm{Formula}^L(12519)) \wedge^L (\textrm{Formula}^L(12519) \to^L \textrm{Formula}^L(4)) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(4 \times 2^4(2 \times 12519 + 1) + 1) \wedge^L \textrm{Formula}^L(4 \times 2^{12519}(2 \times 4 + 1) + 1) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(1602497) \wedge^L \textrm{Formula}^L(2^{12521} \times 9 + 1) \\ & = & \forall^L x^L(1), \neg^L(\textrm{Formula}^L(1602497) \to^L \neg^L \textrm{Formula}^L(2^{12521} \times 9 + 1)) \\ & = & \forall^L x^L(1), \neg^L(\textrm{Formula}^L(1602497) \to^L \textrm{Formula}^L(8 \times (2^{12521} \times 9 + 1) + 3) \\ & = & \forall^L x^L(1), \neg^L(\textrm{Formula}^L(1602497) \to^L \textrm{Formula}^L(2^{12524} \times 9 + 11) \\ & = & \forall^L x^L(1), \neg^L(\textrm{Formula}^L(4 \times 2^{1602497}(2 \times (2^{12524} \times 9 + 11) + 1) + 1) \\ & = & \forall^L x^L(1), \neg^L(\textrm{Formula}^L(2^{1602499}(2^{12525} \times 9 + 23) + 1) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(8 \times (2^{1602499}(2^{12525} \times 9 + 23) + 1) + 3) \\ & = & \forall^L x^L(1), \textrm{Formula}^L(2^{1602502}(2^{12525} \times 9 + 23) + 11) \\ & = & \textrm{Formula}^L(8 \times 2^1(2 \times (2^{1602502}(2^{12525} \times 9 + 23) + 11) + 1) + 7) \\ & = & \textrm{Formula}^L(2^{1602507}(2^{12525} \times 9 + 23) + 375) \end{eqnarray*}

Now let \(N \in \mathbb{N}\) be a natural number such that \(\textrm{Formula}^L(N)\) defines \(\mathbb{N}^L\) without occurence of \(x^L(i)\) for any \(i \in 5\) and with free variable \(x^L(5)\). Then the formula \begin{eqnarray*} \forall^L x^L(1), \forall^L x^L(2), \left( \begin{array}{l} (x^L(1),x^L(2))^L \in^L x^L(0) \\ \leftrightarrow^L \exists^L x^L(3), \exists^L x^L(4), \left( \begin{array}{cl} & x^L(1) =^L (x^L(3),x^L(3))^L \\ \wedge^L & x^L(2) =^L (x^L(4),x^L(4))^L \\ \wedge^L & \forall^L x^L(5), \textrm{Formula}^L(N) \to^L x^L(4) \in^L x^L(5) \\ \wedge^L & x^L(3) \subset^L x^L(4) \end{array} \right) \end{array} \right)\end{eqnarray*} defines a recursive well-order on the diagonal subset of \(\mathbb{N}^L \times^L \mathbb{N}^L\), whose ordinal type is \(\omega^L\).

Given an \(N \in \mathbb{N}\) such that \(\textrm{Formula}^L(N)\) defines a recursive well-order \(\leq_S\) on a recursive subset \(S \subset^L \mathbb{N}^L \times^L \mathbb{N}^L\), it is easy (and also tiresome) to define its successor by replacing \(S\) and \(\leq_S\) by \(\{(0^L,0^L)^L\}^L \cup^L \{(x \cup^L \{x\}^L, y \cup^L \{y\}^L)^L \mid (x,y)^L \in^L S\}^L\) and \(\{(0^L,0^L)^L\}^L \cup^L \{(0,y \cup^L \{y\}^L)^L \mid (y,y)^L \in^L S \}^L \cup^L \{(x \cup^L \{x\}^L, y \cup^L \{y\}^L)^L \mid (x,y)^L \in^L S\}^L\). I denote by \(\textrm{Succ}(N) \in \mathbb{N}\) the natural number corresponding to the successor. If \(N = \textrm{RWO}^L(n,d)\) for some \((n,d) \in \mathbb{N} \times \mathbb{N}\), then \(\textrm{Succ}(N)\) might be one of candidates of \(\textrm{RWO}(n+1,d)\).

It is not easy to determine whether the Goedel code of a given definition of a well-order corresponding to a recursive ordinal number well-known here, e.g. \(\varepsilon_0\), \(\textrm{LVO}\), \(\psi_0(\psi_I(0))\), \(\psi_0(\varepsilon_{K + 1})\) and so on, lies in the image of \(\textrm{RWO}\). However, \(\textrm{RWO}\) partially diagonalises the proof theoretic strengthen of the \(\textrm{ZFC}\) axiom. Therefore I expect that the limit of this ordinal notation is greater actually than that of another ordinal notation which is provably well-founded under \(\textrm{ZFC}\) axiom.

= 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 \(\textrm{RWO}(n,d) \neq 0\). Then the set of pairs \((n,d)\) of standard form forms a recursive subset of \(\mathbb{N} \times \mathbb{N}\), and the restriction of \(<_{\textrm{RWO}}\) 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 \begin{eqnarray*} \mathbb{N} & \to & \mathbb{N} \times \mathbb{N} \\ s & \mapsto & (n,d)[s] \end{eqnarray*} 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 \(<_{\textrm{RWO},d+s}\), 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 \(<_{\textrm{RWO},d+s}\).
 * 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 \(<_{\textrm{RWO},d}\).
 * 8) Otherwise, then \((n,d)[s] = (n-1,d)\).

If \((n,d)\) is of standard form, then the alternative sequence \(\textrm{Standard}((n,d)[s])\) 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 \(f_{(n,d)}(s)\) in the following recursive way: The domain of \(f_{(n,d)}\) is \(\mathbb{N}\) as long as \(n\) and \(d\) are meta natural numbers, because the fundamental sequence of \((n,d)\) consists of pairs of meta natural numbers. Otherwise, the domain of \(f_{(n,d)}\) might be empty. At least, the correspondence \((n,d,s) \mapsto f_{(n,d)}(s)\) is a well-defined partial function on \(\mathbb{N} \times \mathbb{N} \times \mathbb{N}\).
 * 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 theory.

ja:ユーザーブログ:P進大好きbot/ZFCのPTOの「証明論的類似」