User blog:P進大好きbot/BMOCF

Throughout this blog post, I follow the terminology in the 2/6/2018 version of koteitan's classification of BMS.

I construct a notation system \(T\) of predicate logic form whose subsystem is equivalent to BM2.3. A similar construction works for all versions of BMS which employs Upper-Branch-Ignoring-Model as the Bad root searching rule, i.e. BM1.1, 2, 2.1, 2.3, 3.1, 3.1.1, 3.2, and 4. I call a system \(\psi\) of function symbols in \(T\) "BMOCF" by pacring UNOCF. I emphasise that BMOCF is not an actual OCF (set theory) but is just a system of function symbols in a notation system (arithmetic). I appreciate any corrections.

= Outline =

Following The Hydra Diagram based on Upper-Branch-Ignoring Model for understanding BM4 structure, I extended a rough relation between weighted hydra and Buchholz's notation system to one between vector-weighted hydra and a notation system \(T\). Using the relation, I directly converted the expansion rule of BM2.3 into a system of fundamental sequences of \(T\). Then by the construction, BM2.3 (restricted to standard forms) is equivalent to a subsystem of \(T\).

For example, I have the following correspondence, where \(\psi\) denotes BMOCF:

= Notation =

I denote by \(\mathbb{N}\) the set of natural numbers, and by \(\mathbb{N}_{+}\) the set of positive integers.

For a set \(X\), I denote by \(X^{< \omega}\) the set of finite arrays of elements of \(X\).

For a finite array \(a\), I denote by \(\textrm{Lng}(a) \in \mathbb{N}_{+}\) the length of \(a\).

For a finite array \(a\) and a \(j \in \mathbb{N}\) with \(j < \textrm{Lng}(a)\), I denote by \(a_j\) the \(j+1\)-st entry of \(a\).

For a finite array \(a\) and \((j_0,j_1) \in \mathbb{N} \times \mathbb{N}\) with \(j_0 \leq j_1 < \textrm{Lng}(a)\), I denote by \((a_j)_{j=j_0}^{j_1}\) the subarray of \(a\) consisting of \(j\)-th entries with \(j \in \mathbb{N}\) and \(j_0 \leq j \leq j_1\).

For a finite array \(a\), I denote by \((a_j)_{j=0}^{-1}\) the empty array.

= Formal Language =

I denote by \(\Sigma\) the alphabet \(\{\psi_a \mid a \in \mathbb{N}_{+}\} \cup \{,+,0\}\), i.e. a countable set of symbols.

I define the notion of a "BM-term" in the following recursive way:
 * 1) The symbol \(0\) is a BM-term.
 * 2) For any \(a \in \mathbb{N}_{+}^{< \omega}\) and any BM-term \(t\), the string \(\psi_a(t)\) is a BM-term. Further, I call such a BM-term a "principal BM-term".
 * 3) For any BM-term \(t_0 \neq 0\) and any principal BM-term \(t_1\), the string \(t_0 + t_1\) is a BM-term.

I denote by \(T \subset \Sigma^{< \omega}\) the recursive subset of BM-terms, and by \(PT \subset T\) the recursive subset of principal BM-terms.

I call the system \(\{\psi_a \mid a \in \mathbb{N}_{+}^{< \omega}\}\) "BMOCF".

I abbreviate \(\psi_{}\) (the BMOCF whose suffix is the empty array) to \(\psi\), and \(\psi_{(n)}\) to \(\psi_n\) for any \(n \in \mathbb{N}_+\).

= Arithmetic =

I define an injective recursive map \begin{eqnarray*} \textrm{operator} \ulcorner \urcorner \colon \mathbb{N} & \hookrightarrow & T \\ n & \mapsto & \ulcorner n \urcorner \end{eqnarray*} in the following recursive way:
 * 1) If \(n = 0\), then \(\ulcorner n \urcorner := 0\).
 * 2) If \(n = 1\), then \(\ulcorner n \urcorner := \psi(0)\).
 * 3) If \(n > 1\), then \(\ulcorner n \urcorner := \ulcorner n-1 \urcorner + \psi(0)\).

I denote by \(NT \subset T\) the image of \(\textrm{operator} \ulcorner \urcorner\).

I define a recursive map \begin{eqnarray*} \times \colon T \times NT & \hookrightarrow & T \\ (t,z) & \mapsto & t \times z \end{eqnarray*} in the following recursive way:
 * 1) If \(z = 0\), then \(t \times z := 0\).
 * 2) If \(z = \ulcorner n \urcorner\) for some \(n \in \mathbb{N}_{+}\), then \(t \times z := t \times \ulcorner n-1 \urcorner + t\).

= Fundamental Sequence =

I introduce a recursive strict partial order \(<\) on \(\mathbb{N}_{+}^{< \omega}\). For any \(a,b \in \mathbb{N}_{+}^{< \omega}\), \(a < b'\) holds if they satisfies the following:
 * 1) The array \(b\) is not empty.\
 * 2) For any \(j \in \mathbb{N}_{+}\) with \(j < \min \{\textrm{Lng}(a),\textrm{Lng}(b)\}\), the inequality \(a_j < b_j\) holds.

For each \(a \in \mathbb{N}_{+}^{< \omega}\), I define the notion of "degree \(< a\)" in the following recursive way:
 * 1) The BM-term \(0\) is of degree \(< a\).
 * 2) For any \((b,t) \in \mathbb{N}_{+}^{< \omega} \times T\) with \(b < a\), \(\psi_b(t)\) is of degree \(< a\).
 * 3) For any BM-term \(t_0 \neq 0\) of degree \(< a\) and any principal BM-term \(t_1\) of degree \(< b\), \(t_0 + t_1\) is of degree \(< a\).

I denote by \(T_a \subset T\) the recursive subset of BM-terms of degree \(< a\).

I define recursive maps \begin{eqnarray*} \textrm{dom} \colon T & \to & \{\emptyset,\{0\},NT\} \cup \{T_a \mid a \in \mathbb{N}_{+}^{< \omega}\} \\ t & \mapsto & \textrm{dom}(t) \end{eqnarray*} and \begin{eqnarray*} \textrm{operator}[] \colon \(\{(t,z) \in T \times T \mid z \in \textrm{dom}(t)\} & \to & T \\ (t,z) & \mapsto & t[z] \end{eqnarray*} in the following recursive way:
 * 1) If \(t = 0\), then \(\textrm{dom}(t) := \emptyset\).
 * 2) Suppose \(t = \psi_a(t')\) for a unique \((a,t') \in \mathbb{N}_{+}^{< \omega} \times T\).
 * 3) If \(\textrm{dom}(t') = \emptyset\), then \(\textrm{dom}(t) := T_a\) and \(t[z] := z\).
 * 4) If \(\textrm{dom}(t') = \{0\}\), then \(\textrm{dom}(t) := NT\) and \(t[z] := \psi_a(t'[0]) \times z\)
 * 5) If \(\textrm{dom}(t') = NT\), then \(\textrm{dom}(t) := NT\) and \(t[z] = \psi_a(t'[z])\).
 * 6) Suppose \(\textrm{dom}(t') = T_b\) for some \(b \in \mathbb{N}_{+}^{\omega}\).
 * 7) Suppose \(a < b\).
 * 8) Then \(\textrm{dom}(t) := NT\).
 * 9) If \(z = 0\), then \(t[z] := \psi_a(t'[0]) \in T_b\).
 * 10) If \(z = \ulcorner n \urcorner\) with \(n \in \mathbb{N}_{+}\), then \(t[z] := \psi_a(t'[t[\ulcorner n-1 \urcorner]]) \in T_b\).
 * 11) Suppose \(\neq (a < b)\).
 * 12) Put \(j_1 := \textrm{Lng}(b)\).
 * 13) Put \(j_0 := \min \{j \in \mathbb{N}_{+} \mid j < \min \{\textrm{Lng}(a),\textrm{Lng}(b)\} \wedge a_j \geq b_j\}\).
 * 14) Denote by \(c \in \mathbb{N}_{+}^{< \omega}\) the concatenation of \((a_j)_{j=0}^{j_0-1}\) and \((b_j)_{j=j_0}^{j_1}\).
 * 15) Then \(\textrm{dom}(t) = T_c\) and \(t[z] := \psi_a(t'[z])\).
 * 16) If \(t = t_0 + t_1\) for a unique \((t_0,t_1) \in T \times PT\) with \(t_0 \neq 0\), then \(\textrm{dom}(t) := \textrm{dom}(t_1)\) and \(t[z] := t_0 + t_1[z]\).