User blog:P進大好きbot/New Large Number beyond MK set theory

I introduce a first order set theory \(\textrm{MK}_+\) which is a conservative extension of \(\textrm{MK}\) set theory, and define an uncomputable large number in \(\textrm{MK}_+\).

I think that it can be a candidate of the largest one among large numbers whose definition contains no ambiguity or no cheating. If there is a typo or an error, please tell me.

= Language =

The language \(L\) of \(\textrm{MK}_+\) set theory consists of the following:
 * 1) Infinitely many variable term symbols \(x_0,x_1,x_2,\ldots\) indexed by meta natural numbers
 * 2) Infinitely many constant term symbols \(c_0,c_1,c_2,\ldots\) indexed by meta natural numbers
 * 3) A \(2\)-ary relation symbol \(\in\)

Then the axiom schema \(A\) of \(\textrm{MK}\) set theory can be described by \(L\) after noting that the equality \(=\) and the notion of a set are definable. Here, the axioms of separation and replacement refer to any \(L\)-formulae.

= Witness =

For any class \(X\), I define a formal language \(L_X\) of a first order set theory in \((L,A)\) in the following way:
 * 1) For any \(i \in \mathbb{N}\), \(\ulcorner x_i \urcorner := (0,i) \in V^2\) is a variable \(L_X\)-term.
 * 2) For any \(i \in \mathbb{N}\), \(\ulcorner c_i \urcorner := (1,i) \in V^2\) is a constant \(L_X\)-term.
 * 3) For any \(c \in X\), \(\ulcorner c \urcorner := (2,c) \in V^3\) is a constant \(L_X\)-term.
 * 4) For any \(L_X\)-terms \(s\) and \(t\), \(\ulcorner s \in t \urcorner := (3,s,t) \in V^3\) is an \(L_X\)-formula.
 * 5) For any \(L_X\)-formulae \(p\) and \(q\), \(\ulcorner p \to q \urcorner := (4,p,q) \in V^3\) is an \(L_X\)-formula.
 * 6) For any \(L_X\)-formula \(p\), \(\ulcorner \neg p \urcorner := (5,p) \in V^2\) is an \(L_X\)-formula.
 * 7) For any variable \(L_X\)-term \(x\) and \(L_X\)-formula\(p\), \(\ulcorner \exists x(p) \urcorner := (6,x,p) \in V^3\) is an \(L_X\)-formula.

The substition of an \(L_V\)-term for a free variable \(L_V\)-term in an \(L_V\)-formula makes sense in the usual way. For any variable \(L_V\)-term \(x\), \(L_V\)-formula \(p\), and \(c \in V\), I denote by \(\textrm{Sub}(x,p,c)\) the \(L_V\)-formula obtained by substituting \(c\) for all free occurrence of \(x\) in \(p\).

The satisfaction of \(L_X\)-formula at a class makes sense because \((L,A)\) allows unbounded quantifiers. For any \(L\)-terms \(p\) and \(c\), I abbreviate to \(\textrm{Sat}(p,c)\) the \(L\)-formula "\(p\) is a closed \(L_V\)-formula and \(c\) satisfies \(p\)".

= Enumeration =

I define an injective map \begin{eqnarray*} \iota \colon L_{\emptyset} & \hookrightarrow & \mathbb{N} \\ a & \mapsto & \iota(a) \end{eqnarray*} in the following recursive way:
 * 1) If \(a = (n,i)\) for some \(n \in \{0,1\}\) and \(i \in \mathbb{N}\), then \(\iota(a) := 2^n 3^i\).
 * 2) If \(a = (n,x,y)\) for some \(n \in \{3,4,6\}\) and \((x,y) \in L_{\emptyset}^2\), then \(\iota(a) := 2^n 3^{\iota(x)} 5^{\iota(y)}\).
 * 3) If \(a = (5,p)\) for some \(p \in L_{\emptyset}\), then \(\iota(a) := 2^5 3^{\iota(p)}\).

I denote by \(D \subset \mathbb{N}\) the subset consisting of \(m \in \mathbb{N}\) such that there is an \(L_V\)-formulae \(p\) with \(\iota(p) = m\)such that \(\ulcorner \exists \ulcorner x_0 \urcorner, p \urcorner\) is a closed \(L_V\)-formula. I define a bijective map \begin{eqnarray*} d \colon \mathbb{N} & \hookrightarrow & D \\ n & \mapsto & d(n) \end{eqnarray*} in the following recursive way:
 * 1) If \(n = 0\), then \(d(n)\) is the minimum of \(D\).
 * 2) If \(n > 0\), then \(d(n)\) is the minimum of \(\{m \in D \mid m > d(n-1)\}\).

I define an injective map \begin{eqnarray*} P \colon \mathbb{N} & \hookrightarrow & L_{\emptyset} \\ n & \mapsto & P(n) \end{eqnarray*} as the composite of \(d\) followed by the inverse of \(\iota |_{\iota^{-1}(D)}\).

= Axiom =

Let \(i\) be a meta natural number. I denote by \(\ulcorner i \urcorner \in V\) the standard natural number defined by applying successors \(i\)-times to \(\emptyset\). For an \(L\)-term \(p\), I abbreviate to \(T(p)\) the \(L\)-formulae "\(p\) is an \(L_V\)-formula and \(\textrm{Sat}(\ulcorner \exists \ulcorner x_0 \urcorner, p \urcorner,V)\)". For \(L\)-terms \(p\) and \(c\), I abbreviate to \(T(p,c)\) the \(L\)-formulae "\(p\) is an \(L_V\)-formula and \(\textrm{Sat}(\texrm{Sub}(\ulcorner x_0 \urcorner,c,p),V)\).

I define an \(L\)-formula \(p_i\) in the following recursive way:
 * 1) If \(i = 0\), then \(p_i\) is the \(L\)-formula \(x_0 = \min \{m \in \mathbb{N} \mid T(P(m))\}\).
 * 2) If \(i > 0\), then \(p_i\) is the \(L\)-formula \(x_i = \min \{m \in \mathbb{N} \mid (\forall x_{i-1}, (p_{i-1} \to (m > x_{i-1}))) \wedge T(P(m))\}\).

I abreviate to \(q_i\) the \(L\)-formula \(\forall x_i, (p_i \to T(P(x_i),c_i))\).

I define \(\textrm{MK}_+\) set theory as the effectively axiomisable first order set theory described by \(L\) whose axiom consists of formulae in \(A \cup \{p_0,p_1,p_2,\ldots\}\).

= Large Number =

I define a large function \begin{eqnarray*} f \colon \mathbb{N} & \to & \mathbb{N} \\ n & \mapsto & f(n) \end{eqnarray*} in the following way:
 * 1) Put \(D_n := \{m \in \mathbb{N} \mid \exists k \in \mathbb{N}, ((k \leq n) \wedge T(P(k),m) \wedge (\forall h \in \mathbb{N},(T(P(k),h) \to (h = m))))\}\).
 * 2) I define \(f(n)\) as the smallest natural number greater than any element in the finite set \(D_n\).

Then \(f(10 \uparrow^{10} 10)\) is sufficiently large. For example, unlike \(\textrm{ZFC}\) set theory or \(\textrm{MK}\) set theory, \(\textrm{MK}_+\) set theory admits a definable well-order on \(V\). Therefore any effectively axiomisable consistent (in the formalised sense) first order theory \(T\) admits a definable set model \(M\) by completeness theorem. Then natural numbers defined by using such a \((T,M)\) are also used to define \(f(n)\), and hence \(f\) diagonalises effectively axiomisable consistent first order theories.