One of the world's biggest numbers, dedicated to one of the world's best foods! My attempt at making the largest valid googologism. Leave any corrections in the comments, and I will make sure to fix them because I won't let a tiny mistake get in my way :)
Language[]
I define the language \(\textrm{๐}_{\textrm{SUSHI}}\). It consists of:
- Countably many numeric variable symbols \(a_0, a_1, ...\)
- Countably many numeric constant symbols \(0, 1, b_0, b_1, ...\)
- Countably many set variable symbols \(A_0, A_1, ...\)
- Countably many set constant symbols \(\varnothing, B_0, B_1, ...\)
- Countably many function symbols \(\cap, \cup, S, \cdot, +, f_0, f_1, ...\)
- Countably many relation symbols \(<, =, \in, \prec_0, \prec_1, ...\)
- Countable many formula variable symbols \(\phi_0, \phi_1, ...\)
- Countably many logical connectives \(\land, \lor, \neg, \rightarrow, \leftrightarrow, \forall, \exists, L_0, L_1, ...\)
- Countably many predicate symbols \(M, P_0, P_1, ...\)
- The brace symbols \((, )\) and \(\{, \}\)
The syntax of the language is defined in the following inductive way:
- The following are valid formulas:
- \(Ma_0\)
- \(a_0 P_0 a_1\)
- \(a_0 P_1 a_1\)
- ...
- \(a_0 < a_1\)
- \(a_0 = a_1\)
- \(a_0 \in A_0\)
- \(a_0 \prec_0 a_1\)
- \(a_0 \prec_1 a_1\)
- ...
- If \(\varphi\) and \(\phi\) are valid formulas, then the following also are:
- \(\varphi \land \phi\)
- \(\varphi \lor \phi\)
- \(\neg \varphi\)
- \(\varphi \rightarrow \phi\)
- \(\varphi \leftrightarrow \phi\)
- \(\forall a_0 \varphi\)
- \(\exists a_0 \varphi\)
- \(\varphi L_0 \phi\)
- \(\varphi L_1 \phi\),
- ...
I abbreviate \(x < y \lor x = y\) as \(x \leq y\), \(x \neg = y\) as \(x \neq y\), \(\neg \exists\) as \(\nexists\) and \(x \neg \in y\) as \(x \notin y\). For a formula \(\varphi\) in the language \(\textrm{๐}_{\textrm{SUSHI}}\), we define its order:
- If it does not include any numerical or set objects, predicates about them, or quantifiers, then its order is 0.
- If it includes any of the above, but does not quantify over predicates, then its order is 1.
- If it quantifies over predicates, but the predicates quantified over don't themselves have predicates or functions as arguments or quantify over predicates or functions, then its order is 2.
- If its order is n, its order is (n+1) if the innermost predicates quantified over or in as the arguments of a predicate themselves have predicates or functions as arguments or quantify over predicates or functions.
A formula is called well-formed if it is made out of the language \(\textrm{๐}_{\textrm{SUSHI}}\) and satisfies its syntax. The theory \(\textrm{SUSHI}\), which is an extension of ZFC, consists of the axioms:
- \(\forall A_0 \forall A_1(\forall A_2(A_2 \in A_0 \leftrightarrow A_2 \in A_1) \rightarrow A_0 = A_1)\)
- \(\forall A_0(A_0 \neq \varnothing \rightarrow \exists a_0(a_0 \in A_0 \land \{a_1: a_1 < a_0\} \cap A_0 = \varnothing))\)
- \(\forall a_0 \forall a_1 \exists A_0 \forall a_2(a_2 \in A_0 \leftrightarrow (a_2 = a_0 \lor a_2 = a_1))\)
- \(\forall a_0(\nexists A_0(a_0 \in A_0) \leftrightarrow \exists A_1(\forall a_4(\exists A_0(a_4 \in A_0) \rightarrow \exists a_5(a_5 \in \{a_3: a_3 < a_0\} \land (a_5, a_4) \in A_1)) \land \forall a_5 \forall a_4 \forall a_6((a_5, a_4) \in A_1 \land (a_5, a_6) \in A_1) \rightarrow a_4 = a_6)))\)
- \(\forall A_0 \exists A_1 \forall a_0(a_0 \in A_1 \leftrightarrow \forall a_2(a_2 \in \{a_1: a_1 < a_0\} \rightarrow a_2 \in A_0))\)
- \(\exists A_0(\varnothing \in A_0 \land \forall a_0 \in A_0(S(a_0) \in A_0))\)
- \(\forall a_2, ..., a_{a_1} \forall A_0 exists A_1 \forall a_0(a_0 \in A_1 \leftrightarrow (a_0 \in A_0 \land \phi_0(a_0, a_2, ..., a_{a_1}, A)))\)
- \(\forall A_0 \exists \prec_0(\forall a_0 \in A_0 \forall a_1 \in A_0 \forall a_2 \in A_0(a_0 \prec_0 a_0 \land (a_0 \prec_0 a_1 \land a_1 \prec_0 a_2 \rightarrow a_0 \prec_0 a_2) \land (a_0 \prec_0 a_1 \land a_1 \prec_0 a_0 \rightarrow a_0 = a_1) \land (a_0 \prec_0 a_1 \lor a_1 \prec_0 a_0) \land (\forall A_1(\forall a_3 \in A_1(a_3 \in A_0))(S \neq \varnothing \rightarrow \forall(a_4 \in A_0)\exists(a_5 \in A_0)\neg(a_4 \prec_0 a_5))))))\)
- \(\forall A_0(\varnothing \notin A_0 \rightarrow \exists f_2: A_0 \mapsto \bigcup A_0(\forall a_0 \in A_0(f_2(a_0) \in A_0)))\)
- \(\forall a_0(a_0 = a_0)\)
- \(\forall a_0 \forall a_1(a_0 = a_1 \rightarrow a_1 = a_0)\)
- \(\forall a_0 \forall a_1 \forall a_2(a_0 = a_1 \land a_1 = a_2 \rightarrow a_0 = a_2)\)
- \(\forall a_0 \in A_0(a_0 = a_1 \rightarrow a_1 \in A_0)\)
- \(\forall a_0 \forall a_1(a_0 = a_1 \leftrightarrow S(a_0) = S(a_1))\)
- \(\forall a_0(0 < a_0 \rightarrow S(a_0) \neq 0)\)
- \(\forall \varphi((\varphi(0)) \land (\forall a_0(\varphi(a_0) \rightarrow \varphi(a_0+1))) \rightarrow \forall a_0(\varphi(a_0)))\)
- \(\exists A_0 \forall a_0(a_0 \in A_0 \leftrightarrow \varphi(a_0))\)
Coding[]
I define a Gรถdel numbering for a formula \(\varphi\) in the language\(\textrm{๐}_{\textrm{SUSHI}}\) like so:
- \(v(0) = 1\)
- \(v(S) = 3\)
- \(v(\neg) = 5\)
- \(v(\lor) = 7\)
- \(v(\forall) = 9\)
- \(v(() = 11\)
- \(v()) = 13\)
- \(v(a_0) = 17\)
- \(v(a_{n+1}) = v(a_n) + 2 \cdot n\)
- \(v(1) = 21\)
- \(v(b_0) = 25\)
- \(v(b_{n+1}) = v(b_n) + 2 \cdot (n+2)\)
- \(v(A_0) = 27\)
- \(v(A_{n+1}) = v(A_n) + 2 \cdot (n+4)\)
- \(v(\varnothing) = 35\)
- \(v(B_0) = 41\)
- \(v(B_{n+1}) = v(B_n) + 2 \cdot (n+6)\)
- \(v(\cap) = 45\)
- \(v(\cup) = 53\)
- \(v(+) = 59\)
- \(v(\cdot) = 63\)
- \(v(f_0) = 65\)
- \(v(f_{n+1}) = v(f_n) + 2 \cdot (n+8)\)
- \(v(<) = 57\)
- \(v(=) = 65\)
- \(v(\in) = 71\)
- \(v(\prec_0) = 75\)
- \(v(\prec_{n+1}) = v(\prec_n) + 2 \cdot (n+10)\)
- \(v(\phi_0) = 72\)
- \(v(\phi_{n+1}) = v(\phi_n) + 2 \cdot (n+12)\)
- \(v(\land) = 89\)
- \(v(\rightarrow) = 97\)
- \(v(\leftrightarrow) = 103\)
- \(v(\exists) = 107\)
- \(v(L_0) = 109\)
- \(v(L_{n+1}) = v(L_n) + 2 \cdot (n+14)\)
- \(v(M) = 108\)
- \(v(P_0) = 116\)
- \(v(P_{n+1}) = v(P_n) + 2 \cdot (n+16)\)
- \(v(\{) = 129\)
- \(v(\})) = 137\)
- If \(\varphi = x_1 x_2 ...\) for symbols \(x_1, x_2, ...\), \(G(\varphi) = S(v(x_1)) \cdot 2^{n-1} + S(v(x_2)) \cdot 2^{n-2} + ... + S(v(x_{n-1})) \cdot 2 + S(v(x_n))\)
Once you have found what \(v(x)\) should be for some \(x\), before saying "Ok, I have found the result", check if the given value coincides with \(v(y)\) for some \(y \neq x\). If it does, then repeatedly apply add one to \(v(x)\) until you reach an unoccupied spot and that unoccupied is the actual value, to ensure all values are unique. Same applies if \(G(\varphi)\) coincides with \(G(\phi)\) for \(\phi \neg \varphi\).
I define small sushi number like so:
I write the length of the formula as \(L(\phi)\), I abbreviate \(\phi\) is a well-formed formula is in the language of \(\textrm{๐}_{\textrm{SUSHI}}\) to \(\phi \in \textrm{๐}_{\textrm{SUSHI}}\) and the maximum variable index occuring in the formula \(\phi\) as \(V(\phi)\). Then small sushi number is ๐ฃ such that \(\textrm{๐ฃ} = \textrm{sup}\{G(\phi): \phi \in \textrm{๐}_{\textrm{SUSHI}} \land L(\phi) \leq 10^{100} \land V(\phi) \leq 10^{100} \land \}\). This is uncomputable, but is likely upper-bounded by some computable number.
Then, I define the function \(\textrm{๐ก}(n)\) as the smallest finite number larger than any number that can be named by an expression in the language of sushi-nth-order set-theory with less than n symbols. I define second sushi number aka fugu fish number as \(\textrm{๐ก}^{10}(10^{100})\). This would be pretty strong already, but because I'm stupid, I decide to diagonalise over it because, you know, it will make it bigger I guess.
I define the language \(\textrm{๐}_{\textrm{SUSHI}}^+\) as the language of \(\textrm{๐}_{\textrm{SUSHI}}\) with the function symbol \(\textrm{๐ก}(n)\). Therefore, I define \(\textrm{๐ก}^+(n)\) as \(\textrm{๐ก}(n)\) but with \(\textrm{๐}_{\textrm{SUSHI}}\) replaced with \(\textrm{๐}_{\textrm{SUSHI}}^+\). This is much stronger, as now \(\textrm{๐ก}(n)\) itself requires only \(1\) symbol instead of \(>> n\). But, we still don't stop there!
Then, \(\textrm{๐}_{\textrm{SUSHI}}^{\underbrace{++...+++}_{n+1}}\) as the language of \(\textrm{๐}_{\textrm{SUSHI}}^{\underbrace{++...+++}_n}\) with the function symbol \(\textrm{๐ก}{\underbrace{++...+++}_n}(n)\). Then, I define \(\textrm{๐ก}^{\underbrace{++...+++}_m}(n)\) as \(\textrm{๐ก}(n)\) but with \(\textrm{๐}_{\textrm{SUSHI}}\) replaced with \(\textrm{๐}_{\textrm{SUSHI}}^{\underbrace{++...+++}_m}\).
Then, I define \(\textrm{ๆ้ซ๐ก}\) (the supreme fugu fish function :) as \(\textrm{๐ก}^{\underbrace{++...+++}_{\textrm{๐ก}^{\underbrace{++...+++}_{\textrm{๐ก}^{\underbrace{++...+++}_{..._{\textrm{๐ก}^{\underbrace{++...+++}_{n}(n)}}(n)}(n)}(n)\), nested \(n\) layers deep.
I define the third sushi number aka supreme fugu fish number as \(\textrm{ๆ้ซ๐ก}^{10^{100}}(10 \uparrow^{10} 100)\). Still not gonna stop there though! I define \(\textrm{๐ก๐ก}(n) = \textrm{ๆ้ซ๐ก}^{\textrm{ๆ้ซ๐ก}^{...^{\textrm{ๆ้ซ๐ก}^{n}(n)}}(n)}(n)\) with \(n\) layers. Using the same method as earlier, I define \(\textrm{๐ก๐ก}^+(n)\), ... and \(\textrm{ๆ้ซ๐ก๐ก}\). Then I define \(\textrm{๐ก๐ก๐ก}(n) = \textrm{ๆ้ซ๐ก๐ก}^{\textrm{ๆ้ซ๐ก๐ก}^{...^{\textrm{ๆ้ซ๐ก๐ก}^{n}(n)}}(n)}(n)\) with \(n\) layers. I define \(\textrm{็ฉถๆฅตใฎใๆใใชใผใใผใถใใใใฎใๅฏไธ็กไบใฎ๐ก}(n)\) as \(\underbrace{\textrm{๐ก๐ก๐ก...๐ก}}_n\). I coin the fourth sushi number aka ultimate, most over-the-top, one and only fugu fish number as \(\textrm{็ฉถๆฅตใฎใๆใใชใผใใผใถใใใใฎใๅฏไธ็กไบใฎ๐ก}^{10 \uparrow^{100} 100}(10 \uparrow^{100} 100)\).
I guess I could keep going, but I'm fine with stopping now. Hopefully the fourth sushi number is the largest valid googologism :)