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I'm surprised that no one has tried to define this before. This is easy to define, so where's the huge flaw?

Definition

The language of First Order Theory Theory (FOTT) consists of statements, theories, and operators.

In FOTT, a variable is a set. A set is a collection of things, like {1,2,4,8,6,327,pi,Canada,Swiss Cheese,{NO U, THAT IT, CLEAR IT, STOP IT, @person TRIGGERED},...}.

A statement is an operator applied to variables, statements and/or theories (you'll see what theories are later). Here are the basic operators in FOTT:

  • a=b is true iff variable a is the same as variable b.
  • a∈b is true iff variable a contains variable b.
  • ¬(a) is true iff statement a is not true
  • (a)∧(b) is true iff statements a and b are true
  • ∃a(b) is true iff there doesn't exist a variable a such that statement b is true

Before we define the rest of the operators, we need to define what theories are.

A theory is a system that can create statements by using operators defined for that system. Here is how we can define theories:

S(n|T) (where n is a non-negative integer and T is a theory) represents any statement in theory T, but S(n|T) always represents the same statement. V(n|T) (where n is a non-negative integer and T is a theory) represents any variable in theory T, but V(n|T) always represents the same variable. T(n) (where n is a non-negative integer) represents any theory, but T(n) always represents the same theory. T(n|T) (where n is a non-negative integer and T is a theory) represents any theory in theory T, but T(n|T) always represents the same theory.

φ(S|T) represents statement S in theory T. This can be used to define theories, because we can define the iff operator in FOTT and then make the statement ''φ(S|T) iff [statement]''. For example, we can define a version of ∈ (the X operator) in T(0) by making the statement ''φ(V(0|T(0))X(V(1|T(0))|T(0)) iff V(0|T(0))∈V(1|T(0))'' in FOTT.

Now, here's the catch. First, T(n) > T(m) if n > m. Second, statements, operators and theories in T(n) are also statements, operators and theories in T(m) if T(n) < T(m). Third, a theory can only make statements using statements and operators in that theory. And finally, statements cannot use T(n) unless T(n) is in the second entry of S(n|T), V(n|T), or φ(S|T). Actually, you can.

Number!

The nth input variable defines what the variable n in a FOTT statement is. For example, if we have the statement 1∈2, the input variables (apple, set of all types of fruits) make the statement true, since apple is a type of fruit, but the input variables (cheese, set of all types of fruits) make the statement false, since cheese is not a type of fruit.

Let 0 = {} and x+1 = x∪{x}. Using this definition of numbers, define FOTT(x) to be the largest value possible that a statement in FOTT x characters long can force the input variable to have so that the statement is true.

Finally, define the number with a 100% totally original name, BIG FOTT, as FOTT10(10100), where FOTT1(x) = FOTT(x) and FOTTy+1(x) = FOTT(FOTTy(x)).

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