http://en.wikipedia.org/wiki/Formal_system
In Max Tegmark's TOE gif, he cites formal systems as being the basis for all of math, including set theory and first order logic. You see the circle down there at the bottom:
I believe that within my development of grammatical systems, which are more general than formal systems, I found a way to prove that something is true of all theorems within a formal system. Could this hint at the nature of truth if we consider theorems to be, in any sense, true?
This is still in the preliminary stages but I think that if we can prove the following about a property P then property P must be true of all theorems in a formal system:
1. P holds for all axioms in the formal system
2. If P is "closed under all inference rules"
then
P holds for all theorems.
To unpack (2) a bit, it's a little complicated. A very specific example of an inference rule, one used in first order logic, is modus ponens. Modus ponens is a binary inference rule, meaning it has two inputs. Very simply, inference rules are functions that input n inputs, with n at least one, and output a single statement, called the conclusion of the inference rule. Modus ponens for instance takes two statements of the form
(a) Statement A
(b) IF A THEN B
and modus ponens outputs
B.
Basically modus ponens is saying that if A is true and if "IF A THEN B" is true, then one can conclude "B".
So for P to be closed under all inference rules in a formal system means that if R is any n-ary inference rule and P is true of all theorems T1, .., Tn, then P is also true of R applied to the n-tuple (T1,...,Tn).
IOW, If P is true of every term in the n-tuple (T1,...,Tn) and if P is true for R(T1,...,Tn) then we say that P is closed under all inference rules.
Again, if we can show that P is true for all axioms and if P is closed under all inference rules, then P is true for all theorems (i.e., true statements) within a formal system.
I am still working on giving some interesting examples of this phenomena.
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