“Our reach should exceed our grasp,or what’s a Heaven for?” — Tennyson

Gödel showed that not every truth is provable in formal mathematics, and not every falsehood can be disproved. This is well illustrated in the diagram in Hofstadter’s “Gödel, Escher, Bach”, p. 71.

Some statements are undecidable as to their truth, e.g. “All Cretans are liars” said by a Cretan, or “This sentence is false” (self-reference). Fuzzy logic would call them half-true and half-false, i.e. truth value = 0.5 if true = 1 and false = 0. Other fractions are possible, like the dimensionality of a fractal.

“Undecidable” also refers to problems like the Travelling Salesman (finding the shortest route among a number of cities); it would take even the fastest computer so long to solve as to exceed the age of the universe. (Actually Turing pondered whether there are algoriths that never stop at all.) Yet in practice we use reasonable approximations, just as Arrow’s theorem of the impossibility of combining individual utilities into collective utility in a democratic way does not prevent us from practicing a reasonable approximation to (imperfect) democracy. We use something like fuzzy logic to operate in the real world.

As in the diagram mentioned above, we penetrate into truth only partially like a fractal tree, proceeding from axioms to theorems and lemmas. This fractal tree never covers the whole area of truth, just as a Koch snowflake never attains the dimensionality of 2, i.e. never covers the whole area of its enclosing square.

However, we can often guess at or approach or intuit unprovable truths in practice, albeit not with deductive certainty. Computers can only do algoriths, but human minds can reach beyond (Roger Penrose). Some non-computable truths are grasped by intuition, which is a higher human mental faculty, beyond both deduction and induction

So could there be “intuitional axioms”? (Axioms do not need to be proved.) I would suggest one: the moral principle of “shared essence”, which induces us to believe in the moral obligation not to harm other human beings, and by extension a reverence for all life (though compromises must be made in practice).

Probably knowledge in the sense of certainty has definite limits for human beings. Approximation and fuzzy logic can take us a step further, and perhaps intuition (but what is that exactly?) another step. Yet there is still a sense of Gödelian incompleteness. Karl Popper’s “World One” (Kant’s “things in themselves”) will always remain inaccessible. “World Two” (subjective sense experience or phenomenology), is knowable, as is (with some limitations like Heisenberg’s uncertainty principle) “World Three”, the objective scientific interpretation of the world we perceive with our senses and then reason about, with some inter-subjective agreement among ourselves).

To say that “God is Truth” (or “Truth is God” as Gandhi would say), implies that God is transcendent (unprovable, in fact inconceivable) as well as immanent (provable, visible in His creation). He (and Truth) is simply greater by far than we can conceive by reason. If intuition might be a candidate we cannot yet say.