Gödel’s gap between truth and proof refers to a fundamental idea in logic and mathematics, introduced by the mathematician Kurt Gödel. The basic idea is that in any complex system of logic or mathematics, there will always be true statements that cannot be proven within that system.
ELI5 Explanation:
Imagine you have a very powerful rulebook for a game, where every possible move can be described by the rules. Now, let’s say you’re playing the game, and you find a move that you know is correct (it feels right), but the rules don’t give you a way to prove that this move is allowed in the game.
In Gödel’s terms, the game is like a formal system of mathematics, and the move is like a mathematical statement. What Gödel showed is that in any system that’s complex enough (like arithmetic or geometry), there are some true mathematical statements that can’t be proven just using the system’s own rules. It’s as if the truth of the statement exists, but there’s no way to “prove” it by following the rules of the system.
This is what we call Gödel’s incompleteness theorem. It tells us that there’s always a gap between truth (what’s true in the universe of mathematics) and proof (what we can actually prove using a set of logical rules).
The Big Idea:
- Truth means something is correct, but we can’t always show it using the system’s rules (proof).
- Proof is the logical step-by-step explanation or verification that something is true.
- Gödel showed that no matter how powerful our rules or system are, there will always be truths that are beyond our ability to prove.
So, Gödel’s gap means that in mathematics, there are true things we just can’t prove—there will always be limits to what we can know for sure.
ex Goldbach’s Conjecture is the oldest conjecture in number theory. It states that every even integer strictly greater than 4 is the sum of two primes. There have been many empirical verifications of it, up to astronomic numbers, but it has remained unproven since 1742. Goldbach’s conjecture is like a giant puzzle that looks correct for all the pieces we’ve seen, but we can’t yet prove it works for the infinite pieces we haven’t checked (no formula).
How It Ties Into Religion
For a Theist:
Theists often use Gödel’s incompleteness theorem to argue that there are truths beyond human reasoning, which suggests that there may be an ultimate source of all truth—God—whose knowledge transcends human limits. Just as Gödel showed that some truths cannot be proved within a system, theists argue that faith in God allows access to truths that reason alone cannot prove, supporting the idea of a higher, divine intelligence
Counter? - Turing Completeness Theorem
Turing’s ideas don’t directly counter Gödel’s theorem but shift the focus from unknowable truths to what is computationally achievable. An atheist could use Turing’s perspective to argue that logical and computational limits are natural features of reality, not evidence for a higher intelligence, thus reinforcing a worldview grounded in naturalism.