Gödel's Theorem by Torkel Franzén5/13/2023 ![]() The proof of Goedel's theorem is usually done in a meta-theory, but it could be formalized with theorem-provers. Goedel's first theorem states that for every consistent theory that is strong enough to satisfy the Representation theorem (The Presburger arithmetic does not satisfy it : In fact, it is known to be both complete and consistent) there are statements that can be formulated within this theory, but neither be proven nor disproven within this theory. I know I’m probably overlooking something big and obvious, but I need help to see it. So you see, I am a little confused, because it seems to me that Peano arithmetic exhausts truth about arithmetic, and it seems that its theorems should be able to be enumerated with a straightforward computer program. How is the truth about the arithmetic of natural numbers defined? Isn’t it defined in terms of axioms, like the Peano axioms? Does Gödel’s theorem state that Peano’s axioms are not a complete formulation of arithmetic? Do we have some other means of ascertaining truth except by axioms? (Maybe intuition?) Can the theorems of Peano arithmetic (I mean, the theorems that follow from the Peano axioms) be enumerated? I would think that in any axiomatic system with a countable number of axioms, the theorems could be enumerated. an algorithm) is capable of proving all truths about the arithmetic of natural numbers.” From Wikipedia, Gödel’s first incompleteness theorem states that “no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. ![]()
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