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I am not a mathematician and have not read Godel's Proof. My knowledge about Godel's theorem is limited to this. So my question is this (please correct me if I have misinterpreted or misunderstood the meaning):

According to Godel's theorem, every statement is either true or false. And there are statements which cannot be proven true or false with the axioms of that mathematical system. Godel's Theorem assumes that all statements can only be true or false. Now as obvious as this statement might be but does this not bound Godel's theorem to mathematical systems which have statements which are either true or false. I mean there might be a system of mathematics where statements might either be false or true or something else all together (like undeterministic statements).

Now I am not a physicist either, but (according to my understanding) a mathematical system which has statements which can either be true or false only should not be able to classify statements like 'This electron is placed at a distance 10 nano meter.' because it is undeterministic ( Heisenberg's Uncertainity Principle).

Please share your thoughts on this. Also do let me know if my understanding of this problem is correct or not?

Thanks!

Yash Jain
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    You can see Gödel’s Incompleteness Theorems for a good introduction to the topic. – Mauro ALLEGRANZA Nov 11 '19 at 12:04
  • G's Incompleteness Th produce a sentence $G$ of formal arithmetic such that neither $G$ nor its negation $\lnot G$ are provable from the axioms of formal arithmetic. "Obviously", according to current "common sense" understanding of mathematics, one of $G$ and $\lnot G$ msut be true, i.e. must express a fact about natural numbers that holds of them. – Mauro ALLEGRANZA Nov 11 '19 at 12:07
  • The key points are : the Th is relative to formalized consistent mathematical systems regarding arithmetic; the sentence $G$ is specific for a formal system: different systems "produce" different sentences. This means also that the conclusion : "sentence $G$ is "absolutely" unprovable" is incorrect. – Mauro ALLEGRANZA Nov 11 '19 at 12:09
  • A statement gets only true or false with an interpretation. We can prove a statemtent within some theory if and only if it is true with respect to every interpretation. Similarily, we can disprove a statement within some theory if and only if it is false with respect to every interpretation. If a statemtent is true in some and false in some interpretation, it can neither be proven nor disproven. We call a theory incomplete, if it contains at least one such statement. We call it inconsistent, if we can derive some statement and its negative within the theory. – Peter Dec 09 '19 at 13:09

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