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The principle of non-contradiction say that p and $\neg$ p can not be both true.

  1. If you consider the law of excluded middle as relevant, then either p or $\neg$ p is true, and thus the other is false by the principle of non-contradiction.
  2. But if you consider the excluded middle as not relevant, just like intuitionistic logic, what would avoid some p and $\neg$ p to be both false ?

Actually, if true is the same than provable, then such a proposition is just a Gödel one : p is not provable and neither $\neg$ p is.

I think this is a very good interpretation and a big demystification of Gödel theorem : it is not about some true that we can not reach, it is about some absurdity that is still an absurdity when you negate it.

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    $\begingroup$ It is not possible, if by "$p$ is false" you mean "$\lnot p$ is provable". Indeed, then "$p$ and $\lnot p$ are both false" translate to "$\lnot p$ and $\lnot \lnot p$ are both provable", which is a contradiction. But "$p$ is false" is different from "$p$ is not provable". $\endgroup$
    – Zhen Lin
    Mar 30 '21 at 14:13
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    $\begingroup$ No; the derivation of $\bot$ from $A \to \bot$ and $(A \to \bot) \to \bot$ is intuitionistically valid. $\endgroup$ Mar 30 '21 at 14:15
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    $\begingroup$ About the purported "big demystification of Gödel theorem" we have to note that G's Th is sound from an intuitionistic point of view... but it is "obvious". It is a cornerstone of Brouwer's view that Hilbert's assumption that every mathematical problem is decidable was not tenable. $\endgroup$ Mar 30 '21 at 14:17
  • $\begingroup$ I want to define a recursive logic where false is no longer define as $\vDash \neg p$ but as $\nvDash p$. $\endgroup$
    – François
    Mar 31 '21 at 9:14
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The statement "$p$ and $\neg p$ are both false" is formally expressed as $$\neg p \land \neg \neg p,$$ which is false by the law of non-contradiction, which states that for any $q$ the proposition $q \land \neg q$ is false (take $q = \neg p$).

More importantly, these consideration do not "demistyfy Gödel", as they refer to proposition with free propositional variables in them. Gödel constructed a sentence, i.e., a logical formula without free parameters, which is neither provable nor falsifiable (in a given formal system).

To put it another way, it is very easy to find a formula which is neither provable nor falsifiable, if we allow free variables, for example $$x < 42$$ where $x$ ranges over the natural numbers (say, in Peano arithmetic) is not provable, but neither is $$\neg (x < 42).$$

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