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Is There a “Loophole” in Gödel's Interpretation of His Formal Reasoning and its Consequences?

We formally define a “mathematical object” and “set”. We then argue that expressions such as “(A)()”, and “(E)()”, in an interpretation M of a formal theory P, may be taken to mean “() is true for all in M”, and “() is true for some in M”, respectively, if, and only if, the predicate letter “” is a mathematical object in P.In the absence of such a meta-proof, the expressions “(A)()”, and “(E)()”, can only be taken to mean that “() is true for any given in M”, and “It is not true that () is false for any given in M”, respectively, indicating that the predicate “()” is well-defined, and effectively decidable individually, for any given value of , but that there may be no uniform effective method (algorithm) for such decidability.We show how some paradoxical concepts of Quantum mechanics can then be expressed in a constructive interpretation of standard Peano's Arithmetic.