Why We Need to Revise Current Interpretations of Cantor's, Gödel's, Turing's and Tarski's Formal Reasoning

I show—contrary to common beliefs tolerated by the " bosses " —that any interpretation of ZF that admits Aristotle's particularisation is not sound; that the standard interpretation of PA is not sound; that PA is consistent but ω-inconsistent; that a sound finitary interpretation of PA is definable in terms of Turing-computability; and that PA cannot be consistently extended to ZF. 1 Preamble: The semantic and logical paradoxes We are all familiar with the semantic and logical paradoxes 2 which involve— either implicitly or explicitly—quantification over an infinitude. Where such quantification is not, or cannot be, explicitly defined in formal logical terms—eg. the classical expression of the Liar paradox as 'This sentence is a lie'—the paradoxes per se cannot be considered as posing serious linguistic or philosophical concerns 3 , except to illustrate the absurd extent to which 2 Commonly referred to as the paradoxes of 'self-reference', even though not all of them involve self-reference, e.g., the paradox constructed by Yablo [Ya93]. 3 It would be a matter of serious concern if the word 'This' in the English language sentence 'This sentence is a lie' could be validly viewed as implicitly implying that: (i) there is a constructive infinite enumeration of English language sentences; (ii) to each of which a truth-value can be constructively assigned by the rules of a two-valued logic; and, (iii) in which 'This' refers uniquely to a particular sentence in the enumeration. In [Go31] Gödel used the above perspective: (a) to show how the infinitude of formulas in a formally defined Peano Arithmetic P ([Go31], pp.9-13) could be constructively enumerated and referenced uniquely by natural numbers ([Go31], p.13-14); (b) to show how PA-provability values could be constructively assigned to PA-formulas by the rules of a two-valued logic ([Go31], p.13); and, (c) to construct a PA-formula that interprets as an arithmetical proposition that could be viewed as expressing the sentence 'This P-sentence is P-unprovable' (cf. [Go31], p.37, footnote 67) without inviting a 'Liar' type of contradiction.