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Defining Complete and consistent formal systems of mathematical logic
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peteolcott
May 9, 2019, 11:28:41 AM5/9/19
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Within the sound deductive inference model True(x) is formalized as:
(a) Axioms stipulated as expressions of language having the semantic
value of Boolean true.
(b) Theorems stipulated as the consequence of conventional formal
mathematical proofs with True(x) premises.
When the formal proofs of mathematical logic are adapted to the
sound deductive inference model all of the undecidable sentences
of other formal systems are decided to be deductively unsound within
this complete and consistent system.
Every formal system having a provability predicate and the above
definitions Axiom and Theorem is complete and consistent.
When Peano Arithmetic is defined to have a provability predicate and
the above definitions of Axiom and Theorem both the Tarski Undefinability
Theorem and Gödel's 1931 Incompleteness Theorem proofs fail.
--
Copyright 2019 Pete Olcott
All rights reserved
(a) Axioms stipulated as expressions of language having the semantic
value of Boolean true.
(b) Theorems stipulated as the consequence of conventional formal
mathematical proofs with True(x) premises.
When the formal proofs of mathematical logic are adapted to the
sound deductive inference model all of the undecidable sentences
of other formal systems are decided to be deductively unsound within
this complete and consistent system.
Every formal system having a provability predicate and the above
definitions Axiom and Theorem is complete and consistent.
When Peano Arithmetic is defined to have a provability predicate and
the above definitions of Axiom and Theorem both the Tarski Undefinability
Theorem and Gödel's 1931 Incompleteness Theorem proofs fail.
--
Copyright 2019 Pete Olcott
All rights reserved
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