These papers attempt to correct a major error underlying earlier work. For more years than I would care to recall, I mistakenly believed that Gödel, in his original 1931 paper, had implicitly ignored the possibility that F(x) and (Ax)F(x) may not be formally equivalent. However, I had overlooked Gödel’s explicit, ‘in parenthesis’, inclusion of the Generalisation rule of inference in his definition of ‘immediate consequence’; this implies the above equivalence in his formal system.
The focus is no longer on taking issue with implicit elements in Gödel’s reasoning, but on arguing instead that the Generalisation rule of inference makes his formal system of standard Peano’s Arithmetic abnormally non-constructive. I argue further that this rule of inference can be viewed as formalising Platonism in standard PA.
Accordingly, I argue for constructive formal systems of PA that appear to formalise Dedekind's expression of Peano's axioms more faithfully. The limited aim is to indicate, broadly but not in depth, systems of PA that are not formally committed to Gödel's Platonistically influenced premises.
My grateful thanks to Damjan Bojadziev, Department of Intelligent Systems, Jozef Stefan Institute, Ljubljana, for patiently indicating how the content and presentation of my papers can yet be improved for increased clarity and readability. My particular thanks to him for bringing to light the point that my assertion about Gödel’s undecidable proposition being true in every interpretation needs to be justified. This is now attempted in Chapter 5 #14-17.
Finally my debt to Dr. Chetan H. Mehta continues to grow for his encouragement, as also to friends, associates and family members who have willingly sacrificed conflicting interests so that this work might have a chance to reach some conclusion.
Bhupinder Singh Anand
Mumbai : November 2001
(Updated : Tuesday 13th November 2001 12:14:15 PM by firstname.lastname@example.org)