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Beyond Gödel

Simply consistent constructive systems of first order Peano’s Arithmetic that do not yield undecidable propositions by Gödel’s reasoning : Abstracts

B. S. Anand[1]

(A compiled copy of Chapters 1 to 6 can be downloaded as a .pdf file from http://arxiv.org/abs/math/0201059.)

In these papers, we consider significant alternative systems of first order predicate calculus. We consider systems where the meta-assertion “PA proves: F(x)” translates under interpretation as “F(x) is satisfied for all values of x, in the domain of the interpretation, that can be formally represented in PA”, whilst “PA proves: (Ax)F(x)” interprets as “F(x) is satisfied by all values of x in the domain of the interpretation”.

We are thus able to argue that formal systems of first order Arithmetic that admit Gödelian undecidable propositions validly are abnormally non-constructive.

We argue that, in such systems, the strong representation of primitive recursive predicates admits abnormally non-constructive, Platonistic, elements into the formal system that are not reflected in the predicates which they are intended to formalise.

We argue that the source of such abnormal Platonistic elements in these systems is the non-constructive Generalisation rule of inference of first order logic.

We argue that, in most simply consistent systems that faithfully formalise intuitive Arithmetic, we cannot infer from Gödel’s reasoning the Platonistic existence of abnormally non-constructive propositions that are formally undecidable, but true under interpretation.

We define a constructive formal system of Peano’s Arithmetic, omega2-PA, whose axioms are identical to the axioms of standard Peano’s Arithmetic PA, but lead to significantly different logical consequences.

We thus argue that the formal undecidability of true Arithmetical propositions is a characteristic not of relations that are Platonistically inherent in any Arithmetic of the natural numbers, but of the particular formalisation chosen to represent them.

We finally argue that Gödel’s reasoning essentially formalises the Liar sentence in standard PA by means of a “palimpsest”.

Chapter 1.  First order Arithmetic is not omega-constructive

We take as our primary reference Mendelson’s first order exposition of the essentially second order formal system and various revolutionary concepts introduced by Gödel in his original paper “On formally undecidable propositions of Principia Mathematica and related systems I”.

We also borrow some content, and style of presentation, from Karlis Podnieks’ proof of Goedel’s incompleteness theorem in his e-textbook “Around Goedel’s Theorem”.[2]

We take as our starting point the systems of first order predicate calculus and first order Arithmetic defined by Mendelson (p57 & p102).

We introduce the concept of omega-constructivity, and show that standard first order Arithmetic PA is not omega-constructive.

Chapter 2.  Why is first order Arithmetic not omega-constructive?

In this second section we locate specific non-constructivity in PA and trace it to the particular closed Induction axiom schema of PA, and the strong Generalisation rule of inference of first order predicate calculus. 

Chapter 3.  Is there an omega-constructive first order Arithmetic?

In this third section we formally construct and briefly examine various systems of first order Arithmetic. We argue that we cannot use Gödel’s reasoning to prove that every primitive recursive function (Mendelson p120) can be strongly represented formally in all such systems. Hence it does not universally establish the existence of undecidable propositions that are true under interpretation.

We finally construct an omega-constructive system of first order Arithmetic, omega-PA, where we replace the non-constructive closed Induction axiom schema by a constructive, open Induction axiom schema.

We base omega-PA on a modified first order predicate calculus where we replace the strong Generalisation rule of inference by a weaker omega-Constructivity rule of inference.

We show that every model of standard PA is a model of omega-PA. We show further that every primitive recursive function cannot be strongly represented formally in omega-PA if it is simply consistent. So, in omega-PA, Gödel’s reasoning does not establish the existence of undecidable propositions that are true under interpretation.

We argue in this, and the next, section that Gödel mistakenly concluded (Gödel p190-191) that the existence of undecidable propositions that are true under interpretation must follow from his reasoning in all the above formal systems of Arithmetic.

We begin by reviewing in detail Mendelson’s definition (p103) of the system PA. This is essentially the formal system in which Gödel constructed his undecidable propositions.

Chapter 4.  Are there stronger omega-constructive systems of first order Arithmetic?

In this fourth section we begin by briefly reviewing the main arguments of the earlier sections.

We then consider whether, and how, we may strengthen omega-PA.

We use Parikh’s form of the Kreisel-Parson’s conjecture to constructively qualify quantification. This now allows a constructive formal system to refer to selective properties of non-constructive elements under interpretation

We finally construct a strong omega2-PA, and show that the axioms, interpretations and models of strong omega2-PA, and the axioms, interpretations and models of PA, are identical, but have significantly different consequences.

We conclude that the formal undecidability of Arithmetical propositions that are true under interpretation is a characteristic not of any relations that are Platonistically inherent in any Arithmetic of the natural numbers, but of the particular formalisation chosen to represent the Arithmetic.

Chapter 5.  Gödel’s argument for undecidable propositions

We have argued in these papers that though Gödel’s reasoning for the undecidability of true propositions is valid in all the above systems of Arithmetic, the inferences that can be validly drawn from it depend on the choice of the particular system.

We now highlight the conditional nature of Gödel’s conclusions in a general Arithmetic GA.

We also argue that Gödel’s undecidable propositions have the form of palimpsests that translate as ill-defined sentences in every interpretation.

Chapter 6.  Conclusions

◄ Index           ▲ Abstracts           Contents ►           Chapter 1 ►         Chapter 2 ►

Chapter 3 ►         Chapter  4 ►           Chapter 5 ►           Conclusions ►           Web_links ►

(Updated : Friday 11th January 2002 2:18:54 AM by re@alixcomsi.com)


[1] The author is an independent scholar.

 

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(Updated : Friday 11th January 2002 2:19:41 AM by re@alixcomsi.com)