Abstracts†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† Chapter 1

Beyond GŲdel

Simply consistent constructive systems of first order Peanoís Arithmetic that do not yield undecidable propositions by GŲdelís reasoning

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


Chapter 1.First order Arithmetic is not omega-constructive


1.0.  Introduction

1.1.Introducing omega-constructivity : omega-PA

1.2.The significance of omega-constructivity

1.3.Constructive PA should be intuitive

1.4.Differentiating between omega-constructivity and omega-completeness

1.5.The relevance of omega-constructivity

1.6.prf'(x, y) is a Turing decidable predicate

1.7.The Representation Theorem

1.8.Can PA be both simply consistent and omega-constructive?

1.9.Consequences of the omega-constructivity of omega-PA

1.10.The meta-mathematical inconsistency in omega-PA

1.11.  A simply consistent PA is neither omega-constructive, nor omega-complete

1.12.  What is the significance of the omega-incompleteness of PA?

1.13.  The GŲdelian argument

1.14.  omega-provable propositions in PA

1.15.  What kind of propositions are omega-provable?

1.16.  The Platonistic axiom of reducibility and PA

1.17.  What is the significance of: (Ax)F(x) is omega-provable?

1.18.  Why is first order Arithmetic not omega-constructive?

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


2.0.  Introduction

2.1.The Representation Theorem

2.2.What is the formal representation of f(x, y) in PA?

2.3.GŲdelís Beta function

2.4.Can F(x, y, z) create a logical paradox in PA?

2.5.The ĎLiarí paradox

2.6.The ĎRussellí paradox

2.7.Creation through definition

2.8.When can PA+F(x1, x2, x3) be assumed consistent?

2.9. What does PA proveconstructively by (E1x3)F(k, m, x3)?

2.10.Mendelsonís proof of ďPA proves: (E1x3)F(k, m, x3

2.11.What does (E1x3)F(x1, x2, x3) assert?

2.12.The introduction of Platonism into PA

2.13.How does PA prove the Platonistic assertion (E1x3)F(x1, x2, x3)?

2.14.At which point does Platonism enter PA formally?

2.15.Is PA necessarily Platonistic?

2.16.Is every model of PA necessarily infinite, but not denumerable?

2.17.Is there a constructive PA?

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


3.0.  Introduction

3.1.The primitive symbols of a general first order predicate calculus K

3.2.The logical axioms of K

3.3.The rules of inference of K

3.4.GŲdelís formal system PA

3.5.General Arithmetics: weak-GA

3.6.General Arithmetics: strong-GA

3.7.Peano Arithmetics : weak-PA

3.8.Peano Arithmetics : standard PA

3.9.Closed Induction implies open Induction in PA

3.10.Can PA admit models of transfinite ordinals?

3.11.omega-constructive first order Arithmetics

3.12.omega-constructive Arithmetic : omega-GA

3.13.The omega-constructive Peano Arithmetic : omega-PA

3.14.Every model of PA is a model of omega-PA

3.15.The essential difference between PA and omega-PA

3.16.Can omega-PA admit models of transfinite ordinals?

3.17.Can we strengthen omega-PA

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


4.0.  Introduction

4.1.An omega-constructive first order Arithmetic


4.3.Some features of omega-PA

4.4.The difference between PA and omega-PA


4.6.The conflict within model theory

4.7.Introducing the omega-Specification rule of inference

4.8.A yet stronger omega-calculus : omega2-PA

4.9.The Generalisation rule of inference holds for omega2-PA

4.10.We cannot infer GŲdelís conclusions from his reasoning in omega2-PA

4.11.Does GŲdelís reasoning hold even when his conclusions do not?

Chapter 5.GŲdelís argument for undecidable propositions


5.0.  Introduction

5.1.Every recursive function can be constructively represented in any Arithmetic

5.2.Every representation of a recursive function has a unique GŲdel-number

5.3.GŲdelís recursive definition of provability

5.4.GŲdelís Turing-computable q(x, y)

5.5.GŲdelís constructive self-reference lemma

5.6.Hilbert and Bernayís representation lemma

5.7.The GŲdelian premise

5.8.G$ is not provable

5.9.~G$ is not provable

5.10.The essence of GŲdelís reasoning

5.11.GŲdelís undecidable proposition GUS

5.12.GŲdelís reasoning in PA

5.13.GŲdelís reasoning does not establish his conclusions in most Arithmetics

5.14.The Palimpsest syndrome in standard PA : Overview of the argument

5.15.The Extended Representation Lemma

5.16.The Palimpsest syndrome in standard PA : The proof


Chapter 6.Conclusions


6.1.The non-constructive nature of GŲdelís first order Arithmetic

6.2.Simply constructive first order systems

6.3.Strongly constructive first order systems

6.4.The roots of GŲdelís Platonism

6.5.The conflict with model theory


6.7.GŲdelís undecidable proposition is an ill-defined sentence

Index†††††††††† Abstracts†††††††††† Contents†††††††††† Chapter 1†††††††† Chapter 2

Chapter 3†††††††† Chapter4†††††††††† Chapter 5†††††††††† Conclusions†††††††††† Web_links

(Updated : Friday 11th January 2002 2:23:54 AM by re@alixcomsi.com)