Beyond
Gödel
Simply consistent constructive
systems of first order Peano’s Arithmetic that do not yield undecidable
propositions by Gödel’s
reasoning
(A compiled copy of Chapters 1 to 6 can be downloaded as a .pdf file from http://arxiv.org/abs/math/0201059.)
Contents
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
Gödelian 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.
Gödel’s Beta function
2.4. Can F(x, y, z) create a logical paradox in PA?
2.5. The ‘Liar’ paradox
2.6. The ‘Russell’ paradox
2.7. Creation through definition
2.8. When can PA+F(x1, x2, x3) be
assumed consistent?
2.9.
What does PA prove constructively by (E1x3)F(k, m, x3)?
2.10. Mendelson’s proof of “PA 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. Gödel’s 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
4.2. omega-PA
4.3. Some features of omega-PA
4.4. The difference between PA and omega-PA
4.5. omega1-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 Gödel’s
conclusions from his reasoning in omega2-PA
4.11. Does Gödel’s reasoning hold even when his conclusions do not?
Chapter 5. Gödel’s 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 Gödel-number
5.3. Gödel’s recursive definition of provability
5.4. Gödel’s Turing-computable q(x, y)
5.5. Gödel’s constructive self-reference lemma
5.6. Hilbert and Bernay’s representation lemma
5.7. The Gödelian premise
5.8. G$ is not provable
5.9. ~G$ is not
provable
5.10. The essence of Gödel’s
reasoning
5.11. Gödel’s undecidable proposition GUS
5.12. Gödel’s reasoning in PA
5.13. Gödel’s 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
5.17. Conclusion
Chapter 6. Conclusions
6.1. The non-constructive nature of Gödel’s
first order Arithmetic
6.2. Simply constructive first order
systems
6.3. Strongly constructive first order
systems
6.4. The roots of Gödel’s Platonism
6.5. The conflict with model theory
6.6. Consistency
6.7. Gödel’s undecidable proposition is an ill-defined
sentence
◄ Index ◄ Abstracts ▲ Contents Chapter 1 ► Chapter 2 ►
Chapter 3 ► Chapter
4 ► Chapter 5 ► Conclusions
► Web_links ►
(Updated : Friday 11th January 2002 2:23:54 AM by re@alixcomsi.com)
