Constructivity06

◄ Chapter 5 ◄ Index Web_links ►

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.)

Chapter 6. Conclusions

Contents

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 within model theory

6.6. Consistency

6.7. Gödel’s undecidable proposition is an ill-defined sentence

6.1. The non-constructive nature of Gödel’s first order Arithmetic

We have argued in §1 and §2 that the non-constructive nature of Gödel’s system of standard first order Arithmetic PA admits omega-provable well-formed formulas that may translate into propositions that refer to ill-defined totalities (interpreted Platonistically) such as those involved in the logical antinomies.

6.2. Simply constructive first order systems

In §3, we defined various systems of first order Arithmetic in which well-formed formulas that may correspond to ill-defined predicates under interpretation cannot be strongly represented if the systems are simply consistent. We argued, accordingly, that Gödel’s reasoning does not hold in most simply consistent systems of Arithmetic.

We defined a simply constructive system, omega-PA, of first order Peano’s Arithmetic PA, and argued that every model of PA is a model of a simply constructive omega-PA. We argued further that omega-PA, unlike PA, admits Cantor’s ordinal-Arithmetic CA as a model.

6.3. Strongly constructive first order systems

In §4, we extended the scope of the above argument by defining stronger constructive systems, omega₁-PA and omega₂-PA, of first order Peano’s Arithmetic PA.

We argued that, even here, well-formed formulas that may correspond to ill-defined predicates under interpretation cannot be strongly represented if the system is simply consistent, and so again Gödel’s reasoning does not hold.

However, we have also argued that the axioms, interpretations and models of PA and the axioms, interpretations and models of a strongly constructive omega₂-PA are identical.

6.4. The roots of Gödel’s Platonism

We thus suggest that Gödel’s Platonism may not have been the result of a faith-only belief in an abstract world of absolute mathematical ideals.

If the arguments of this paper are substantive, then Gödel’s Platonism may have been an intuitive reflection of the non-constructive, and implicitly Platonistic, nature of the first order predicate calculus chosen by him for defining his formal system of Arithmetic PA.

6.5. The conflict within model theory

We have also highlighted the conflict with classical model theory, and argued that the consequences within a model are not Platonistically determined absolutely by the interpretation of only the axioms of a formal system in the model, but are consequences of the rules of inference that we select for the system.

6.6. Consistency

We conclude with the tentative remarks that, from the above, it can be reasonably argued that consistency may not be an inherent feature of an axiomatic system.

It may, instead, be viewed as a feature of specification that we design into the definition of a system through our choice of appropriate rules of inference in order to specify those elements of the system, and its interpretations, that reflect what we intend the system to communicate faithfully.

6.7. Gödel’s undecidable proposition is an ill-defined sentence

We argue in §5 that Gödel’s undecidable formula GUS, if assumed well-defined under interpretation, translates as a true sentence in every interpretation of standard PA. It follows that there can be no consistent, non-standard, interpretation in which GUS is false.

Since, by virtue of Gödel’s completeness theorem for a first order predicate calculus, the above argument contradicts the essential unprovability of GUS in standard PA, we argue further that, if standard PA is assumed simply consistent, then Gödel’s undecidable formula GUS translates as an ill-defined sentence that mirrors the Liar sentence in every interpretation of standard PA.

◄ Index ◄ Contents ◄ Chapter 1 ◄ Chapter 2

◄ Chapter 3 ◄ Chapter 4 ◄ Chapter 5 ▲ Conclusions Web_links ►

(Updated : Friday 11^th January 2002 2:39:00 AM by re@alixcomsi.com)

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