An introduction to the ontological foundations of Gödel's incompleteness theorems

Abstract

In the present thesis, we carry out an ontological investigation of Goders incompleteness theorems in view of which we obtain an interpretation of Godel's incompleteness results. In our ontological investigation, we determine the basic elements whIch are used in the statement and the proof of G6del's incompleteness theorems as 'formal objects', 'recursive functions and relations', and 'expressibility' and 'representability'. Then, we clarify the grounds on which these elements rest and obtain the result that Godel's incompleteness proofs rest ultimately on the order of natural numbers. In view of Godel's first incompleteness theorem, we are led to the following result: that one cannot prove all the true propositions of arithmetic as theorems of a formal theory is restricted by the fact that no formalized theory of arithmetic can be obtained without the 'order' of natural numbers (being independently present). This result clarifies, and in a sense restricts, the meaning of the first incompleteness theorem. That is to say, the first incompleteness theorem no longer expresses an inability of a most rigorous axiomatic employment of logic to capture all truths of arithmetic, as generally accepted. The first incompleteness theorem, we conclude, expresses an inability of a certain enlployment of the 'order' of natural numbers to capture all true propositlens concerning themselves. Similarly, in view of oUf ontological investigation, Godel's second result that the consistency of a consistent formalized theory of arithmetic cannot be proved within that theory is restricted by the fact that no formalized theory of arithmetic can be obtained without the 'order' of natural numbers. Thus, we conclude that the second incompleteness theorem expresses an inability of a certain employment of the 'order' of natural numbers to prove the consistency of any consistent formalized theory of arithmetic. We also note.that our results have interesting consequences in view of . Church's thesis. We claim that Church's thesis holds because all the procedures constructed for computation rest ultimately on the order of natural numbers. It appears that this idea, if mathematically developed, would lead to a proof of Church's thesis.