Proving Incompleteness Theorems Formally

Prologue. We are told that we should not trust our natural language arguments when it comes to mathematics, and we should follow the formal derivation rules of predicate calculus, if we want to prove any thing rigorously. Yet, when it comes to the most serious matter of incompleteness of Elementary Arithmetic (EA) of Goedel, what we do is to talk in plain English and conclude for certain that the theory cannot be complete.

Adding three derivation rules. The issue we want to consider is whether it is possible to rectify this flaw and make all our logical arguments totally formal. A little investigation suggests that this can be done by adding three more derivation rules to Predicate Calculus. For details of these derivation rules, click here. Of course, adding derivation rules in a theory is a risky enterprise, and to make it worse, Goedel tells us that there is no logical way to prove the consistency of a significant axiomatic theory. The time tested method is to wait and see whether a given theory eventually turns out to be inconsistent. Recall, Bertrand Russell's devastating letter to Gottlob Frege about set theory.

Introversion and profundity. If what we are suggesting is indeed a solution, every formula in a theory would fall in one of four categories: theorem, falsehood, introversion, profundity. To give a rough idea of the classification, an example of an introversion is the Consistency of EA, and an example of a profundity is the Continuum Hypothesis of Zermelo-Fraenkel theory. Introversions arise when we attempt to use the theory to investigate itself, and profundities are the profound concepts which we are not sure to choose or not to choose as axioms.

Elevating Goedel's Incompleteness Theorem. It is worthwhile to note that what Goedel has shown is that there are introversions in EA. A more significant result would be to show that there will always be profundities in any significant theory. Whether this is what Goedel really wanted to show is a moot question. Shankara, a philosopher of eighth century, in what he calls "maya" and what I call Shankara's Incomprehensibility Thesis, claims that humanity will have profound questions facing them perennially and our understanding of nature will always be imperfect.

Inescapable Question. By allowing four categories of statements, we are in essence giving freedom of expression to a theory. Throughout civilization humans have indulged in introversion, mostly with disastrous results. Wise men have always been preoccupied with profundities and have enjoyed the consequent agony and ecstasy. A conspicuous example is provided by Mahatma Gandhi's autobiography, The Story of my Experiments with Truth. What if, we decide to whole-heartedly accept and live contented with Goedel's incompleteness theorem and Shankara's incomprehensibility thesis. The fact is that we will still have no peace of mind when the following question about religion, in the best sense of the word, crops up. It is as though God wants us to be permanently on tenterhooks when it comes to important issues.
Humanity's Inescapable Question: Is religion, an introversion or a profundity?

Epilogue. Those who may consider this as a frivolous question should look into some facts of history. It is easy to see that, if an introversion is accepted as an axiom in a theory, the theory will be destroyed by contradictions. From the gruesome violence prevalent in the world today in the name of religion, it is tempting to classify religion as an introversion. But on the other hand Mahatma Gandhi, deeply religious, spent all his life experimenting with Truth, which he considered the same as Reality or God. His autobiography, makes it clear that he considers religion as a profundity.