What happened with respect to Fermat's last theorem and four-color conjecture forces us to change our view on the significance of Godel's incompleteness theorems.

1. Significance of
Godel's Incompleteness Theorems
and its
Implications on Mathematical Logic
an unconventional view of Godel's discovery
Kannan Nambiar
Contention here is that in evaluating the significance of Godel's
incompleteness theorems, we have erred considerably. The
presentation explains why I hold this view.
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2. Four-color Conjecture
Recall what was once called the four-color conjecture (first proposed
in 1852), which says that four colors are enough to color any map
with two adjacent regions getting different colors. In 1976 the
conjecture was proved and now it is called the four-color theorem.
But there is an interesting fact we have to note here. When a
hundred years passed with no solution in sight for this conjecture,
some of the frustrated mathematicians began saying that the
conjecture could be an undecidable problem, invoking the
incompleteness theorems for the wrong purpose.

3. Fermat's Last Theorem
Recall Fermat's Last Theorem, actually a conjecture, first
proposed in 1637 and eventually proved in 1995. When the
conjecture remained unresolved for more than three centuries,
again there were some who claimed that the problem could be
an unsolvable one.
The story repeats with the present apprehension being about
the Riemann Hypothesis, a difficult conjecture stated in the
simplest terms possible. What do we make out of all this? Can
we get out of this persistent farcical situation? I think we can.

4. Zermelo-Fraenkel Theory
To be specific we will take Zermelo-Fraenkel (ZF) set theory as the
basis for our discussion, accepting that there are no
contradictions in it. In this axiomatic theory, the assumption was
that its grammar would generate only meaningful sentences
pertaining to set theory and the chosen set of axioms would
show every sentence in it to be either true or false. In the event a
particular sentence or its negation could not be derived, the
accepted method was to enlarge the set of axioms to make a
derivation possible.

5. Godel’s Seminal Discovery
In 1931 Godel showed that this method would be a never ending
one and that there would always be formulas in set theory for
which neither the formula nor its negation can be derived. The
method Godel adopted was to convert a paradox of ordinary
language into a strictly logical formula F and show that neither F
nor ~F can be derived within the axiomatic theory. Thus, the
incompleteness theorems of Godel showed that there will always
be underivable formulas in any worthwhile axiomatic theory. This
was indeed a significant discovery, but I want to point out that
there is even a more significant incompleteness, which can be
recognized from the discussion below.

6. Sentient Zermelo-Fraenkel Theory
We will make our terminology clear before we proceed further.
First, we call derivable formulas of ZF relative truth and propose
an extension of ZF called Sentient ZF (SZF) wherein the derivable
formulas will be called absolute truth. SZF is defined as the theory
we get when we allow meta mathematics in our arguments.
Further, we assume that law of the excluded middle with all its
consequences are not applicable in SZF. Thus, proof by
contradiction is not a legitimate method in SZF, even though it is
in ZF. It is assumed that the vocabulary of ZF and SZF are the same.

7. Categories of Formulas
Today we have a better understanding of axiomatic theories and I
have a point of view that I would like the reader to consider. I
believe that we can avoid a lot of confusion if we classify four
categories of formulas in ZF set theory: theorems, falsehoods,
introversions, and profundities (names introversion and profundity
are mine). Briefly stated, introversions are logical formulas
corresponding to the paradoxes of ordinary language, also called
Godelian sentences. Profundities are candidate axioms available
for ZF theory without creating contradictions in it.

8. Definitions of Categories
Here are the definitions of these categories: F is a theorem, if a
derivation exists for F, but not for ~F. F is a falsehood, if a derivation
exists for ~F, but not for F. F is an introversion, if a derivation exists
for ~F when F is assumed, and a derivation for F exists when ~F is
assumed. F is a profundity, if a derivation exists for neither F nor ~F,
and it is not an introversion. Note that according to our definition,
Continuum Hypothesis is a profundity and "Consistency" of ZF theory
is an introversion. These are deep results, but we will not get into the
details.

9. Genuinely Significant Undecidable Formulas
It is about the significance of the introversions or Godelian sentences
of ZF theory that I have reservations about. The existence of
introversions merely shows that there is a class of formulas which
cannot be derived within ZF theory (since contradictions will
immediately follow). As I said before, introversions in ordinary
parlance represent absurdities and paradoxes of natural language,
and by necessity, a robust ZF theory would not provide a derivation
for them. Hence, I conclude that the only genuinely significant
undecidable formulas are the profundities.

10. Profundity belongs to Meta Mathematics
If one accepts what I have said so far, it is unlikely that any person
would even remotely claim that Riemann hypothesis is an
introversion. That leaves us with the possibility of Riemann
hypothesis turning out to be a profundity and thus undecidable.
But let us note that profundity belongs to meta mathematics and it
usually requires years of cogitation by logicians on set-theoretic
concepts to come up with a profundity. It needed a Godel and a
Cohen to show that the continuum hypothesis is a profundity.

11. Profundity in Regular Mathematics
It is my understanding that meta mathematics is the study of the
structure of reality like the continuity of space, and ordinary
mathematics on the other hand, investigates the visible reality.
From this I conclude that a profundity will never occur in regular
mathematics and what eventually happened in cases like that of
four-color conjecture and Fermat's last theorem gives credence to
my point of view.

12. Meta mathematics subsumes Mathematics
Finally, here is a question I want to ask the reader: Can we accept it
as a fact that undecidable problems can never occur in regular
mathematics, as we commonly understand the term? Still better,
assuming meta mathematics subsumes mathematics, can we
define mathematics as that part of meta mathematics where
undecidable problems cannot occur? While deciding the answer,
please remember what Dante said: The hottest places in Hell are
reserved for those who in time of moral crisis preserve their
neutrality.