Linking Godel's Theorem with Russell's Paradox Is a Display of Ignorance

Recently, several people have been relentless, always trying to use Godel's incompleteness theorem to catch this young lady in a mistake. Unfortunately, these people don't understand it themselves and have dug up various random sources -- some so-called Indian expert, some mediocre popular science writer. This afternoon, this young lady has nothing to do, so let me give these gentlemen a lesson in the most accessible language possible. But this young lady has never been a teacher, much less a kindergarten instructor, so please forgive any impatience.

However, this young lady absolutely cannot muster the patience to start from the infant-level concept of sets, so set-related issues will be skipped. The so-called Russell's paradox concerns the problems that arise when discussing "the set of all sets." Before Russell's paradox was discovered, people talked about sets freely and loosely. Russell's paradox showed that such loose talk leads to contradictions. Russell's paradox can be avoided through the axiomatization of set theory. The main starting point is to disallow discussion of things like "the set of all sets" involved in Russell's paradox. Under axiomatized set theory, sets are no longer things that need no definition, but rather things prescribed by a large system of axioms. Under this system of axioms, Russell's paradox no longer exists. So Russell's paradox is really not such a big deal.

Godel's incompleteness theorem, however, has a completely different background from Russell's paradox. Nowadays, people generally only know about Godel's incompleteness theorem, but there is also a Godel completeness theorem that probably fewer people know about. This theorem primarily proves the completeness and consistency of the first-order predicate calculus system. Loosely speaking, completeness means being both complete and consistent. Complete means that every proposition in the system is decidable -- meaning you can determine whether it is true or false. Consistent means the system cannot derive any two contradictory propositions, which can generally be reduced to the inability to derive a conclusion like 1=0. Godel's completeness theorem proved that the first-order predicate calculus is just such a complete and consistent system.

Due to the emergence of non-Euclidean geometry, proving the completeness and consistency of mathematical systems became extremely important. Through the efforts of a great many people, these issues could all be reduced to the completeness and consistency of the arithmetic system. The so-called arithmetic system, for the average person, can be roughly understood as just integers and the addition operation (multiplication and subtraction can be defined through addition; for the ring of integers, division need not be considered).

In other words, if one could prove the completeness and consistency of the arithmetic system, all of mathematics -- including all of science (since physics can actually be axiomatized and is essentially a branch of mathematics) -- would have a rigorous foundation. Since the first-order predicate calculus is already a complete and consistent system, and the arithmetic system is the simplest non-formal system, the whole world was watching this final step. Unfortunately, success breeds failure -- it was that damned Godel again who proved that any non-formal system containing the arithmetic system is incomplete, meaning completeness and consistency cannot both hold simultaneously. What makes this fellow detestable is that he first ignited the hope of the entire world, then destroyed that hope for all to see. Although this made even Einstein treat him with utmost deference, he apparently starved to death in the end -- karma's net is wide indeed.

Because of this, the idea of reducing mathematics to formal systems -- that is, to logic -- was thoroughly shattered. Instead, everyone came to understand that logic is just one part of the entire mathematical game. There are infinitely many kinds of logic, each differing only by personal preference. For example, some mathematicians refuse to accept real numbers and the law of excluded middle, and this does not mean they are more peculiar than those who do -- it's simply a matter of taste. Of course, most people accept real numbers and the law of excluded middle, because with them the game is much more fun. That's all there is to it.

More importantly, God does not exist -- not even within a local non-formal system. In any non-formal system, there are either undecidable propositions or contradictory propositions. This has nothing to do with ability. Some half-informed people say things like "mathematics is no longer rigorous" or "such-and-such has been lost," when in reality they have no idea what is going on. Mathematics is still rigorous -- it's just that God does not exist, omniscience and omnipotence do not exist. That's all.