SR6: Incompleteness

Recently, I read the awesome graphic novel "Logicomix''. It is based on the life story of Bertrand Russell, a famous mathematician and philosopher. God used the book to affect me in a profound way, but I will reserve my thoughts for the next blog. Here, let me summarize the story.

[Spoiler alert!]

Russell grew up in a painful environment with family tragedies and religious control, and was troubled by fears of genetic insanity. So from a young age, he turned to mathematics for stability, because it provided objective truth he could believe in, and offered rational means to make good decisions.

In 1901, early in his career, Russell discovered a paradox which challenged the logical foundations of mathematics. In his attempt to patch this gap, he wrote three volumes of "Principia Mathematica'' but was not satisfied with the solution.

The problem was this: mathematicians want all theorems to be deduced using logical rules from a small number of axioms. Such a system of rules and axioms should be complete (produces all true theorems) and consistent (not produce any false theorems). Many systems were invented to patch Russell's paradox, but nobody could prove that their system was consistent. Someone could come along and find another devastating contradiction.

In 1931, these attempts were dealt with a fatal blow when Kurt Gödel proved his Incompleteness Theorems:
1. For every consistent system, there will always be some true theorem about numbers that cannot be deduced from the system.
2. If a consistent system is powerful enough to deduce some basic facts about numbers, then it cannot prove its own consistency.

As a result, Russell's childhood dreams of finding stability in mathematics were dashed. Nonetheless, these developments spurred the invention of the computer, arguably the greatest technological breakthrough of the last century.

[To be continued...]