The Logic of Logic

Is there anything in this world that’s completely logical?

How about mathematics?  This is one area in which every problem has a precise, finite solution.

Or so many people thought.  The seminal work Principia Mathematica by Alfred North Whitehead and Bertrand Russell  sought to describe a set of axioms and rules which, in principal at least, proved all mathematical truths.  Also, German mathematician David Hilbert proposed to establish all existing theories to a finite and complete set of axioms and prove they were consistent.  In other words, he wanted to show that all mathematics could be reduced to consistent, basic arithmetic.

But there was a problem.  There are certain logical problems that do not have simple solutions.  For example, a library decides to compile a list all of its books and bind them into a book.  Should that “inventory book” be included in the list as one of the library’s books?  Or assume a person has never made a mistake.  If that person thinks he has made a mistake, but is wrong, does that count as a mistake?

One person who asked questions like this was Kurt Godel of Austria. He showed that any consistent system could never be complete.  His arguments became known as Godel’s Incompleteness Theorms and were published in 1931.

Obviously, this type of question is not the kind an average person would even have time to think about.    But if you are really smart and can become a logician, mathematician and philosopher (like Godel), you can solve these kinds of problems.




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