Can we trust logic – revisited
A few days ago (well, more than a few days ago, but some technical problems got in the way since I wrote this post) I wrote about how logic cannot be conclusively shown to be correct. I mentioned that the argument is explained better in the book GÃƒÂ¶del, Escher, Bach. I knew it was actually an application of GÃƒÂ¶delÃ¢â‚¬â„¢s theorem.
That night I went to bed, and picked up my book, Ã¢â‚¬Å“The music of the primesÃ¢â‚¬Â. I turned to the next chapter, and what do I discover, an analysis of GÃƒÂ¶delÃ¢â‚¬â„¢s theorem! So I can now explain with a bit more accuracy the history.
I had thought that the theorem was originally presented in terms of sets, but in fact it was originally presented in terms of mathematical axioms. Maths, like any logical system is defined by axioms and rules. The axioms set out a list of facts, things that are “true” by definition. The rules allow you to modify the axioms to create theorems, statements that can be derived from axioms. For a long time, one of the biggest questions for mathematicians was Ã¢â‚¬Å“are our axioms consistentÃ¢â‚¬Â. That is to say, is it possible to construct two different theorems from the axioms that contradict each other? This is another way of asking if mathematics works. If maths allows us to produce two theorems that contradict each other we cannot trust either of them. We would not be able to allow ourselves to trust any result in mathematics. Unless we simply accept that the axiomÃ¢â‚¬â„¢s are consistent.
GÃƒÂ¶delÃ¢â‚¬â„¢s theorem, in its original form, shows that no set of axioms and rules, can be used on their own, to prove that those axioms are consistent. Mathematics alone cannot prove that mathematics is consistent, that it is valid, and that it works. A greater system is needed, one that encompasses the axioms of mathematics, maybe it produces them as theorems, or has them as axioms itself. Logic is this system. But how to we to be sure the axioms of logic are consistentÃ¢â‚¬Â¦