You will never be able to prove every mathematical truth. For me, this incompleteness theorem, discovered by Kurt Gödel, is one of the most incredible results in mathematics. It may not surprise everyone — there are all sorts of unprovable things in everyday life — but for mathematicians, this idea was a shock. After all, they can construct their own world from a few basic building blocks, the so-called axioms. Only the rules they have created apply there, and all truths are made up of these basic building blocks and the corresponding rules. If you find the right framework, experts long believed, you should therefore be able to prove every truth in some way.This is a typical explanation, and I know what they are trying to say, but it is misleading.But in 1931 Gödel demonstrated otherwise. There will always be truths that elude the basic mathematical framework and are impossible to prove. And this is not a purely abstract finding, without implications for practical situations. Shortly after Gödel’s groundbreaking work, the first provably unprovable problems emerged. For example, it will never be possible to clarify how many real numbers exist within the mathematical framework currently in use.
In spite of this, it may well be possible to prove every mathematical truth from the axioms, in some way. How would we ever know that it is the truth, unless it is proved in some way? It just might not be provable within a particular system.
People read this and conclude that the method of mathematical axioms and proofs does not work. But it does. Every provable statement is true in every model, and every statement true in every model can be proved from the axioms. Goedel proved that. That is what justifies using first order logic as the basis for mathematics.
ZFC serves as a suitable axiom system for all of mathematics.
Even before Goedel, it was known that set theory had countable models, so a model of the real numbers might have only countably many reals in use. Yes, it seems strange, but it does not undermine the use of axioms and proofs to find truths about reals. The set that enumerates those reals is not in the model.
People say it is shocking that a mathematical system cannot prove, from within the system, its self-consistency. But not really. Such a proof would not make much sense anyway. Inconsistent theories can prove their own consistency, as they can prove anything.
Goedel's incompleteness theorem was indeed a profound and important theorem, but popularizations of it are so misleading as to be not helpful.
Why is everyone so surprised you can't create a logical system with numbers attached big enough to contain itself? After all, the math you do certainly doesn't contain YOU, and the 'YOU' part is the main ingredient required to even have a reason for the damn math to be done (or exist) in the first place. What Godel is touching upon is the paradox of having a logical system sans the external system that allows it to function, namely you. Think about basic causality for a moment.
ReplyDeleteEven in a precious subset of logic called math, an observer (or prime mover) is still a requirement, just like the rest of science for the very same reasons. To think otherwise is to believe a ungrounded mysticism, that another otherworldly plane of existence sustains and informs math independently of those who perform the math. Don't get too upset though, you aren't the first. Descartes had the very same problem and it drove him insane.
Math is an activity... and dear readers, it doesn't do itself...
much like making your bed, doing your laundry and cleaning your room. Just because you didn't notice someone else (probably your mother) doing your housekeeping doesn't mean it could be done without their efforts.
Time to grow up.