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The second theorem says that the consistency of a system cannot be proved within that system (sufficiently strong, etc). If a contradiction is discovered in se
by arakyd 18y ago
The second theorem says that the consistency of a system cannot be proved within that system (sufficiently strong, etc).
If a contradiction is discovered in set theory, either it will be fixed by a change to the axioms, or another set of foundational axioms altogether will become the standard foundation of mathematics. Perhaps the canonical example of this is naive set theory itself - paradoxes were discovered, but set theory was interesting enough that mathematicians looked for alternate definitions that did not contain the paradoxes, and they succeeded.
There are two things to note here. First, axiomatization was a useful tool in distinguishing various types of set theory and defining the ones that did not contain the (known) paradoxes. Second, the full formal treatment of these set theories came relatively late to mathematics, as did widespread use of formal, axiomatic methods in general. Rigorous axiomatic formalisms are very useful mathematical tools, but mathematics got along without them for a long time (not even Euclid counts, by modern standards) and can do so again if necessary. It is unlikely that this will happen, not because axiomatics can be proved to always work or because mathematics cannot get along without it, but because it is too useful a tool to abandon easily.
This was really Godel's point: mathematics is not identical with formalism. They stand and fall separately. This is not to say that mathematics could never collapse for any reason, only that it would take a lot more than finding a paradox at the center of ZFC to make it happen.
- Tichy 18y agoHm, I thought set theory already contains arithmetic, so wouldn't Goedel's proof show that such axioms without paradoxes can't be found? I thought that is why he is so famout - for years it had been the goal of mathematics to find the ultimate axioms, and then Goedel showed that it can't be done, ever.
- cchooper 18y agoThe second theorem tells us that we can't prove the consistency of set theory without using an even more powerful system. This doesn't mean that set theory is inconsistent, it just means that we can't prove its consistency in any meaningful way. So yes, Gödel indeed shows that there are no 'ultimate axioms' that contain everything of interest in mathematics.
- mstoehr 18y agoAlthough it does lead to a mathematically uninteresting paradox: if you let A be the axioms of set theory and you add an axiom P which states that A proves x and not x, (i.e. set theory is inconsistent) then A' = A and P is still consistent.
- mstoehr 18y ago"This was really Godel's point: mathematics is not identical with formalism. They stand and fall separately. This is not to say that mathematics could never collapse for any reason, only that it would take a lot more than finding a paradox at the center of ZFC to make it happen." That's significant because another great mathematician David Hilbert challenged mathematicians to come up with a complete and consistent set of axioms for all of mathematics. This was a great hope at one time that was shattered by Godel. Most people don't really talk about this old program anymore (except as a historical curiosity) because there is utterly no hope in it at all. The other implications that people try to draw from Godel's work is probably a consequence of the fact that his work sounds like it says so much more than it actually does when it's translated into normal English (and out of math-speak).
- d0mine 18y agoConsistency is overrated. General relativity and quantum mechanics are inconsistent with each other but at the same time they are extremely precise when applied to an appropriate domain (I remember watching a lecture where Feynman wrote down a number with 20 or so digits representing conformity of theory prediction with experiment).