Posted on 02/24/2004 6:38:39 AM PST by dead
"There is a set of numbers 'smaller' than the reals and 'bigger' than the natural numbers" (aka the Continuum Hypothesis) is undecidable but is easily grasped by most with a short explanation of what 'bigger' and 'smaller' mean in this context.
There is an important assumption that is often overlooked: the methods of proof in the system must be "finitistic." IOW if you allow something like transfinite induction then you can prove completeness and consistency.
Of course then the question is, is a method like transfinite induction intuitively obvious?
It seems intuitively clear to me if you make the assumption that people (i.e. beings capable of formulating number theory) are a natural consequence of physical law.
I'll grant you what's not clear is whether the incompleteness theorem has anything interesting to do with physics.
Actually section 1 (esp. 1.1) of that page is pretty short and easy-to-read and the rest is not material to understanding CH.
It's not clear to me at all. Number systems, formal as they are, were arrived at via human reasoning, which is not a formal system. Human reasoning is, I maintain, complete, in that all possible truths are in principle available to it, including the truth or falisity of formally undecidable Gödel statements. The price paid for this is inconsistency: some wrong statements will seem to be true, and vice-versa. I don't think I need to make a case that human reasoning is inconsistent.
Nothing more than the fact that the precise mathematical statement of the axioms gives a rigorous definition of a topological space. The moment one tries to water that down to a layman-level understanding, one sacrifices rigor (or "precision") for understandability.
Sorry about the headache....
Caution must be execised to avoid confusing "undecided" hypotheses from "undecideable" hypotheses; the CH is undecided -- that is, we don't know if it is true or not, and it also makes no difference to standard set theory whether it is true or false (IOW, standard set theory is independent of the CH), but no one to my knowledge has demonstrated the CH can NEVER be decided. IOW, no one has shown the CH is a Gödel statement.
IMO, this is a very, very horrible way to get the idea across. For starters, it's flat out false. It makes the claim in an even more positive fashion than I put it, and is just going to end up confusing a lot of people, despite his rather insufficient attempt later to correct himself. I do agree with everything else you've said, Physicist -- no argument there -- but I guess I just don't think it's a good idea to make false statments, as he's done here. It only adds to the confusion, skepticism and ignorance. We can almost guarantee there are going to be people running around after reading this article claiming that math is a joke, very likely including NEA types who think there are no "right" answers anyway.
You're most welcome. But I hope you were talking notes, because there will be a "pop" quiz next period!
;-)
I think you're misreading it, but let's be explicit: what is wrong about that statement, in your reading?
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.