(no subject)

[prog]

Heh heh, pages upon pages of people going No... NO! Aaaaaagh you're all wrong shut up about the .999... thing. This is worse/more amusing than the time that the Monty Hall problem was AOTD.

I find it interesting that the text of the article actually predicts the belief-path the doubters take... when faced with simple and easily graspable proofs, they change their minds and state that obviously this means that the number system is broken.

Comments

[wrog]

it's a self-consistent system

actually, we don't know that. Nor can we know that given Gödel's Theorem, which basically says that once you have a proof system that's rich enough to handle arithmetic, the only way you can prove its own consistency in terms of a bigger system, at which point it's turtles all the way down.

Best we can say is we have yet to find an inconsistency, or, rather, none of the inconsistencies found thus far (e.g., Russell's set of all sets that don't contain themselves) has proved to be irreparable. ZFC (everybody's favorite set theory) has held up pretty well over the last 80 years or so; on the other hand, the Axiom of Choice leads to a bunch of awfully strange theorems.

The real point of mathematics is that it's a way of thinking that (thus far ) has been remarkably successful at screening out bullshit. YMMV

[chocorisu.livejournal.com]

You're talking rubbish. Even Godel's theorem wouldn't work if the rules of the system were inconsistent. maybe it's incalculabale or there are things you can do to delibreately break it, but as far as actually gettnig useful results out of it, if we start making up silly rules just because we don't like the things that fall out from following proofs, we may as well just give up and go back to living in caves. SORRY ABOUT NONSENSE, VERY DRUNK.

[radtea.livejournal.com]

Even Godel's theorem wouldn't work if the rules of the system were inconsistent.

Which is why the response to Godel's theorem is ususally to say that mathematics is incomplete rather than inconsistent. What Godel showed is that any consistent axiomatic system that is rich enough to contain arithmetic contains true theorems that cannot be proven via deductive manipulations within the system. Such a system is said to be "incomplete".

To experimental scientists this was not exactly a big surprise, as we always viewed the kind of knowledge we created as more than just a hack to get at things that the theorists could get at by other means. The big surprise is that fifty years later pure mathematics has an even greater hold over theoretical physics than it did in Godel's day.

[chocorisu.livejournal.com]

Incomplete, certainly. But definitely not inconsistent.