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BehindCurtai
Joined: May 25, 2004 Posts: 12589 Status: Offline 
So ... Well, there's a pair of wikipedia articles. Axiom of Choice, and Axiom of Determinacy. http://en.wikipedia.org/wiki/Axiom_of_choice http://en.wikipedia.org/wiki/Axiom_of_determinacy Yes, up til now I've used classic set theory. I know that you have to ensure that defining a set by a property requires you to determine that the set does not have that property (a set cannot contain itself), and that construction by negation is illegal under the Russel Paradox. But ... Well, I don't understand ZFset theory. And then there's ZFC set theory. I mean, choice seems to make sense. But the consequences of determinacy makes even more sense. Is there is decent introduction for someone that used to be a math/logic major 20some years ago? And don't get me started on ... http://en.wikipedia.org/wiki/Inaccessible_cardinal http://en.wikipedia.org/wiki/Category_theory http://en.wikipedia.org/wiki/Pointless_topology ... so apparently you can divide a sphere into 5 parts, each of which has the same measure as the full sphere? http://www.irregularwebcomic.net/2339.html ... my brain hurts.  "We're trying to find the error bars on that number"



Dylan
Joined: Jul 21, 2003 Posts: 10005 Status: Offline 
To me (after a similar gap) Choice remains natural, whereas Determinacy seems very much like an unfounded assumption. Axioms should (must) be consistent with themselves, so asking questions like "does the set of all sets excluding itself exist" is pointless masturbation. But you don't have to use every axiom you can think of to form a mathematical theory (= "if we assume these axioms, what do we end up with?") If two axioms are mutually contradictory, then there is NOTHING wrong with picking one OR the other, and forming two theories (and many conflicting theorems) from those. As you know, mathematics is sometimes about turning your brain off "intuitive" explanations, and exploring the purely logical. Turns out (Go"del) that we can't prove everything, even in a supposedly complete system. Suspension of disbelief! 


Alfwyn
Joined: Nov 24, 2005 Posts: 974 Status: Offline 
Axiom of Choice  sure makes sense Wellordering theorem  no way that can hold (for R^2 for example) The two are logical aquivalent  oops
But then again this only happens if the parts form sets with most mathematicans would call nonmeasurable. Or the power set of the reals is just too big to make sense, one should take more sane subsets such as collections of Borelsets to measure things. And now to some stuff I was thinking about, but which may be completely bogus too: Ok, all this mess just starts to crop up if you fool around with uncountable sets. Why not just ditch them completely, are they really needed? The rationals are countable. The larger set of algebraic numbers is still countable. The still larger set of all numbers you can describe in a finite way (one would need and probably could define this in a rigorous way) is countable, since the descriptions are. But this last set contains all interesting numbers (and some more) you can think of (given only finite time). 


Dylan
Joined: Jul 21, 2003 Posts: 10005 Status: Offline 
There is nothing "better or worse" about the difference between countable infinity (aleph zero) and uncountable (the powerset of it, aleph one). However to ignore one completely (and thereby, solutions to infinite series, as opposed to limited polynomials) is to throw out the baby with the bathwater. 


Alfwyn
Joined: Nov 24, 2005 Posts: 974 Status: Offline 
Oh, I didn't plan to ditch all those, only solutions to infinite series you can't even write down. Those you can write down somehow describe numbers that have a finite description (the series you wrote down). You can sort all finite descriptions (that actually unambigiously describe a number) "alphabetically", hence the generated numbers form a countable set. 


Dylan
Joined: Jul 21, 2003 Posts: 10005 Status: Offline 
Ok, what is the smallest number that can't be described in one more character than this post of mine? 


BehindCurtai
Joined: May 25, 2004 Posts: 12589 Status: Offline 
Do you want that answered seriously? :). Alephzero, or alephnull, or omega/wubbleu: The number of integers. The countables. Approximated by A(g,g) or {0, 1, 2, 3, ...  } (Do I have that notation correct?) Alephone, or Cantor's number (?): The number of reals. Alephtwo: The number of functions from reals to reals. Alephthree: ??? That's how I understand it / had it explained to me. Do I have it accurate? Whether there is a number between Alephnull and Alephone: Arbitrary. Most modern mathematicians apparently want to assume that there are numbers between them, but don't ask me why. (Or so I'm told.)  "We're trying to find the error bars on that number"



REJBELLS
Joined: Oct 16, 2005 Posts: 1435 Status: Offline 
Sounds like ye all are ..trying to go where no man has gone before..(picture Captain Kirke in the first series..) Perhaps 'we' just have not gotten there...yet. Perhaps The Dreamer has not Dreamed.  Wildsrose Of: Cerulean, Meridian & Emerald The Titan. The Kraken. 


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