QuoteOriginally posted by: msg908Isn't Godel the one who says that you can't really prove anything?msg.Godel only said that there are TRUE things than cannot be proven. However, as discussed elsewhere in this forum, this never gets in the way. It´s nothing to worry about. Philosophers always make a big deal about it though. I challenge anyone to prove me wrong!!!If you want another example of building uncountable things it would be with language: The set of symbols of any language if finite, but the things that can be expressed in infinite (expressive power of language).

what about the Lowenheim-Skolem paradox :there exists a countable model of ZFC set theory, so how can anything be uncountable?(to address the original question, how do you define "construct"?)

If the means of construction is "countable" then there are serious problems with trying to construct an uncountable set. E.g. Baire's Theorem:"A non-empty complete metric space cannot be represented as a countable sum of its nowhere-dense subsets." (R. Baire, Ann. Mat. Pura Appl. , 3 (1899) p. 67.) Roughly speaking, you can't construct a real number line out of a countable union of points.René Baire was another guy who made it from a poor background. Seehttp://www-history.mcs.st-andrews.ac.uk/~history/Biographies/Baire.html

QuoteOriginally posted by: CactusManIf you want another example of building uncountable things it would be with language: The set of symbols of any language if finite, but the things that can be expressed in infinite (expressive power of language).Don't confuse uncountable with infinite!The things that can be expressed in a language with a finite set of symbols is infinite but countable.

Set of sequences of integers:

Are the primes countable (because they are a subset of the integers)?

Even if you don't know what countable is: guess!If the integers weren't countable, then this question would be useless. The set of integers have to be countable, it is implied by the question.Now do you think a subset of a countable set is countable? Even without knowing the definition, you could give it a guess...

sorry, double post.

QuoteCan you construct an uncountable set from countable sets?Def.: A finite set is always countable. An infinite set S is countable iff there is a bijective map from N (the set of integers) to S. Well, I do not know what do you mean by "from". If "from" means" as a subset", the answer is no. R, the set of real numbers, is uncountable. R can be written as Q U I, where Q is the set of rational numbers and I are the irrationals. Any irrational i can be expressed as the limit of a rational numbers sequence, so we can construct an uncountable set (R) from an uncountable one.

QuoteOriginally posted by: adgyEven if you don't know what countable is: guess!If the integers weren't countable, then this question would be useless. The set of integers have to be countable, it is implied by the question.Now do you think a subset of a countable set is countable? Even without knowing the definition, you could give it a guess... Of course I could "guess". But a guess is worthless especially when it turns on some detail of the definition. How do I know the mathematicians didn't fiddle with the definition to get a different answer?

QuoteOriginally posted by: manolomDef.: A finite set is always countable.Any closed subset of R is "finite" (has finite Lebesque measure), but is uncountable.

Does the notion of "countabllity" have any PRACTICAL value in finance? Give me an example where countability plays a critical role.

QuoteOriginally posted by: INFIDELQuoteOriginally posted by: manolomDef.: A finite set is always countable.Any closed subset of R is "finite" (has finite Lebesque measure), but is uncountable.Well, when I talk about finite/infinite sets, I refer to the number of elements. Counterexample to what you said: S = {1} is a closed subset of R and not is uncountable.

QuoteOriginally posted by: KackToodlesDoes the notion of "countabllity" have any PRACTICAL value in finance? Give me an example where countability plays a critical role.I think only in proving some minute details of some financial theory. Practical value = 0.

QuoteOriginally posted by: KackToodlesDoes the notion of "countabllity" have any PRACTICAL value in finance? Give me an example where countability plays a critical role.Countability is a paramount concept in Math, so is in mathematical finance. Black-Scholes, for instance, assume continuity in the evolution of stock prices, continuous hedging, ... so the notion of countability (the set of the prices of a stock following a non-constant continuous evolution is uncountable) is present in almost anything.