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Dragonfall
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Does there exist an uncountable set of positive reals such that every countable decreasing sequence from that set sums to a finite number?
Beat me to it.Tac-Tics said:EDIT: Or maybe the Cantor set is a counter example to the last assumption I made. It is uncountable with zero measure. (http://en.wikipedia.org/wiki/Measure_zero)
Actually, that's not guaranteed. For example, consider the infinite set containingVKint said:(Indeed, if [tex] S \cap [a,b] [/tex] has infinitely many elements, we can choose a countably infinite, decreasing sequence, which necessarily diverges, since all terms are [tex] \geq a [/tex].)
Every well ordered (by <) set is countable.
VKint said:Not if you allow the axiom of choice...I think.
But then an uncountable well-ordered set would be a counterexample to your contradiction.
Every uncountable subset of R has an uncountable accumulation point
I don't remember precisely what I was thinking -- but I had convinced myself my first proposition was either an easy consequence of what you had proved, or that your proof of your lemma contained most/all of the bits needed to prove my proposition.VKint said:I don't see how this follows directly from what I proved.
Hurkyl said:Define an uncountable accumulation point of a set X to be a point such that every neighborhood of that point contains uncountably many points of X. (I doubt this is a standard term)
The proof you give of my second and third propositions is flawed -- accumulation points of S don't actually have to be elements of S, so you don't get the contradiction you were relying on.
Preno said:How about simply this...
Elucidus said:If one permits the Axiom of Choice, the reals can be well ordered.
An uncountable set with this property refers to a set that contains an infinite number of elements that cannot be counted or enumerated. This means that the elements in the set are not able to be put in a one-to-one correspondence with the natural numbers.
A set is considered uncountable if it contains an infinite number of elements that cannot be counted or enumerated. This can be determined by using the Cantor's diagonalization argument or by showing that there is no bijective function from the set to the natural numbers.
The property of being an uncountable set is important in mathematics because it helps distinguish between sets that can be counted and those that cannot. It also has implications in areas such as set theory, measure theory, and topology.
Yes, an uncountable set can have a countable subset. This is because a subset of a set is a set that contains only elements of the original set. Therefore, if the original set is uncountable, it can still have subsets that are countable.
Yes, there are different types of uncountable sets with this property. Some examples include the set of real numbers, the set of all subsets of natural numbers, and the set of all functions from natural numbers to real numbers. Each of these sets has different properties and characteristics, but they are all considered uncountable.