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  1. R

    How are the Real Numbers distributed?

    They are distributed how you want them to be. A set does not have an intrinsic distribution. For instance you may define a probability density function p(t) by p(t) = 1/2 for t in [1,3] and p(t)=0 otherwise. If what you want is a uniform distribution, then no such thing can exist on the real...
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    Why can't the well ordering of the reals be proven with Dedikind cuts without AC?

    See http://en.wikipedia.org/wiki/Well-ordering_theorem" [Broken] on Wikipedia. Basically you form the set of all well-ordered subsets of the real numbers, then you invoke Zorn's lemma to conclude that there is a maximal such well-ordered subset, and you then note that it must actually be a...
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    Why can't the well ordering of the reals be proven with Dedikind cuts without AC?

    This makes no sense. It's like saying all integers of the form 4n with n integer satisfies the requirement that every integer is even. It's true that every integer of the form 4n is even, but not that every integer is even. In the same way it's true that with the usual order [a,b) has a minimal...
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    Why can't the well ordering of the reals be proven with Dedikind cuts without AC?

    a) How exactly would that help you? There are many well-ordered subsets of the real numbers that are easy to construct. For instance the positive integers. What you need is a well-ordering of an uncountable subset of the reals, but this cannot be done without AC. b) That paper never seems to...
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    Difference between least, minimal element

    (Assuming you're replying to me. There was another reply shortly before mine, but it seems to have been deleted. If you meant for that poster to get the reply disregard this post.) I mainly learned the very basics of set theory from the various introductions that many introductory math books...
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    Difference between least, minimal element

    A least element is an element smaller than all other elements. I.e. x is least if for all y we have, x \leq y A minimal element is one that is not larger than any other element. I.e. x is minimal if for all y, either x and y are incomparable or x \leq y. If a poset has a least element, then...
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    Why is Nat(hom(A,-),F) a class?

    Thanks for the answer. That makes sense. For a fixed object B in category C both hom(A,B) and FB are sets so the class of functions from hom(A,B) to FB is a set. Hence if \eta is a natural transformation from hom(A,-) to F, then \eta_B is a set and thus \{\eta_B | B \in ob(C)\} is a class in...
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    Why is Nat(hom(A,-),F) a class?

    I'm trying to read a bit up on category theory, but I'm a bit confused about one aspect of the proof of Yoneda's lemma. Suppose we have a locally small category C, a functor F : C \to \textrm{Set} and an object A in C. Now according to Yoneda's lemma there exists a bijection from Nat(hom(A,-),F)...
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    Family vs. Set

    They use the term family when they want to emphasize that they are all subsets of some set (U_a \subseteq X). Formally there is no difference, but you rarely speak about a family of integers because we don't think of 5 a set (even though according to most definitions it is). I have also heard it...
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    Kuratowski's Definition of Ordered Pairs

    There are many ways to define such tuples, but the important thing is that they behave similarly. Anyway for finite n-tuples we often simply define it as the ordered pair where the first coordinate is a (n-1)-tuple and the second is an element of the underlying set. For countably infinite tuples...
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    Set theoretic definition of a singleton.

    Why don't you think that {x,x} has a single element? When people are informally introduced to set theory they often don't see why {x} or {x,y,z} may not be sets (they turn out to be though, but only because of a number of axioms which are rarely mentioned or even thought of by beginners). In the...
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