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    Can a nontrivial quotient space of R be homeomorphic to R?

    With the additional constraint, the set of image points with more than one origin cannot be uncountable, since each preimage contains an open interval. From separability of \mathbb{R} there is no uncountable set of pairwise disjoint open intervals.
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    Mathematical induction

    Greater and Greater-Equal The propisition is true since it says 2^{k+1} > k + k \ge k+1 and together 2^{k+1} > k+1.
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    Can we apply non-linear smoothing to a linear looking like data ?

    It really sounds like a question in the science of statistics. For some series it'll make the linear correlation stronger, for some weaker. It probably depends on the effect you want, or the features you want to find in your data. Statistics can also help you choose the correct coefficients...
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    Simply set theory question: all standard forms of ZFC imply power set of {naturals}?

    One thing to note here is that the power set P(\mathbb{N}) might not be the same in all models, some may contain only some of the subsets. (indeed, it's also possible to create models where \mathbb{N} is different, but that's much less common)
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    Paradox with elementary submodels of the constructible tower

    I see what you mean... So, it seems that for every ordinal \alpha, the set \{\delta < \omega_{1} \mid L_{\delta} \prec L_{\alpha}\} is closed w.r.t taking limits. I thought about it some more and it's not hard to see this set is unbounded for \alpha = \omega_{1}, since for each \beta <...
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    Help with ordinal numbers

    Actually, it was shown by Gödel that consistency of ZFC follows from consistency of ZF. So you can add another axiom to your list :smile:
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    EM-radiation (light) realllllly have gravity?

    I'm a physics hobbyist as well. From what I read in blogs and popular books, there is no direct evience as to whether EM energy attracts other objects gravitationally. There is a considerable amount of indirect evidence like the two kinds mentioned earlier in this thread. This is the same...
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    Can a model be countable from its own perspective ?

    A standard model is one where the elements of the universe are sets, and the membership relation is the normal membership relation. In symbols, \mathcal{M} \models x \in y is true iff x \in y/itex] I'm not sure about my English here. I meant a phenomenon like A_{1} \ni A_{2} \ni A_{3} \ni...
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    Paradox with elementary submodels of the constructible tower

    This is an argument I thought up after a class on combinatrical properties of the model \textbf{L}. Our course is about set theory, not logic, so this paradox desn't seem relevant in its context. Can you help me figure out where I got it wrong? The constructible heirarchy of sets is a series...
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    Can a model be countable from its own perspective ?

    A model can be countable from its own perspective, in some sense. It depends what other axioms it satisfies in addition. If a model (which is a set for the universe and a 2-relation on it for membership) contains the set ω and a function from ω to the universe it can be said to be countable...
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    Set of real numbers in a finite number of words

    It is a misunderstanding, but one that is very easy to have. This is why I referred to Skolem's paradox - even he got mixed up in this logic. Of course. This thread has gone beyond freshman level, I think, after the OP got an answer that being able to describe a set concisely doesn't mean...
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    Set of real numbers in a finite number of words

    Consider a countable model of ZFC, or a countable elementary submodel of your model-of-choice-for-sets. (this exists by Skolen_Lowenheim) Then you can consider sets as labelled by natural numbers, and then you have 1-1 correspondence between the sets of algorithms (as that term means in the...
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    A very probably flawed attempt at CH

    Actually, you argument also doesn't explain why the set F is uncountable. You quote Cantor's argument, but you don't explain why it works for this set. (For example, a diagonalization might give you a rational number not in the list, but that is not in F so the argument doesn't work) This is...
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    Co-prime of vectors

    I don't know if there's a name for this operation. But I can recommend that you move the thread to the Number Theory forum, it seems to belong there more.
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    This summation sums to zero. Why?

    I don't think that is correct. Define \lambda(m)=m, and pick L=r=2. Then...
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    Derivative of a vector

    I never encountered this convention, and I can't see where it can benefit the presentation. This seems like a perfectly good thing to ask your prof.
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    This summation sums to zero. Why?

    In general there is no equality. It must depend on the definitions of λ, r and L. Can you provide more details?
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    Vectors and spanning

    There is actually a trick that can give a simple formula to know if there is a solution or not. This is based on observing that the equation set micromass showed near the end of her post is very similar for different vectors: only the numbers on the right side of the equals sign change. The...
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    Prime Number Algorithm

    I believe number theory involves a great deal of algebraic geometry nowadays. It's not at all like the approach in your paper, but if this leads to something, it'll be wonderful. Even if you don't prove new theorems, elementary proofs of existing theorems are ofter enlightening. PS. I liked...
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    Fermat's equation solution in Z

    There are many solutions where one of the numbers is 0: two examples are 64+04=64 and 53+(-5)3=03. There can be no solution with non-zero numbers, and this can be inferred from non-existance of positive solutions. For even n, this is trivial: since an=(-a)n, a solution with negatives is also...
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    Help with logic exercises

    So, if you're not allowed to use truth tables explicitly, what tools can you use? Did your course define any axiom system for propositional calculus?
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    Temperature defined in terms of entropy and energy

    @juanrga I can vaguely recall from my Thermo course some time ago that there are discrete systems, where you have N particles that can be in either base or excited states. In those systems once the energy is >N/2, temperature is negative. Am I right here? Or is it an axion that the...
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    Temperature distribution of neutrino background radiation

    Hello. I wonder - is the CνB energy curve expected to follow the Boltzmann distribution? In the 2 seconds neutrinos were in equilibrium with matter, did they bounce off enough to even out the energy? I'm asking because a back-of-envelope computation gives cosmic neutrino speeds of ~0.09c...
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    How reasonable to assume a prime gap of at least 10 before a pair of Twin Primes?

    My 2 cents on you first question. I would be very, very surprised if there were an infinite number of prime twins and almost all of them had another prime close to them (say ±10 like in your example). By the pigeonhole principle, for one of the 10 distances you would have an infinite...
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    Prime Number Algorithm

    The article is quite long, so I skipped a few parts. If I understand correctly, your observation is that any odd composite number N is a sum of a series of consequtive integers of length < √N. This is a nice property, I for one didn't know it, and it wasn't covered in my number theory course...
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    Co-finite topology on an infinite set

    To complete the classification, you can see that if X is finite and has n elements, all of its subsets are finite, and there are 2n of them, which is more than n. So there is never a 1-1 correspondence between τ and X in the finite case.
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    Light Speed and life

    Actually, the light will dim slightly as it goes. This happens for 2 major reasons. For one, the outer space isn't completely empty. It is actually a very thin gas, but it still blocks some of the light. The second reason is that a laser is not perfectly collimated, instead the light spreads...
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    Dark energy <=> negative energy?

    Thanks a lot for the info, it really answered my questions. The cosmology tutorial is especially helpful. (it's also interesting to learn that the CMB measurements tell us the universe is closed)
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    Dark energy <=> negative energy?

    Sorry if this sounds a bit mixed up. When I was growing up, in the late 1990's, popular science books about cosmology use to describe the average mass-energy density of the universe, especially comparing it to the critical density. Those books used to say that visible matter gives too-low...
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    Probability of Existence

    The probability you describe is a very interesting quantity, which tells us basically how complete our description of the nature is. It is not calculable, however. For one thing, there is no accepted Theory of Everything, a basic theory of physics that is applicable to all situations. Even if...
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