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.
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...
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)
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 <...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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...
@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...
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...
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...
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...
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.
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...
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)
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...
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...
Measuring the energy of a single particle isn't too hard. If it doesn't interact with the surrounding material there are only kinetic and mass energy. If that material is a translucent, it emits Cherenkov radiation in a small amount. The angle of the radiation cone gives you a measurement that...
I'm not sure I understand you correctly. True, after measuring the energy the system collapses to an eigenvector of the Hamiltonian, and this can easily be seen to stay this way. But is energy not conserved in every circumstance? That's certainly the experience from classical physics, so there...
I was actually wondering about the system together with its environment.
Is the total energy of both parts still conserved? Is there any way to even state conservation of energy, momentum, spin, charge and so on? Since the observer interacts with the system, she should be included in the...
Is conservation of energy, momentum, and other physical properties absolutely true in quantum mechanics, or only on average?
As an example, think of a single particle in free space. Measure its energy, and write down the result. Then look at where it is, and measure the energy again. You'll...
A physics hobbyist such as myself, trying to understand high-energy experiments from the recent decades, often hears about symmetries in the model, conservation laws, Feynman diagrams and so on. These are all intuitive properties but very far from a basic world view of "what happens...
@stephen: In addition to Josh's comments, there is another and more direct problem with your construction. If you build your sets with geometric series like you suggested, and the factor is rational, all the numbers in all the sets are rational, so the union must be countable.
@josh: Your...
As micromass hinted, it is hard to give an example without ordinal numbers. What I'll try to do in the mean while, is give some intuition how this sort of chain can happen.
A similar question to the one you asked, is if there can be a union of a chain of finite sets which is infinite. This...