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.