I think we have different ideas of what counts as 'countable'. Mine, not being a set theorist and therefore probably being the completely wrong one, is not one intrinsic to the model, but to the 'standard' model that everyone uses. I am not denying there is no bijection between N and P(N) within...
Oh boy. Did you even look up Skolem's paradox? All that ZFC implies is that there is no bijection between P(N) and N, where we take N to be the inductive set that ZFC states exists. Note that this requires you to define the power set (ZFC doesn't say what it is) and also note that all we can say...
The proof is not incomplete - the reader is supposed to do the steps to verify it.
The axioms of ZFC are very weak, in some sense. You cannot prove a lot directly from them: they don't even assert that the real numbers exist, so you're not going to deduce any analysis from them even. ZFC...
May I simplify?
We define a function on the strictly positive real numbers:f(x)=x if x >=1,
f(x)=1/x otherwise.
It probably isn't widely used because no one has needed to use it.
Try with a smaller example, like 3 elements {a,b,c} to begin with - or just try writing out a few symmetric relations and trying to see what needs to be true about them.
You notion of 2^6 implies that a relation (of some type) is purely defined by being a subset - if that were true then it...
No (correct) proofs are incomplete, or they are not proofs. The proof of the theorem will follow from the hypotheses of the theorem. Whether or not the hypotheses are ever satisfied is essentially unimportant to whether or not the proof is rigorous.
Of course a writer may omit 'obvious' steps...
To add to what Halls said: you have not take the negation of A => B correctly. The positioning of the quantifiers is very important: the negation of A => B is A and not(B), so the "for all" in there is not changed. You have negated A=>B and gotten, well, goodness knows what in relation to A...
"So" means "hence" - there should be a deduction: we choose X so that Y is true.
Or something like that, anyway. 'Such that' can also be used to mean 'satisfying' where it wouldn't be sensible to use 'so that'.
http://www.dpmms.cam.ac.uk/~wtg10/galois.html
is a useful link to expand on Hurkyl's idea.
Or.
Define x to be that expression above, what is x^2? what is x^2 - 2 - 5?
In what way is that not in a useful quantifier form (for those who like things like that)? Really, we shouldn't even bother with the quantifier "for all a,b", and putting things in the full on formal abstract quantifier notation just makes things far more opaque than they need to be.
You're...
This is ripe for a proof by contradiction approach (which may even yield a direct proof).
If there is a c such that B(f(x),b) is not contained in f(B(x,c)) what does that mean?
That isn't right. You have your quantifiers all kludged up into one it should have read
(for all a,b)(ae^x + be^2x=0 for all x => a=b=0)
the negation of which is
(there exists a,b)(ae^x + be^2x=0 for all x AND NOT(a=b=0))
The for all x is not part of the definition of linear (in)dependence...
But you've defined the modulus function to be the distance. You didn't need to do that; I can define |x| to be negative and then claim that distance is -|x|. As I said before it is entirely perverse to do so, but would also have been legitimate.
You are not seeking a proof that is anything...