Recent content by aravantv

Interesting, I never saw that definition before. Indeed the proof is much more torturous that way! The nowadays-generally-used definition of intuitionism just excludes proof by contradiction, but not proof by recursion (see for instance Intuitionistic Logic @ Stanford Encyclopedia)! Indeed...

That's my point: I do not see how you risk an "infinite-regress" if you talk about axioms of a theory. If you talk about the axioms of the logic, it does make sense (and it is a consequence of Godel 2nd incompleteness theorem), but for the axioms of the theory I do not see when we can have such...

Indeed, and this is precisely the role of the axiomatization (typically the one of Peano as I mentionned in my post). Well we generally introduce a symbol "succ" to denote the operation "+1". So that you define 2 as succ(1), and 1 as succ(0) (0 being a primitive symbol as well). Then you can...

Precisely. Indeed, it seems so (my knowledge in measure theory is very low, so I cannot confirm this for sure, but it seems very probable since few things require anything more than set theory - well, none to my knowledge).

It does not mean much actually. You can build measure theory on top of some more basic theories like set theory (but note that set theory remains a theory, not a logic). Hence the connections that can be found between them (like the one you mentionned with the Continuum hypothesis). The most...

Well that's not really a matter of belief rather than a matter of opinion. Indeed. Though I don't really see how all this helps answering the question here. Could you make the connection? What do you mean by "descriptive"? The main reason why we need axioms is because we need to define...

Not sure this is really needed to answer your question, but what do you call a "line" in a Turing program? Actually how do you define a Turing program? (I never saw the expression "Turing program"; "Turing machine" ok, but "Turing program" never; is it different?) What is B in your statement...

But you wrote that the operation was valid only if one complex vector only was involved, right? Ok, so what does this tell me about the dot product?

1+1=2 can be proved by giving some axiomatization of arithmetic (say Peano arithmetic, "PA") in first order logic ("FOL"). FOL is a language which says pretty much nothing unless you give it some axioms to work with. Well, to be fair, either you add some set of axioms A, or you prepend every...

I think I've seen in some contexts some complex wavevectors \vec k. Then, from what you say, it means that \vec{k} \cdot \vec{E}=0 does not hold in such a context?

Do you mean that we are indeed in the situation that I described in my second post? So you mean that \vec E\cdot\vec B=0 holds only for the real parts of both vectors? It seems contradictory to what Antiphon just wrote, isn't it?

Am I right by observing that all complex vectors that we meet in electromagnetism can be written as z\vec{x} where z\in ℂ but \vec{x}\in ℝ^{n}? In this case the complex cross and dot products can be easily reduced to their real counterparts: z_1\vec{x_1}\odot...

Hi, I have a hard time finding a justification that electric and magnetic fields are still orthogonal when presented in complex form. As far as I know the notion of orthogonality for complex vectors is not as intuitive as the one for real vectors. Notably, \vec{x}\cdot\vec{y}=0 does not imply...