Well one thing they do is constrain operators to be self-adjoint. This forces real-answers and is what gives "quantum superposition". So arguably you need completion if only to give you the spectral theorem
@amWhy: I in fact wanted to say that I now realize you were right so many months ago. But I decided it wasn't necessary because it is obvious.
@DavidReed Strangely, your question was better received on Phys SE than mine, despite mine being rather much more logically precise. I eventually deleted it and reposted it on MO, where it was immediately understood and answered by Terence Tao.
@XanderHenderson Currently I believe ACA is meaningful (or at least very close to it) in the real world. And reverse mathematics has shown that practically all real analysis and more more can already be done within ACA. Uncountable stuff cannot be reasoned about in ACA, of course.
@XanderHenderson But hey, you already doubt a complete countable infinity, so... =)
And I too doubt the 100% real-world meaning of ACA (hence why I said "very close to it"), for the reasons mentioned here:
This question was born out of a discussion Is the real number structure unique? on Math SE, but since it is more philosophical than mathematical I decided to ask here.
From Gödel completeness and incompleteness theorems, assuming the standard axioms of math being consistent, the set of natural n...
As of now, I haven't thought much about what can be a viable alternative that would be consistent with a finite world and yet recover most of the actual theorems that have practical implications. In particular, I've yet to see any cogent explanation of why many theorems like Fermat's little theorem seem true at human scales, if we are to reject meaningfulness of PA.
Ah that's interesting. So we have a mathematician (Terence Tao) and a QM researcher (Daniel Sank) who both say pretty much the same thing, though Terence goes into far more mathematical detail (after all it was on MO).
I'd also wait and see what he says about my claim, since that wasn't about quantum mechanics, which is what also makes your question not quite the same as mine on MO.
@DavidReed I know, but my question is asking about the Lebesgue measure's real-world meaning, if any, whereas yours is asking about its use in physics (not necessarily meaning).
From a logical point of view, yours allows for a conservative extension over a physically-meaningful formal system, where the Lebesgue measure exists in the extension.
Yes, I had to clarify my question and found it was difficult to articulate. I wanted basically to know whether things like the quantum computer would have come to exist if the notion of L measure had never been conceived
I tried to capture that with the phraseology "practically necessary"
You need the space to be complete to give you the spectral theorem, which opens up all the sturm-liouville theorems regarding things being expressible as a linear combination of eigenfunctions
T. Tao touched on this
For finite dimensional V spaces you don't need to impose such fancy requirements
I believe the crucial point behind the physical significance of Lebesgue measure as opposed to Jordan measure boils down to the issue of completeness, as Gerald Edgar remarked in part (2) of his answer. I'm answering here anyway because I want to elaborate a bit on this.
Since a bounded set is J...
Well my point regarding the logical phenomenon stands; but I'm not sure you got what I was saying.
I believe the arbitrariness is an artifact of using ZF. In a constructive setting, the quotient type may not be constructible, and so using it as a complete whole is impossible, but many of the results involving it can be rephrased in terms of the original type and the equivalence relation, so there is a concrete distinction between what seems arbitrary from the point of view of a stronger theory like ZF. Coq for instance uses setoids which are one manifestation of this constructive distinction. — user21820May 7 '16 at 5:27
Piled Higher & Deeper by Jorge Cham www.phdcomics.com title: "Nuclear War Explained" - originally published 12/11/2017 For the latest news in PHD Comics, CLICK HERE!
Piled Higher & Deeper by Jorge Cham www.phdcomics.com title: "On the side" - originally published 12/22/2017 For the latest news in PHD Comics, CLICK HERE!
Piled Higher & Deeper by Jorge Cham www.phdcomics.com title: "Number of Side Projects" - originally published 1/8/2018 For the latest news in PHD Comics, CLICK HERE!
Piled Higher & Deeper by Jorge Cham www.phdcomics.com title: "Did you leave it on?" - originally published 1/12/2018 For the latest news in PHD Comics, CLICK HERE!
Piled Higher & Deeper by Jorge Cham www.phdcomics.com title: "Let them know" - originally published 1/17/2018 For the latest news in PHD Comics, CLICK HERE!
@SimplyBeautifulArt: Hmm does subscribing to a feed grab the last couple of posts to that feed? But hmm I thought it would post the actual comic, like XKCD.
@user21820 That could be. But my second sentence was not suggesting such behavior from Simply. Just all the threads in which our friend user17**39 can be found, over the last few days, trying to recruit folks who are sympathetic to him, and his worship of another user. I've included transcripts in CRUDE, perhaps will in the mod's chat. In anycase, hope he realizes the same kind of walking on ice that's he's inflicted on others, realizing that his words, too, are subject for broadcast.
He's here, in Crude, in Constructive Feedback,in the Math mod's office, in Archive, in Philosophy of math, now a gallery, in Misc. and in Random Discussion. Also created a chat yesterday to recruit Martin Sleziak, and also take Martin's words out of context. Pretty sad.
@MartinSleziak I recall we discussed something like this before and in some cases it will silently omit some close-voters who didn't pick a particular reason.
In response to your comment, close but not quite. First off $\mathbf{b-a}$ is not a linear transformation so $\Vert \mathbf{b-a} \Vert$ makes no since here.
The sequence, which makes use of the Cauchy-Schwarz inequality, goes like this:
$$|\mathbf{g}'(t)| = | \mathbf{f}'(\gamma(t)) \cdot (\math...
case in point
Actually I have just realized my answer is wrong
which is a testament to how confusing the book can be at times
@XanderHenderson: By the way, if you're interested in discussing more about the foundational issue about infinity, feel free to drop by the Logic room some time! =)
@user21820 It has a lot of really slick proofs and is rightfully considered a reference that every mathematician should have on his shelf. It is confusing in this particular instance because his notation is outdated in a way. It is confusing in general for beginners because he makes "leaps" and expects persons to have the mathematical maturity to fill in some of the missing steps for themselves
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…