btilly

26 ms·

451.
by btilly 1y ago
The purpose of https://www.random.org/draws/ (which is unfortunately currently down), is to do exactly that. Also it is possible for any group to agree that they will all sign messages at a given time about a given sou
452.
by btilly 1y ago
Can you illuminate what uses public randomness has in cryptography? If I think about it, I can come up with some. But they seem pretty niche relative to secret keys.
453.
by btilly 1y ago
It isn't quantum, but as far as I know https://www.random.org/ is sufficiently random for any purpose that I can think of for publicly verifiable random numbers. (Most of the demand for random numbers, of course, comes
454.
by btilly 1y ago
Then you likely know more set theory than I do!
455.
by btilly 1y ago
Anything by https://en.wikipedia.org/wiki/Gregory_Chaitin , for example https://www.amazon.com/Exploring-Randomness-Discrete-Mathema... .
456.
by btilly 1y ago
My apologies for having missed this request. I don't have good suggestions for a good set theory textbook. Grad school was 30 years ago, and I didn't specialize in logic. The best set theory book that I read was https:/&#x2F
457.
by btilly 1y ago
We definitely are talking past each other. The existence of a Gödel number in a model of PA, does not imply the existence of a proof corresponding to that Gödel number. PA can only prove the existence of the Gödel number. Now consider a fun
458.
by btilly 1y ago
No, it is not different from the earlier claim. If you think that it is, then you were misreading me. My earlier claim is that for each n, "PA proves that it proves that G(n) terminates". But in logic there is an important distinc
459.
by btilly 1y ago
It is now fixed. :-)
460.
by btilly 1y ago
The Gödel number for all of the standard statements in that collection will indeed exist. But if it is an infinite collection, then a nonstandard model of PA will have statements in that collection that are not in the standard model, and th
461.
by btilly 1y ago
For every n, the function will return a function, that PA proves has a proof in PA. Suppose that we're in a nonstandard model. For all standard n, there really will be a proof. For nonstandard n, there may or may not actually be a proo
462.
by btilly 1y ago
You are being unnecessariky rude. I have not forgotten what it says. There are pairs of physical properties, connected by a Fourier transform, that cannot both be known at once. Therefore each property is, in general, only known to be in a
463.
by btilly 1y ago
I think we are saying the same thing. If PA proves that it proves a statement, PA cannot conclude from that fact that it proves that statement. If PA proves that it proves a statement, and then fails to prove it, PA is unsound. There exist
464.
by btilly 1y ago
Thanks! I am away from my computer for the day, but I will fux it later.
465.
by btilly 1y ago
> If PA proves "PA proves X" then PA can prove X. Not true. From PA we can construct a function that can search all possible proofs that can be constructed in PA. In fact I outlined one way to do this at the end of my answer. W
466.
by btilly 1y ago
> Of course it can! For example, if some logic theory contains the theorem that a particular realisable polynomial-time algorithm that can solve an EXPTIME-hard problem, that would contradict physical reality. It would only contradict ph
467.
by btilly 1y ago
I disagree with calling them all failures. Everything that is known about numerical analysis is based on discreteness, instead of continuity. Every useful predictive model that we have about the world, for example for forecasting weather, d
468.
by btilly 1y ago
Are they? The idea that uncountable means more comes from a bad metaphor. See https://news.ycombinator.com/item?id=44271589 for my explanation of that. Accepting that uncountable means more forces us to debatable notions of
469.
by btilly 1y ago
The Heisenberg uncertainty principle says that basic physical quantities do not have unique values. They have ranges of values. And therefore it is not always possible to compare two physical properties. For example you can't always co
470.
by btilly 1y ago
I knew that this sort of stuff was possible, but it is fun to see it. When it comes to bootstrapping a programming language from nothing, the two best options are Lisp and Forth. Of the two, I find Lisp easier to understand.
471.
by btilly 1y ago
Thank you for the correction! And I'm glad that someone liked some of my attempts at humor. As my wife sometimes says, "I know that you were trying to make a joke, because it wasn't funny."
472.
by btilly 1y ago
Your first paragraph is contradicted by the Heisenberg uncertainty principle.
473.
by btilly 1y ago
If I understand what I just Googled correctly (definitely not guaranteed), the reason why Cantor's argument can fail in NFU is that NFU does not necessarily allow you to build a function that returns things in X, out of a function that
474.
by btilly 1y ago
It slows down by releasing a large light sail in front of it, designed to reflect light back to a much smaller light sail behind it. The laser then pushes the large sail away, and as the sail goes it pushes the smaller sail (and ship) back.
475.
by btilly 1y ago
It's just a straight up liar's paradox. If enumerate is a function that works as advertised, then unenumerated is as well. If enumerate tries to list unenumerated, then enumerate can't work as advertised. For the argument tha
476.
by btilly 1y ago
It should be true in the stronger sense. Suppose that you've written down a function enumerate, that maps all natural numbers to functions that themselves map all natural numbers to booleans. We then can write down the following progra
477.
by btilly 1y ago
Now we're getting a little beyond the detail that I feel comfortable making statements about. ChatGPT tells me that PA+"PA is consistent" is not quite enough. I believe that it has digested enough logic textbooks that I'
478.
by btilly 1y ago
I have absolutely no idea how physical reality can contradict a statement about a formal symbol game. I am also convinced that what you are saying is not what Hilbert actually meant. But he died before I was born, so I can't ask him. B
479.
by btilly 1y ago
I believe that you are misrepresenting Hilbert here. If ZFC led to a result that doesn't agree with physical reality, Hilbert would not reject that result. Instead, at worst, he would simply move it from the category of being a real fo
480.
by btilly 1y ago
When you talk about the point of constructivism, do you mean currently, or historically? For me, personally, the point of constructivism is to wind up talking about mathematics in a language that corresponds with what I want words to mean.

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