btilly

57 ms·

841.
by btilly 2y ago
That ties to one of the standard axioms of the real numbers in classical mathematics. There exists a set P which is closed under addition and multiplication, such that for every real number x, one of these three things is true: x is in P,
842.
by btilly 2y ago
That goal is what I meant by the phrase "that all could accept". You're right that I could have expanded on it though. That said, Gentzen's result is new to me. Though I have seen other results that are somewhat similar.
843.
by btilly 2y ago
Yes.
844.
by btilly 2y ago
My ex got a device to discourage ants. Then forgot where she placed them. Then found out that it was in my daughter's hearing range. Finding the last one was a nightmare. Here is the device. https://www.amazon.com/Ultra
845.
by btilly 2y ago
Oops, thank you for the correction! I learned something today. Though I'm sure you can see how I thought that the other Robinson fitted in the conversation,
846.
by btilly 2y ago
Read https://plato.stanford.edu/entries/logic-higher-order/ if you wish. I view the "for all properties" bit as handwaving. See particularly section 6 of that article, which at one point gives: The sema
847.
by btilly 2y ago
If you start with NPA it is identical as PA. You can apply all the same logic and all the first theorems. The problem with multiplication is not that the proofs fall apart, it is that multiplication does not mean what you want it to mean. W
848.
by btilly 2y ago
How hard could that be? It seems like it should be straightforward. But it's mathematically impossible. Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which wo
849.
by btilly 2y ago
Robinson arithmetic is an excellent example here. Using ZFC, Robinson can take a standard model of the reals and construct a nonstandard model, and a function from the standard model to the nonstandard model. Every possible statement of fir
850.
by btilly 2y ago
In any model of PA, all natural numbers will be finite by this definition. Even if we, from outside of the model, know that many of them aren't. You've been presented with two different ways to prove this, and a standard textbook.
851.
by btilly 2y ago
You don't define it. You list things that you want to be true of it. The Peano Axioms are examples. But no matter how long your list gets, it can always get longer. For example we can ask whether a particular polynomial has integer roo
852.
by btilly 2y ago
I guess I calibrated your understanding of formal logic to the wrong place. Let me try again. PA is a set of axioms satisfied by the natural numbers. As Gödel showed, all of first order logic can be encoded into the natural numbers. And que
853.
by btilly 2y ago
Some other fraud in the same subfield lead to an amoral douchebag becoming President of Stanford, where he destroyed the campus culture (read https://www.palladiummag.com/2022/06/13/stanfords-war-on-soc... to
854.
by btilly 2y ago
Well, you can define something meaning "finite" in first order logic. It just won't mean what we want it to mean. Take your example. "All natural numbers are finite." How do we define the natural numbers in first or
855.
by btilly 2y ago
How much explanation do you want? Any consistent first order set of axioms that has a model, has multiple models. In the case of the Peano Axioms, there is one and only one model that we're interested in - the finite integers. All the
856.
by btilly 2y ago
Exactly the same for the one that I did, except that the ad serving was in Python. But golang is probably the better choice.
857.
by btilly 2y ago
Serre is indeed a top mathematician. (I'm actually surprised to find out that he's still alive!) At this point Tao only has 3/4 his number of publications, similar numbers of textbooks, a similar number of awards (using http
858.
by btilly 2y ago
According to logicians, it always is an arbitrary choice. But we have a second-order notion of what "finite integer" should mean. And within that notion the idea might be false. Here's how that plays out. Suppose that the RH
859.
by btilly 2y ago
No matter what you think, best living mathematician is what other mathematicians say about him. But I'll humor you. How would you prefer we say things? - Highest IQ on record. - One of only 3 people to score over 700 on the Math SAT be
860.
by btilly 2y ago
When you've been stuck for 80 years, any improvement at all seems like a big deal. But more importantly, a new idea like this often points the way to variations that may also create improvement. Therefore as other researchers pile on,
861.
by btilly 2y ago
When I did that on a project I solved that problem by replicating redis to the same container that the ads were being served from. Replication was very fast, and all that was blocked was the local redis. I worked to make the matching rules
862.
by btilly 2y ago
The fact that the audience is so far is part of why I think that the attitudes of skepticism are more important than any particular set of facts. But it is also important that you make it easy to follow your explanation without having to ch
863.
by btilly 2y ago
Well, distributions like the Dirac delta qualify. But they can also be viewed as approximations of a continuum. Outside of that, maybe if you look somewhere like string theory? Still, in general, the continuum is enough. And while it takes
864.
by btilly 2y ago
Complete does not mean uncountable. Complete means that any sequence that looks like it is converging, actually is converging. Related to Hilbert spaces, consider the set of analytic functions with period 1. Under the L2 norm, this is an in
865.
by btilly 2y ago
I see your goal. But I question how you're going about it. First, I'm all for convincing people that the wave function is something real. If you believe that strongly enough, then you're pushed towards accepting the Everett I
866.
by btilly 2y ago
I got a lot out of your comment, but I don't know that Mars and Earth are directly comparable. - Earth has nearly 10x the surface area of Mars. This decreases the rate of escape. - Mars has lower gravity meaning escape velocity is roug
867.
by btilly 2y ago
If you know more than you let on, can you summarize the positions of the Platonists versus the Formalists versus the Constructivists? And can you explain how the word "exists" means very different things in each of those schools o
868.
by btilly 2y ago
> That depends entirely on whether or not real numbers can be physically represented in our universe. That is unknown. At least some real numbers can be. Take the construction of reals from Cauchy sequences of rationals. We can easily wr
869.
by btilly 2y ago
> It's pretty clear you didn't actually read all the way to the end. It's pretty clear that you aren't actually familiar with the historical philosophical debates from a century ago about the foundations of mathematic
870.
by btilly 2y ago
While interesting, that doesn't matter for what I said. An example of a key question is whether mathematical constructions can use the absolute truth of statements in mathematical constructions. Even if those absolute truths are not th

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