wtf

LOL. Nevermind. Here's the answer you were hoping for. "Congratulations. Enclosed find check for US $1,000,000 for solving the P = NP problem. Your Fields medal is in the mail too." All the best.

I suggest starting with the link I gave in my previous post, about countable and uncountable sets. After that you might consider learning some basic computer science. I'm afraid I can't add much to what I've written. You need to master the basics.

If P is infinite then what can "Compa(p) = Order(P) - 1" possibly mean? Since P is infinite, is Order(P) - 1 any different than Order(P) + 47? Do you understand that by definition all algorithms consist of a finite sequence of steps? You are demonstrating a complete lack of basic knowledge of the subject areas you are attempting to work with. Why not just pick up the basics? Start here. https://en.wikipedia.org/wiki/Countable_set. We'll work up to basic computer science later.

Have you considered studying the very basics of computability theory and infinitary set theory before tacking P = NP? Do you understand my point about the countability of the set of Turing machines?

Aren't each of P and NP countable? After all there are only countably many Turing machines.

Your entire chain of reasoning is based on the sets being finite. But the complexity classes P and NP are infinite. How do you know any of your prior reasoning still applies? And why won't you answer the most basic notational questions?

You won't even answer direct questions about your ambiguous, confusing, and undefined notation. Why should I read yet another version of your paper?

In definition 1, can you define the notation [math]x|y[/math]? I'm guessing it's not the usual divisibility relation, but I can't figure out what it is. Also, having spent most of the paper discussing the case of finite sets, you then pivot to the P = NP problem, which involves infinite sets. How do you justify that?

Actually that's true with probability 1, but it could still happen. Just as you can toss infinitely many coins and have them all come up heads. The probability is zero, but it could still happen. After all, the probability of ANY particular outcome is zero. Say we let the monkeys type away for an infinite amount of time. We look at their exact output. The probability of that exact output is ... zero!! It's no more likely than the complete works of Shakespeare! The psychological reason we don't appreciate this point is that we are comparing the probability of gibberish, which is 1; to the probability of the works of Shakespeare, which is zero. But whatever specific outcome does happen to occur, the probability of that exact outcome was also zero.

Glad I was able to express myself more clearly. I'll leave y'all to the calculations. Did you know that under laboratory conditions, coin flips are not random? People make hidden philosophical assumptions about probabilities, without even realizing it. There's nothing random about a coin flip.

I don't mean to interrupt the math fest. But I am making a point about the philosophy of probability. Suppose I say to you that it's astronomically unlikely that you exist. I point out the incredible unlikeliness that earth formed, that life developed, that your parents met, that a particular sperm cell fertilized a particular egg, that you were carried to term and lived to your current age. Why, the odds must be something like .00000000000000000000000001 or even smaller. And you reply, "Don't be silly. Of course I exist. Here I am!" Who is right? The person that says it's unlikely that you exist? Or the person who observes that it's certain that you exist, you're standing right there. This is not a trivial issue in probability theory Now if you want to calculate the odds that a randomly selected sequence of characters represents the works of Shakespeare, the mathematical arguments are valid. But if you want to say it's astronomically unlikely that Shakespeare did what he already obviously did (modulo historical arguments that say he didn't write his own plays) you have a harder case to make. So: A philosophical diversion, to be sure. But a joke? Not in the least. A rather serious point. Bayesians versus frequentists. https://stats.stackexchange.com/questions/22/bayesian-and-frequentist-reasoning-in-plain-english

The first confusing remark came pretty early: > We define [math]f[/math] as a function [math]f_B(x) : A \to f_B(A)[/math] * You didn't define the notation [math]f_B[/math]. * What is [math]x[/math]? * What's the difference between [math]f_B(x)[/math] and [math]f_B(A)[/math]? * How am I supposed to interpret this notation? Also why do you say that [math]B[/math] is a set of size [math]v[/math] then never use [math]v[/math] again I mention in passing that the moderator has already asked you to express your ideas here and not solely via an external link, so I don't mean for my comment to be encouraging you in flouting the moderator's instruction. But frankly your exposition is incoherent and the problems start right at the beginning.

You're joking, right? How long did Shakespeare take to write his plays? Is he a primate or not? Am I stretching a point? Not by much. The posterior probability that a primate typed out the complete works of Shakespeare in less than 50 years is 1. It already happened.

No I didn't say that. All heads has zero probability. Exactly zero. It's just that probability zero events can occur. In fact since the probability of any particular sequence whatever is exactly zero, a probability zero event MUST occur. I haven't followed the thread so I can't comment on your original question. I only jumped in to clarify probability zero events.

It's the standard approach to probability. Think of it this way. First think about 3 flips. The odds are 1/8 of getting any particular possibility: hhh, hht, hth, htt, thh, tht, tth, ttt. The odds of any particular sequence are the same. Now what if there are infinitely many flips? The odds of any particular sequence are the same. Zero. But some sequence must occur. Therefore it's possible for probability zero events to occur. This is basic to standard probability theory. https://en.wikipedia.org/wiki/Probability_axioms Another way of imagining this is to pick a random real number in the unit interval. The probability of picking any particular real number is zero, but all the real numbers are there and some real number must get picked. In infinitary probability theory, probability 0 events may occur and probability 1 events might not. That's what the phrase "almost all" or "almost surely" means.

50 years. William Shakespeare's lifetime. Or 4.5 billion years, the time it took evolution to produce Shakespeare.

Consider a Mac. Unix under the hood which you'll need to know for cybersecurity. Of course you can always dual-boot your Windows system to Linux but you should look at some Macs anyway. They have some nice laptops and some people much prefer the look and feel of the UI to Windows. Also you can run Windows on Macs too. Just something to consider.

Will this be on the test?

Ok, I feel better now. Somebody using your handle wrote: But now you are saying you WILL teach them the quadratic equation. And if we're going to teach them the quadratic equation then we MUST teach them completing the square, because that's the trick that makes it possible to UNDERSTAND where the magic formula came from. You can't fault me for reading what you wrote, right? But if you're putting the quadratic formula back into the curriculum, along with the method for re-deriving it if we happen to forget it, then we're on the same page. I agree with the OP that math education is terrible. I don't really know what to do about it.

It'a a problem. Not every kid will be a mathematician, physicist, or an engineer. Or a biologist, or economist. Anything quantitative. But SOME kids will go on to those fields. I don't think you want to separate kids at age 8, "This one goes to math classes, and that one goes to trade school and learns to make change." Or whatever "practical" skills are. Should we not teach kids Shakespeare because not everyone will teach English lit? Anyway Shakespeare's a dead white European male so nobody needs to read him. If you wouldn't teach high school math to high school students, what would you teach them? And why would you eliminate the learning path for the future engineers of the world? We all have to drive over the bridges they build. We should educate them, don't you think? Am I misunderstanding your point?

http://www.dcode.fr/lagrange-interpolating-polynomial

That seems a little vague. If a function is [math]C^{10000}[/math] but not [math]C^{10001}[/math] then it's not smooth, but we'd be hard pressed to say it has corners or changes abruptly. Of course a function with a big obvious corner like the absolute value is not differentiable at zero, but infinite differentiability is much stronger than mere differentiability. And the difference is not always visible to the naked eye.

Of all orders. [math]C^\infty[/math]. Differentiable once, twice, thrice, etc. Even Xerxes and I agree on that But note that there is a [math]C^\infty[/math] function that is not real analytic. In other words it has derivatives of all orders at some point, and we can form its Taylor series at that point, but the function is not equal to the Taylor series. So if you need smoothness it's good to say whether you mean [math]C^\infty[/math] or real or complex analytic. https://en.wikipedia.org/wiki/Non-analytic_smooth_function

Two short points in lieu of a much longer post. * Math can not prove anything about the world. Math models Euclidean geometry and non-Euclidean geometry. They can't both be true. They're both logically consistent. Math has no way to say which might be true about the world. Physics is the discipline that seeks to determine truth about the world. Physics uses math to build models. But the math doesn't care. You can use math to model multiple realities that are inconsistent with each other. The only requirement of a math theory is that it's internally consistent and interesting. True or false doesn't apply. * Simulation theory is bunk. Many reasons. If that's your question, it certainly deserves a thread. But it's a separate question from whether math tell us what's true or false in the world. But now that you mention it ... one of the strongest arguments against simulation theory is that it IS essentially a theological or metaphysical claim. Simulationists are seeking God and pretending to be rationalists. "God isn't the Baby Jesus, it's a big computer in the sky." Give me a break.

> I knew Prince William would have a boy and name him George. Yes but do you remember the things you believed that DIDN'T happen? The only scientific test is to formally write down predictions on paper and then see how you do after the fact. Otherwise you are subject to selective memory bias. I remember the longshot horse I saw in the paddock and just KNEW he'd win so I put down five bucks and won a hundred. Of course I don't remember all the run of the mill losers I've had. Ask any gambler, they all think they're precogs too.