viraghr

>It is more natural to talk about primality as a test that can be done independent of any assumptions about other primes, but rather as a matter of whether it can be expressed as the multiple of some number other than 1 and itself

This is clearly not what I was doing. I clearly referred to having no smaller prime factors. Anyway this aside is tiresome, it's like poking me for saying "every positive integer has a prime factorization" and then asking, okay, so what about 1 or something. I think my proof is fine and I'm not going to defend it anymore.

100 possibilities are really easy to test, though, so P must be fully available near the end. We can all agree it's not available at the beginning (subject to the methodology.) So it becomes more and more available over time, but that is not what the EMH says -- at all. If you are saying market prices will reflect information as a function of how "readily" available it is (for example how smart you have to be to see it), that is just not EMH.

For the drug trial example, again, I like it but perhaps such an expert is theoretically excluded? Your statement might be like saying "suppose there's a hypothetical O(n) halting algorithm that answers the halting problem in constant time for the length of the program it's testing." That's nice but there's a proof that what you've just hypothesized is impossible[1].

I am not exactly saying that an ideal expert in human biochemistry who can predict the results of drug trials is impossible, but how do we know it's possible? I mean, is chemistry guaranteed to be fully computable or something?

For your last point, I think that what I asked to assume (that the $1B transfer is credible, that the computer is secure) are nowhere near the size of assumption of something like "assume there exists someone who can predict biochemical reactions in humans without testing." Basically the legal instruments and secure computers I included are easy, solved problems that I ask the reader to assume are being implemented following best practices.

[1] https://en.wikipedia.org/wiki/Halting_problem

Andrew, I perfectly see what you're saying, but I am telling you it is absolutely no problem that there's a false statement.

Remember that we are arguing from the false assumption that you start with the finite set of all primes, up to some largest prime, L. Call this false assumption A. It's fine to make false statements that follow from A, no problem at all. Indeed this is the point.

One definition of a prime is that there are no primes smaller than it in its prime factorization. We'll call this definition NSPIPF - no smaller prime in prime factorization. Is NSPIPF an OK test for primality? Sure.

So when you get to new_number, produce the prime factorization. Does it meet the definition of NSPIPF? Yes, because we made it relatively prime to every prime (under assumption A).

It passes primality test NSPIPF. Under A. And therefore is prime. Under A. (This is the part you and others object to, but it's absolutely flawless application of the NSPIPF primality test.) It doesn't matter that in some other way I could also get to a contradiction. For now this is what we do.

Under A, using NSPIPF primality test, we just proved new_number is prime.

Next we show that this new_number definitely wasn't in the set of all primes, since it's larger than L, having had L and a bunch of other positive integers as factors and then one added to that for good measure. It is at this point that we show explicitly that A cannot be true, because we used A to produce a prime that wasn't in the set of all primes.

It doesn't matter if that was a false statement!

I think all this is in my original comment and it is a flawless proof by contradiction. I asked you to "Suppose there are just finite primes, up to some largest" and you gave up when you started seeing false statements, rather than at the end of the paragraph where I presented the conclusion in black and white.

Sure, but you can't "amend EMH" to fully reflect all the facts that are "feasible to compute" any more than you can amend it to say that that asset prices fully reflect the "best feasible analysis". That's not EMH.

For the prime proof, the new number is prime as long as as it has no prime factors < itself, which we assumed at the start. This guarantees it's prime under our assumption and this falsifies the assumption. Doesn't matter if it's really prime.

Thanks, very charitable reading but doesn't work because next sentence I said that the new prime is larger than largest. As other poster points out if 2,7 are all your primes you produce 15, whose primes are not larger than 7 so the "new prime" wasn't referring to one of them.

But that's not how I argued. Instead I showed that 15 is the new prime because it's relatively prime to all the primes. (By the assumption we had started from all the primes.) And this is what falsifies the assumption.

I think my argument is fine and I don't need to address whether 15 is really a prime.

If you can show an inefficient market experimentally whenever you want, that's good enough to disprove EMH. The EMH doesn't say, "All information is normally factored into the price, except for technicalities."

I like your drug trial example but a "hypothetical ideal expert" isn't assured to exist. Do you have an existence proof?

On the other hand, unique prime factorization is assured, we know that given infinite time anyone can factor any large number.

Suppose 2 and 7 are the only primes < 15. Then 15 is prime because its prime factorization couldn't contain any other number (2 * 7+1 makes it relatively prime to both 2 and 7 which under the supposition are the only primes < 15.) This is true if we suppose 2 and 7 are the only primes < 15, and that's just what the word "suppose" asks you to entertain.

This little mental exercise proves that 2 and 7 can't be the only primes in existence. It doesn't really matter what 15 really is or isn't. No need to add noise about it. We only care about the supposition, which you cut out of your quote.

Hope this explains why I didn't deal with the true status of new_number! It just doesn't matter.

However, I'm disappointed in you, in the other respondent, and in the moderation here, and I'm glad I made this alternate account.

Have another account here but don't want to take the karma hit for being a crank. I published this 2-page proof that the EMH is false:

https://arxiv.org/abs/1011.0423

(didn't set the date so in the document it's wrong, this was published 2010.)

It doesn't depend on P = NP, it's simply a rigorous proof that EMH is false.

Let's switch gears a second. Here's a famous elementary proof[1] that there are infinite primes. Suppose there are just finite primes, up to some largest. Multiply them together and add one. No prime divides the new number (because "every" prime leaves a remainder 1), so you've just produced a new prime. This new prime is larger than the largest prime in your finite set because you multiplied that by the rest of them and added one to get it. So this is a contradiction, you couldn't have had finite primes up to some largest.

Anyway if you think there might be a largest prime after what you just read, it just means you don't understand the proof. If you believe EMH might be true it just means you don't understand the proof that it is false.

Of course, nobody ever even hypothesized that academia was efficient :)

[1] https://en.wikipedia.org/wiki/Euclid%27s_theorem#Euclid's_pr...

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EDIT: no mistake in my comment