I am saying there is no reason to think they are independent.

Right....I am saying they prob don't exist so nothing you say about their nature is useful.

Fine. But Enrique said that these kind of statements if true have been proved to be provable. That is presently incorrect right?

Not totally sure what "these kind of statements" are but if you mean number theory statements that can be disproven by a single counterexample that can be checked algorithmically then their must be some that are unprovable. Specifically, there exists a Diophantine equation that will have no solutions in the integers but it is impossible to prove that in ZFC.

Are you referring here to maybe the MRDP theorem/Matiyasevich's theorem;

eg see here; https://en.wikipedia.org/wiki/Diophantine_set

"Matiyasevich's theorem says:

Every computably enumerable set is Diophantine.

A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x1, ..., xk) such that an integer n is in S if and only if there exist some integers x1, ..., xk such that f(n, x1, ..., xk) = 0.

It is not hard to see that every Diophantine set is recursively enumerable: consider a Diophantine equation f(n, x1, ..., xk) = 0. Now we make an algorithm which simply tries all possible values for n, x1, ..., xk, in the increasing order of the sum of their absolute values, and prints n every time f(n, x1, ..., xk) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x1, ..., xk) = 0 has a solution in x1, ..., xk"

and

"Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's theorem with earlier results known collectively as the MRDP theorem implies that a solution to Hilbert's tenth problem is impossible"


finally

"Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable.

One can also derive the following stronger form of Gödel's first incompleteness theorem from Matiyasevich's result:

Corresponding to any given consistent axiomatization of number theory,[5] one can explicitly construct a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization.

According to the incompleteness theorems, a consistent theory is incomplete, meaning the truth of some propositions cannot be established. The statement above says that this incompleteness must include the solvability of a diophantine equation, assuming that the theory in question is a number theory."

And my contention is that if you start checking for counter examples by hand you will get diminishing "chances" for success where the sum of those chances converge.

In other words you argue that if there is some Diophantine equation (or other number theory claim) that is of the Fermat (sparse) type (but not exactly that of course since we need it to be unprovable) in the way i previously described where for higher exponents the "chance" to hit gets astronomically small quickly, hinting no solution, where no counterexample can be found in early searches beyond which the heuristic non-rigorous probability argument would have you believe there is no solution (by "estimating a tiny probability" of a solution like eg 10^-1000 or less) and it also happens that this Diophantine equation cannot be shown to not have solutions within eg ZFC (some Diophantine will be like that eventually) then you argue that the higher theory within which is provable will indeed verify that there is no counterexample and complete the proof, therefore vindicating the probability heuristic claim?

But if that conjecture you make is true it can only be shown as a true theorem within the extended theory to be true not the original theory within which the original problem was posed. Because if it can be shown within eg ZFC or Peano (whatever the original was) we may have a problem if such example of properly sparse Diophantine equation appears unprovable and yet your conjecture that is now theorem can be used to settle it, rendering it provable and therefore ZFC itself inconsistent.

I don't think I mean what you just wrote. I'm saying that dense conjectures like Euler's are either provable or false. But you can't say that about sparse conjectures (unless you can say that about all conjectures.) This higher theory, inside ZFC stuff, seems like a side issue.

(I am still not clear about something. Did Godel imply that regarding specifically FLT, you could NOT (before it was proved) say for certain that:

you will either find a counterexample to disprove it in a finite number of trys OR

there is a finite sequence of English words that does prove it.)

In other words was it ELIGIBLE to be in the third category of no counterexample but no proof?

Did you mean that Euler conjecture? https://en.wikipedia.org/wiki/Euler'...ers_conjecture

I am trying to get an example of what you see as dense enough even if i am sure most of these examples will likely be provable (even if hard any time soon).

I want to see how you relate probability heuristics with the concept of dense in a real example to get your meaning to be seen explicitly even if the example wont be unprovable in all likelihood.

Also about FLT i dont think what Godel's theorems were saying anything about FLT.

It does eventually say though through work others continued about it that there will be a Diophantine equation eventually in any axiomatic number theory system without solutions that is unprovable to not have solutions within that system. (eg see my previous posts and links on page 2 that i think describe what dessin d'enfant was likely talking about)

Yes that's the conjecture I was speaking of. And its dense, because with simplistic thinking I get that each time you try out a new fourth power, say 10,000 the number of combos of three smaller powers, sort of 729 in this case, is a slowly decreasing fraction of that fourth power. Slow enough to get a divergent series. Its like monkeys trying to type Shakespeare, where each failure results in a slightly weirder typewriter.

I'm frustrated that it seems more technicalities than necessary are being brought into this.

So lets go back to "random" walks. They eventually return to the origin, in one dimension and I believe two as well. But not always in three. There they return only 34% or so.

So lets say George asks me for instructions as to how to walk in one or two dimensions. I tell him to use the decimal expansion of some number, digit by digit. My contention is that IF he never returns, there is something about that number that caused it and that there is a finite sequence of words that explains that cause.

But if he uses similar instructions to go on the three dimensional walk and he never returns to the origin might it be simply because 66% of all decimal expansions would result in this? So IF it is unprovable that this decimal expansion will never return to the origin my contention is that we can't be talking about one or two dimensional walks.

Unfortunately I think it's way more central than you realize....which causes misunderstandings like

All Godel proved was that there were certain true statements in say PA that can't be proved in PA. Godel statements in PA are trivially easy to prove in ZF (and Godel statements in ZF are trivially easy to prove in ZF +large cardinals etc) Sorry I, (and im guessing Enrique) might have mislead you earlier. I assumed somebody thinking about this stuff over the course of multiple years as you claimed would have known the logic 101 facts. In my defense even philosophy majors usually get it in a couple weeks .

But the metaphysical category of cannot be proven/disproven in a finite number of English words doesn't exist. Even AC or CH might be obviously true or false with better self evident axioms. When we talk about provable or not we are always referring to a specific semantic/syntax schema.

Dont be discouraged just yet in the possibility there is some value to your ideas if modified a bit.

Why was it necessary to talk about unprovable problems? I dont think most major open number theory problems have been shown to be unprovable. I think in fact Godel theorem is feared (to those not very seriously exposed to its true applications with examples) to be pervasive in major problems more than it deserves really.

I am hesitant in taking probability heuristics too seriously as we cannot treat every complex thing we do not yet understand its higher internal connections as random in ways we treat other things.

Still there is research on seeing how far some heuristics can be taken by studying properties of arithmetic functions (eg the distribution of primes). I wouldnt be surprised if it is possible under some proper treatment and after proper theorems to arrive as a machinery that can allow one in certain spots that pass a test to use such arguments up to point to establish bounds or to point to places that there might be an explanation (the finite number of words argument). That doesnt have to be related to Godel actually. It may still be applicable to simply unsolved problems to offer hints or to produce bounds.

I wouldnt be discouraged just yet. Just more careful. If indeed heuristics point in one direction there may be some arguments to the existence of a higher reason indeed for any discrepancy. That reason may not need an extension to be claimed provable.

Furthermore until a proper example of a Diophantine equation (like that in Matiyasevich's theorem) that cannot be proven it doesnt have solution within ZFC is offered to examine (what some probability heuristic might say about it) it may not be entirely unthinkable that unprovable cases share some statistical properties anyway (after care is taken to define these things like "dense" better).

Again, I think its lack of the basics you somehow see as needless "technicalities" that are causing you to be wrong. You keep saying that sparse conjectures could possibly be independent a la Godel statements. But you seem to miss that Godel statements are true and are fairly trivial to prove if you just add the axiom of infinity,large cardinals etc. So what you are effectively saying is that sparse conjectures are more likely to have really basic proofs, which I think is the opposite of what you think you are saying.

So why are all the unsolved number theory problems that I am aware of sparse?

For the same reason as why, until recently, we thought that viruses are all very small.

Because they would be solved if they were independent of PA like Godel statements.

I'm not saying that "sparse" unsolved number theory conjectures likely have simple proofs, you are.

Does whatever you are saying apply to conjectures about the digits in the decimal expansion of transcendental numbers?

of course

I didn't listen to the talk, but this is called True arithmetic and there are no unprovable and true statements. Of course there won't be a mechanical set of prove verification operations like you have for PA etc.

There are a couple of ways this can happen....depending on exactly what is meant by unprovable and true. Godel statements are sort of an example...."PA cannot prove this statement" which is true if PA is consistent, but this can be proven in ZF so unprovable only means "unprovable in PA".

Another way would be if something like Goldbach was independent of PA....that could mean that Goldbach is true for the standard natural numbers 1,2,3,4.....but you could construct a non standard model of PA 1',2',3',4'.... where Goldbach would be false. But this would basically amount to a proof that Goldbach is true, because Goldbach is a statement about the standard integers and not about all systems that are models of PA.