I am not bothered by Godel's incompleteness theorem. First of all, it is better to have an incompleteness theorem where we know that our axiomatic system cannot prove everything than to not have an incompleteness theorem and not know whether our axiomatic system can prove everything or not. The correct response to Godel's incompleteness theorem is to look at Godel's second incompleteness theorem to tell us how to strength our axiomatic systems so that we will be able to prove more results. Godel's second incompleteness theorem states that no axiomatic system stronger than Peano arithmetic is allowed to prove its own consistency. Godel's incompleteness theorem is good because it gives us a direction to go to strength our axiomatic systems. If we want to strength an axiomatic system A, then a natural thing to do would be to work in the system A+Con(A) which states that A is consistent. We can iterate this process to obtain A+Con(A)+Con(A+Con(A)), and so on and so forth. While iterating this process does yield stronger axiomatic systems, this process of iterating the consistency is rather inefficient and cumbersome. First of all, if we iterate the process that gives us A+Con(A)+Con(A+Con(A)) finitely many times, we won't get very far, so we will need to iterate this process transfinitely. But we can do much better than this. Godel's completeness theorem states that an axiomatic theory is consistent if and only if it has a model. Therefore, by combining Godel's second incompleteness theorem with his completeness theorem, we conclude that a strong axiomatic theory cannot prove that it contains a model of itself. Therefore, in order to strengthen an axiomatic theory, we can add an axiom from which we can obtain a model of that theory. And by adding axioms about models, we can get better strengthenings of our axiomatic theory. For example, if we add an axiom stating that there exists a well-founded model of ZFC to the ZFC axioms, then this new axiomatic theory is stronger than what we would obtain by iterating the process of ZFC+Con(ZFC) transfinitely. We can do even better than this though. Large cardinal axioms are much stronger strengthenings of the standard ZFC axioms that easily encapsulate the process of transfinitely iterating the consistency hierarchy A,A+Con(A),A+Con(A)+Con(A+Con(A)) and much more. Not only do large cardinal axioms provide strengthenings of ZFC, but one can prove interesting theorems from these large cardinal axioms including theorems about finite structures. The only catch is that if we go too far with large cardinal axioms, then we will end up with an inconsistency (such as Kunen's inconsistency). I am personally confident that all large cardinal axioms up to rank-into-rank cardinals are consistent. If anyone is able to prove that the existence of n-huge cardinals for all n is inconsistent, then I will forfeit all of my cryptocurrency.
I do not know much about the JWST, but I have no reason to believe that it is a waste of money.
-Joseph Van Name Ph.D.
You truly are a brilliant mathematician Dr. Joseph. I would imagine there's very few mathematicians that would understand all of what you just discussed. Of course, I'm woefully inadequate to understand it but I'll be googling some of the things you talked about to try and get a better understanding but wow. You are one amazing mind and thanks for your 2 cents (200 more cents more like it!). If foundations of mathematics is not your specialty research area then I would be flabbergasted since you sound so authoritative in your knowledge of it. Thanks so much for your lengthy response, I really do appreciate it.
All I can say about the incompleteness theorems in my limited understanding of them is I just don't understand how there could ever be a concrete example of a statement that is true but cannot be proven. Because from what it appears, all that "true" means is "cannot be proven with the set of axioms in the system". So you could either add the statement itself or its negation to your axiom system and I guess it would still be consistent. I don't see how that could work with something like the Twin Prime Conjecture or The Collatz Conjecture. The Twin Prime Conjecture is either true or false. That truth exists we may just not know what it is. So because of that issue, the statement that there are infinitely many twin primes doesn't seem like a feasible candidate to add as a new axiom of arithmetic since we don't know if it is true but the truth exists one way or the other about it so if we choose the wrong version of the statement we are going down a wrong path.
Given some arbitrary statement that one does not know how to prove and seems challenging to prove, how do they go about proving it is not provable in their axiomatic system? I would think that's impossible. For example the Riemann Hypothesis.
One other quick question, have you ever heard of Dr. Norman Wildberger I think that's his name. He has made alot of videos criticizing the "real numbers" as though they really don't exist and he doesn't think there is a valid construction of them, just handwaving. Do you think the real numbers have a solid foundation? or do they have issues. he maintains that things like dedekind cuts and cauchy sequences as ways of constructing the real numbers are flawed. to me that's kind of troubling since higher math is all based on you guessed it, the real numbers!
I have not heard of Norman Wildberger, but there are some mathematicians who criticize basic structures like the real numbers or even the natural numbers. And they do not have much of a reason to do so. We cannot even find an inconsistency with extremely large cardinals, so these extremely large cardinals probably exist. So since extremely large cardinals work just fine in practice, we should accept that there is probably some legitimacy to them and to the real numbers as well. One can say that the real numbers and large cardinals may exist but only in a countable model (this is guaranteed by the Lowenhein Skolem theorem), but that unnecessarily complicates the issue. I do not see any reason why large cardinals would evade all attempts at finding an inconsistency while they only exist in a countable model and cannot extend to larger models.
So far, large cardinal axioms have not helped with problems like the Twin prime conjecture, but I have used rank-into-rank cardinals to produce some falsifiable statements about finite algebraic structures, and I have ran the computations trying to falsify these statements myself, but I have not been able to find any inconsistency even after about a million attempts on a computer. Large cardinal axioms may not allow us to prove everything like the twin prime conjecture, and we may not be able to keep on adding stronger large cardinal axioms since mathematicians have not been able to formulate large cardinal axioms much higher than rank-into-rank; there are some cardinals axioms that are much stronger since they imply models of rank-into-rank, but these stronger axioms are not consistent with the axiom-of-choice. If we do not abandon the axiom of choice, then the only way that I know to make larger cardinals is to take something like limits of cardinals, but this idea does not seem to go very far since we do not know how to get them to be as strong as the axioms that are inconsistent with the axiom of choice.
Why are you begging for money? Get a job scammer.
Everything you say is f@#$ing stupid.
-Joseph Van Name Ph.D.
Says the guy who's using ChatGPT to formulate his topics
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You are worthless. The Lord Jesus Christ hates your soul and will send you to Hell after the next 5 pandemics kill you.
-Joseph Van Name Ph.D.
