porton

@wtf You misunderstood me, I do not want spend time to explain why it is correct. You can check yourself by writing a first-order predicate calculus software. I don't want to spend time on explaining this.

"But {n∈N:n is prime}∈P is NOT a valid statement, because it mixes up the metalanguage with the model." You are wrong: $X\in Y$ where $X$ and $Y$ are sets is always a valid statement.

"Strings talking about sets are not sets." String talking about sets are sets in a metalogic, that is in another "instance" of ZF. So, it seems that we need to differentiate between different instance of ZF... Math logic book authors don't enforce this rule in their metalogic. This is an error. Error of them. "But the string "n in N such that n is prime" is not a set!" True, but the string "{ n in N : n is prime }" is a set comprehension. "I don't think you can ask if "n in N such that n is prime" is an element of the set of primes." The string "n in N such that n is prime" is an element of the set of numbers (if we encode strings in numbers). So, in this sense, we indeed can ask if "n in N such that n is prime" is an element of the set of primes. So, the issue is in which logic we can and in which logic we can't ask this. Textbook authors assume we can ask questions like this in ZF metalogic, and that leads to a contradiction, as I showed below. "Well yes all the statements have Gödel numbers but I don't think it works this way. I don't think my knowledge of Gödel numbering is good enough to point out exactly what's wrong with your argument." I don't understand you. "If I help you can I share the million dollars?" How are you going to help? If the help is great, sharing say $500000 may be reasonable.

I messed the terminology: I should have said "set comprehension" instead of "set-definition". My paradox with set comprehensions is an "improved" form of Russell's paradox. Russell's paradox applies to the pre-ZF wrong "native" set-theory, my paradox applies to the metalogic commonly used together with ZF. BTW, a paradox similar to this applies to predicate theory, too: Moreover, even if we consider just predicate logic rather than ZF, the error is still here: Let P be Godel's (or other) encoding of a one-variable predicate. Then define the one-variable predicate S(Q) encoded by Q as S(Q)(P) <=> not (S(P))(P). Then S(Q)(Q) <=> not S(Q)(Q). Contradiction! "But is a "set definition," whatever that is, and NOT a set" (should be "set comprehension"). You are wrong, because in ZF every object is a set. "Or perhaps a set definition is a specification like {n∈N:n is prime}." Yes! "But in that case a set definition is a piece of syntax, a string in the formal language. It can't be an element of some set that the language is talking about. You're mixing syntax with semantics, formal strings with models." In ZF strings are defined as a kind of sets. So it can be an element of some set that the language is talking about. I do mix syntax with semantics, formal strings with models; but that math logic authors do the same is a security hazard. BTW, I found this "error in logic" while trying to solve P = NP. I proved P = NP, but soon found that I use this erroneous logic. Well, then I rewrote my proof of P = NP without this error and... proved P = NP again! Here is my proof of P = NP (please check for errors): https://math.portonvictor.org/2021/06/28/a-proof-of-pnp-using-a-merkle-tree-a-new-version/ Direct link to the latest version: https://math.portonvictor.org/wp-content/uploads/2021/06/pnp-merkle-4.pdf Humorously, I needed to edit my erroneous (by using the logic error) proof of P = NP rolling changes back after I "simplified" it using this logic error, before I produced the above (probably correct, in my opinion) proof of P=NP. "Accordingly the legend, the way to proof of P=NP passes through a logic error." If it happens that my proof of P=NP is correct, I did two great discoveries at once: bug report to logic textbooks and proof of P=NP. Well, at least about the bug report, I am sure it is a correct bug report.

This shows something that appears to be a contradiction in ZF: https://arweave.net/RJ4DRuRjdVWqJ5RBHB30SXDBXATzqmAV3Qs1b6f1ykw (Note that by set-definition I mean it's numeric code, in an encoding such as Godel's encoding.) What the heck? What's wrong? --- It seems I understood why the error: S is a function in a model of ZF, not in the formal system the proof is written. So, it's illegal to use it in the proof. That's a very beautiful sophism. What happens if we try to define S inside ZF? It should fail (unless ZF is contradictory). But how does it fails? That is an practically useful question, because I really made this error today. --- Digging deeper: If we would be able to prove that S exists, this would be enough to trigger the contradiction in ZF. If we attempt to prove its existence, we try to map the set definition { x in A | P(x) } into the corresponding set, but we to fail to do that. Note that this paradox is pertinent not to only to ZF but even to predicate logic: Let P be encoding of a one-variable predicate. Then define the one-variable predicate S(Q) encoded by Q as S(Q)(P) <=> not (S(P))(P). Then S(Q)(Q) <=> not S(Q)(Q). Oops! Mathematicians commonly do this trick: Let S be a function from some expression X of our formal system to another expression S(X) of our formal system. It is pertinent to logic studybooks! And that was an error! Possibly millions of math logic articles are wrong!

Small errors corrected.

Please check my proof (PDF) of P!=NP for errors. Small error: I should speak about "coherence" of all members of a set rather than of a set. I've found an error myself: It should be log w not w. So the proof is wrong. There were errors in the proof. Here is a corrected (even shorter and more elementary) proof:

"What revolt?" - https://gowers.wordpress.com/2012/01/21/elsevier-my-part-in-its-downfall/

Could you please repeat your question? If you are about my definition of generalized limit, I gave it above two times.

Yes, my limit is kinda (in some sense I myself define) is multivalued. However, that depends on the exact definition (out of several equivalent ways). I am repeating (it is present above) one of my definitions of generalized limit (it is not a multivalued function, speaking formally, but it is multivalued in some important sense that is not easy to explain quickly, or you want me to retype my entire manuscript here?): Generalized limit is a function from ultrafilters (including the improper one) "nearby" a point into limits of the function at these ultrafilters. I do know that the "traditional" limit does not exist in this case. My set of "shifts" (not of shifts but of results of shifts) does not "map" to the real number system. Limits in my system are something like infinite numbers. It is an extension of the real number system (in the case if our space is the real line). For example generalized limits at zero of 1/x, 1/x^2 and 1/x^3 are different "infinite numbers".

> Can you reproduce any piece of known physics with your theory? Let's say Coulomb's law, or Newton's law of gravity, or the like. Coulomb's law - no - its about gravity only. Newton's law of gravity - most likely yes, but calculations need time to spend on. One of the outcomes I consider likely that the "external" (anything except of the point of a singularity) effect of my theory is exactly the same as of GR. The difference may be (likely) that there is information in singularities.

I clearly told that in my system every function has a limit. In this case it's the set of all shifts of the funcoid taking the value 0 at the left neighborhood of 1 (including 1) and 2 at the right neighborhood (excluding 1). Well, I forgot to tell that my funcoid is to be topologically "smashed" vertially.

I am repeating: Here is my discontinuous analysis: https://math.portonvictor.org/binaries/limit.pdf that is based on my another (that one about 400 pages) text. Here is my modified GR that uses discontinuous analysis: https://math.portonvictor.org/2020/01/31/an-infinitely-big-structure-in-the-center-of-a-black-hole/ My 400 pages text did not fail peer review: Here it is published by a reputable scientific publisher: https://znanium.com/catalog/document?id=347707 OK, my theory in short: We can define limit of every (even discontinuous) function in several equivalent ways, for example: Generalized limit is a function from ultrafilters (including the improper one) "nearby" a point into limits of the function at these ultrafilters. Then my text goes into such details as other ways to describe generalized limits and arithmetic operations on generalized limits. For this I define something I call "singularities" (not to mess with the usual usage of this word) that is infinitely bug values like the value 1/x takes near 0 in my theory. We can define generalized (partial) derivatives simply by replacing limit by generalized limit in the definition of derivatives. So, every differential equation could have solutions that could consist of "signularities" (rather than e.g. real numbers). After restricting these solutions in a reasonable way (see the actual text for details), we get a new interpretation of every (partial) differential equation, including a new interpretation of GR. For GR I propose the following (exactly formulated) mathematical model: We require the solutions to be pseudodifferentiable in timelike intervals. (We do not require the solutions to be pseudodifferentiable in spacelike intervals.) So, we have a new modified GR, possibly with some infinite structure in the centers of blackholes. It seems likely that my model preserves all information. "IOW, put up, or shut up !" - treating me like an animal is...? Excuse me, I can't post formulas here, there is no LaTeX! Well, as a part of the well known mathematicians revolt: LHC is maybe a small thing compared to discontinuous analysis.

No, not great: I am a general topologist. I am not a partial differential equations expert. @beecee I think you most likely would solve this much faster than me (even counting the time to read my above mentioned article). "Do... maths" I estimate the probability that in my theory black holes form very similarly to GT as 90%, that my model preserves information 50%. Then assuming my theory is such, the probability that Hawkings's theory is right and that mine is right are equal by my estimation. But they can be both true (I mean their "combination" to be true, and likely this combination is much simpler to find than e.g. QG theory), so 75%. Calculating 90% * 50% * 75% = 33.75% that LHC produces non-bursting blackholes (in the assumption that it produces balckholes - was this already proven?) 33.75% of eating the Earth by blackholes. @beecee calculate faster than me, please.Meanwhile, guys, could you please turn LHC off till my discontinuous analysis publication succeeds? That would be a reasonable outage given the importance of the problem.

I am a general topologist. I have my own theory of preserving information by black holes. (I have formulated my modified math model of general relativity and it is likely in my opinion that in this model information is preserved, but I didn't do calculations whether the model really preserves information yet, because my research topic was general topology, not physics.) The consequences? If we have an alternative explanation, the Hawkings's theory may be wrong. Isn't it so? I hope that both my theory and Hawkings's theory are correct (in the sense that they to be combined in one unified theory.) But if it happens (we don't know) that my theory is the reality and the Hawkings's one is not, then blackholes don't burst (most likely, I didn't calculate yet). I recommend to stop LHC now! https://math.portonvictor.org/2020/01/3 ... lack-hole/ describes my theory, a modification of Einstein's equations (well, not of the equations themselves but of their interpretation). Comment!! https://math.portonvictor.org/binaries/limit.pdf is my theory of "generalized limit" and another meaning of any partial differential equations (including the Einstein ones).

So, I've posted to a physics forum, but it is still pending moderation.

Yet humor: Scientists: Consider limit of a function on an arbitrarily chosen (and impossible to be pointed concretely) ultrafilter except of the principal ultrafilter "near" given point. The result depends on this incomprehensible for finite creatures choice. Me: Consider all limits of a function on all (ultra)filters (including the principal ultrafilter) "near" a given point. Yet humor: Scientists: The properties of operators on a normed space are similar to properties of topological spaces... Operators are actions of semigroup... This semigroup is ordered. Me: Consider actions of ordered semigroups. That's a common generalization of topological spaces and operators on a normed space. Yet: Scientists: There are several kinds of continuity, defined in different ways, having in common, well, the word "continuity". Me: All kinds of continuity are foa<=bof for semigroup elements f, a, b and its operation o. And: What is science development discontinued by unlimited idiotism? When we lost generalized limit defined for every discontinuous function. Yet: Student: Defining Lipshitzs derivative is a complex topic. Me: f'(x) = lim_{r->0}(h|->(f(x+rh)-f(x))/r)). Yet: Hawkings got Nobel prize for finding the only explanation of black holes preserving information. Me: Another explanation (yet not mathematically checked, because I work alone). Oh, a new thought I never had: LHC scientific measurement system produces small black holes that accordingly Hawkings's theory quickly burst and therefore don't devour the Earth. If not Hawkings's but my explantion happens to be right... They most probably don't burst at all... and devour the Earth.

Yes, but the trouble is that nobody (except of Todd Trimble that wrote a short comment) and about two prospective PhDs that referred to me without any quotes and any reason to refer except to refer to somebody to increase the count of literature references in their theses, that doesn't count. To simplify your work further, I say: To verify that I did a big scientific discovery, it's enough to read the very beginning of the PDF, because it is enough to know that I did found a new simple axiomatic system. Discovering a new simple and "elegant" axiomatic system is a big discovery in any case: either if it was thoroughly and correctly researched further or not. I claim that my book researches it correctly (small errors are possible, but that does not invalidate the entire stuff in my book) and rather thoroughly, but that's mostly irrelevant for the sake of this thread discussion. By the way, I found also another simple axiomatic system: Oversimplifying my ideas, I found axioms for "finite and infinite formulas". That's the joke about an old lady (mathematicians) that saw everything except the glasses (formulas) but lost the glasses themselves sitting on her nose (not discovered axioms about formulas). Yet another my discovery is that I am the first who put words "ordered semigroup actions" or "actions of ordered semigroups" (and researched the properties of this three-words phrase), while before me there were only two-words phases "ordered semigroups" and "semigroup actions". That sounds funny, but putting these three words together is a big discovery (but more is that I found a connection between these three words and general topology). You can check this my claim using Google. Not to contribute to the discussion but to add some humor: Scientist: What else research topic to think about? Advisor: Think out of the box! Scientist: Which box? Advisor: You have some mathematical object D. Think out of the box D(x), instead apply it to D itself, so write the formula D(D). Scientist: What D would be exactly? Advisor: Think about as many different kinds of formulas as possible! Me: formula(formula). More humor: Scientist: We have the definition of uniform space: A filter on a binary Cartesian product + some axioms. To make it more general, we should remove some axioms. We are investigating about last 50 years which axioms to remove. Me: A filter on a binary Cartesian product.

Thank you for your reply, I am doing your request that is I am providing details: Here is that my now >400 pages math article: https://math.portonvictor.org/binaries/volume-1.pdf (The article does not contain some of my newest discoveries that I decided to keep to myself because the extrapolation of what I said in the original post witnesses that publishing it further could make things worse.) The thing that is (in a sense) more general than group theory is my definition of "funcoid" using small delta (see the above text). It is more general because it does not use functions (a second class object in ZF) but only sets and relations (first class objects in ZF). However, TBH, my definition has 4 axioms rather than 2 axioms of group theory. Also, funcoid can be defined equivalently using one axiom (but with more high-level objects). The above text misses my later discoveries: discontinuous analysis and "space in general" (well, not quite in general, but in general topology). (I was afraid to publish further because of extrapolating this ill-effect to my future publications.) Here is the Russian peer-reviewed publication of an older version of the same long article: https://znanium.com/catalog/document?id=347707 Another relevant fact is that I was essentially banned from arXiv after their moderators lying to me. (That is probably a result of them being uncareful.) The most relevant aspect of that ban is that they provided no explanation at all of the reason of their effective ban, so I have no idea if they think I am a crackpot or no, etc. Maybe the reason was just that I published too many articles in one day. What else do you want to know? Oh, one more relevant detail to simplify your validation of the facts: Here a famous established expert professor claims (well, implies) that my concepts are mathematically correct: https://ncatlab.org/toddtrimble/published/topogeny Well, this professor does not value my discovery as a big one - opinions of different scientists on importance of some discovery may be different. I claim that he is very wrong in not considering my discovery as a big one and can give persuading arguments. To make your task even easier, I will explain what the above referenced PDF file is: It is absolutely usual research article on the topic of fundamental mathematics except of just two things: It is unusually long. It was put online about the end of 20th century, but it would be a typical 18-19 century text except of its length (no idea how scientists "succeed" to miss this research topic.)

An amateur discovered a theory that in a significant relevant sense is more general than group theory. The amateur wrote a very long scientific article (~400 pages), put Creative Commons on it and then mis-published it (this time instead of publishing in a predatory journal, it was published in a Russian scientific site with no English UI to purchase). So, the long article has very few downloads. Nobody does research on this topic, because scientific priority tradition forbids publishing on others' research topics. To made things worse, it was also discovered discontinuous analysis that relies on this fundamental theory. So the world almost fully lost both this foundational axiomatic theory and discontinuous analysis. This essentially means no future science. If you were a politician with power to decide, what law would you set? Canceling intellectual property laws seems not to help in this particular case: The long article is open access. The main issue seems to be in it being amateurish. So, it looks like that the solution would be to remove the concept of being an amateur. It is equal to removing the concept of scientific degrees. So, should we ban the words like PhD? But somebody would invent another word, so I see no reason that banning word PhD would solve this problem. Your proposals?

My proof contained a big error. Nevermind.

That's a bug of the forum software, @joigus

This proof in details: http://arweave.net/S-o7FdXfiHu9waP0T4SAJ1vpNAQiBQzpSdQkTjpXyOA Oh, sorryL

From Check my proof of P=NP for errors: Let X be NP. Then by *link removed* (np-complete-5) exists X’ in P such that X’(Z)=Y. But X’ in NP. Therefore applying it again, exists X’’ in P such that X’’(Y)=Z. So X’’ produces the same result as X in polynomial time. So, P=NP. #complexity (Here "generalized NP" is assumed: not yes/no, but arbitrary polynomial-sized data for my NP-complete problem.)

@joigus I didn't prove P=NP. I only proved some its consequence. If they to split their million, my discovery is 1/2. But they reward only a complete solution of the entire problem. I won't attach the PDF for licensing issues. The link to PDF: https://math.portonvictor.org/wp-content/uploads/2021/04/np-complete-3.pdf (better indeed use the indirect link above, as the direct PDF link may be updated to a new version of my text from time to time). The algorithm: Let R(X) be the property, whether an arbitrary algorithm X (that takes any input data Y) produces a proof (in a certain formal system) of the statement (for every algorithm Y) X(Y) = Z ⇒ ∃algorithm X' : X' (Z) = Y. In other words, R(X) is provability of invertibility of an algorithm X. R(X) has the convenient property that it is decidable and moreover always provable to be either true or false. The algorithm V(A) means to run algorithm A and then check if the result of A is a logical proof of either R(X) or ¬R(X). R(X) on { X | R(X) or ¬R(X) } is an NP-complete problem. Here is its NP-complete (if P=NP) solution: Enumerating all algorithms An (n ∈ N) run the algorithms V(An) on X in parallel (interleaving these algorithms, and before each step n of the loop adding An to our dynamic array of algorithms) until one of the “threads” F produces “true” or “false” (not “unknown”) (thus having a proof of R(X) or a proof of ¬R(X)). The bold algorithm halts and solves any problem in { X | R(X) or ¬R(X) } because by the assumption some An is NP-complete. Then I prove that the bold algorithm is polynomial-time using the assumption (P=NP) that some algorithm An that produces true or false is polynomial-time. BTW, when for an experiment I removed ads for some time, I didn't decrease bounce rate of my site (however I don't really earn on ads anyway).