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).