for x,y > 7 twins(x+y) <= twins(x) + twins(y)
in [Math]
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From: Jesse F. Hughes on 18 Apr 2010 00:31 Transfer Principle <lwalke3(a)lausd.net> writes: > On Apr 14, 5:10 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Transfer Principle <lwal...(a)lausd.net> writes: >> > Both RF and TO -- the infinitesimal "cranks," have proposed >> > such a theory. MR, another infinitesimalist, has also looked >> > at something similar. But of course, the standard theorists >> > refuse to consider such ideas at all. >> Just go ahead and suggest an interesting theory that you >> think approaches the ideas of Ross, Tony or Mitch. I don't promise >> that it'll get a whole lot of attention -- most mathematicians are not >> looking for unusual theories > > Most standard theorists aren't looking for theories at all -- > since they already have a theory, namely ZFC. The only people > who are looking for _unusual_ theories are those who object to > the _usual_ theories for one reason or another. Oh, nonsense. Have you never heard of logicians? > The reason that I post unusual theories is not necessarily so > that the standard theorists would accept the _theories_, but > so that they'll accept the _posters_ whose ideas are given in > the theories. By "acceptance," I mean avoidance of words like > "crank" and "troll" to describe them. Ah, but can you name any "unusual theories" that you've posted? Up 'til now, it seems you mostly post about posting about theories that justify the statements someone else is posting. It should be obvious that this oddly roundabout strategy doesn't really dissuade use of the term "crank". > What theories will standard theorists accept? We already know > that they'll accept ZFC, of course. So it's reasonable to guess > that standard theorists are more likely to accept theories that > closely resemble ZFC than those that don't. Paucity of axioms, > schemata, and primitives, as well as avoidance of "ad hoc" > axioms, are some reasons given by Hughes and other standard > theorists in the past as to why ZFC is a preferred theory. > > But here's the thing. If a theory is _not_ "ad hoc," then we > should be able to prove powerful results from only a few axioms, > but the proofs might be long -- and then the standard theorists > will grow tired of reading the proofs, since why should they > invest so much time in reading a proof in another theory when > they themselves already have a theory, ZFC? > > Therein lies the dilemma. If I were to simply "ad hoc" assume > the assumptions of RF, TO, MR as axioms, then the standard > theorists criticize the theory for being "ad hoc." But if I > were to post natural-sounding axioms that imply that their > assumptions are true, then the proofs might be so long that the > standard theorists lose interest. But of course, no one ever > said that convincing the standard theorists to accept the ideas > of RF, TO, MR, tommy1729 was going to be easy. Right. Well, there's always the possibility that your heroes are indeed morons. In fact, it's plainly clear that at least some of them are just stupid. At least, it's clear to some of us. -- "Is that possible? Could it be that easy? No way. [...] There must be a mistake. Right? "But I am the top mathematician in the world." -- James S. Harris
From: master1729 on 18 Apr 2010 03:48 lwalke wrote : > On Apr 17, 3:49 pm, master1729 <tommy1...(a)gmail.com> > wrote: > > lol ! H-Li is still wide open today ! > > i didnt see " your disproof " in the media :p > > Technically speaking, tommy1729 is correct that H-Li > is > still an open conjecture, since it depends on the > Prime > Patterns Conjecture that is itself an open > conjecture. > > So of course, what Bau really meant was that one has > already found proofs of the following: > > PP -> ~H-Li > PP -> ~tommy1729-Li > > and that the proof of the former was given just a few > years ago, while the proof of the latter is, by > comparison, trivial. Indeed, the proof that Bau wrote > in the other thread is a proof of PP -> > ~tommy1729-Li. > > To me, it's a bit counterintuitive that tommy1729-Li > would be easier to disprove than H-Li, since "most" > primes are not twin primes. But we see that > nonetheless > tommy1729-Li is far easier to disprove. is it ? care to explain ?
From: Transfer Principle on 20 Apr 2010 13:50 On Apr 17, 9:31 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > The reason that I post unusual theories is not necessarily so > > that the standard theorists would accept the _theories_, but > > so that they'll accept the _posters_ whose ideas are given in > > the theories. By "acceptance," I mean avoidance of words like > > "crank" and "troll" to describe them. > It should be obvious that this oddly roundabout strategy doesn't > really dissuade use of the term "crank". Well, the _direct_ (as opposed to "roundabout") way to dissuade use of the term "crank" would be just to ask people not to use it, such as: "Hughes, please stop using the word 'crank.'" But of course, that won't work, any more than the standard theorists' attempt to convince me to stop defending "cranks" simply by asking me to stop. > > The reason that I post unusual theories is not necessarily so > > that the standard theorists would accept the _theories_, but > > so that they'll accept the _posters_ whose ideas are given in > > the theories. By "acceptance," I mean avoidance of words like > > "crank" and "troll" to describe them. > Ah, but can you name any "unusual theories" that you've posted? Well then, let me post a theory right now. This is in fact a theory to which I've alluded several times in past threads. Since we know that standard theorists find ZF(C) to be acceptable, let's start with ZF as a base. Now we drop the Axiom of Infinity, and replace it with the following two schemata, which both use a new primitive symbol, N: Schema 1: If phi is a one-place predicate that does not mention the symbol N, then all closures of: (phi(0) & Ax (phi(x) -> phi(xu{x}))) -> phi(N) are axioms. In previous threads, I started with 1 rather than 0, since 1-indexing is popular with "cranks." But I found it easier to use finite von Neumann ordinals rather than {1,...,n}, since the ellipsis is unpopular with standard theorists. More importantly, I decided to add that all important phrase "that does not mention the symbol N" -- since without it, standard theorists are able to come up with trivial counterexamples. But first of all, what exactly is "N"? Some standard theorists might say how it's perfectly consistent for "N" to be 0 -- which would result in this first schema becoming trivial. In order to prevent this, we must add another schema: Schema 2: If phi is a one-place predicate that does not mention the symbol N, then all closures of: Ax ((xeN & ~phi(x)) v (~xeN & phi(x)) are axioms. This tells us that N is not definable by any predicate phi that doesn't mention the symbol N. The only way to specify N is just to write "N" itself. So N can't be 0 since 0 is definable without using this new symbol N. Nor can it equal 1, 2, 3, ..., as these are all definable without N. But now the standard theorists might ask, how is this related to anything written by a "crank"? I'd like to try to prove some "crank" claims in this theory, but by the time any of these proofs are completed, the standard theorists will have long stopped reading this thread out of boredom. But hopefully, the standard theorists can see for themselves how certain "crank" claims can be proved -- especially those claims given when "cranks" attempt to use an "induction" schema similar to Schema 1. Is this theory consistent? Some of what I've written sound similar to the construction of a nonprincipal ultrafilter on omega. In ZFC, this requires AC. So ZFC should be able to prove this theory consistent, with ultrafilters being used somewhere in the proof (but we might not be able to prove it consistent in ZF without the Axiom of Choice). Still, some "cranks" who object to AC might prefer this to ultrafilters since AC isn't used in the theory (since AC isn't even an axiom of the theory) -- it's merely used in the _ZFC_ proof of its consistency, whicb isn't the same thing.
From: Transfer Principle on 22 Apr 2010 14:48 On Apr 18, 4:48 am, master1729 <tommy1...(a)gmail.com> wrote: > lwalke wrote : > > To me, it's a bit counterintuitive that tommy1729-Li > > would be easier to disprove than H-Li, since "most" > > primes are not twin primes. But we see that > > tommy1729-Li is far easier to disprove. > is it ? > care to explain ? Bau already posted the proof of PP->tommy1729-Li in the other thread. Let me attempt to paraphrase his proof in this thread. Theorem: The Prime Patterns Conjecture implies the negation of the tommy1729-Littlewood Conjecture. Proof (paraphrased from Bau): Let's first remind ourselves what exactly the Prime Patterns Conjecture is. We expect there to be only finitely many primes p such that p+1 is prime, since one of them must be even. Indeed, there is exactly one such prime. But there is no reason for there to be only finitely many primes p such that p+2 is prime, since both can be odd. And so this is the Twin Prime Conjecture -- that there exist infinitely many primes p sch that p+2 is prime. What about "triplet primes"? If p is odd, then so are p+2 and p+4. Is it possible for all three to be prime for infinitely many primes p? The answer must be no, since at least one of these must be divisible by three. Indeed, there is exactly one such prime. But what about, say, p, p+2, and p+12? Can we have all three of those be prime, for infinitely many primes p? We notice that if p is one less than a multiple of three, then p+2 and p+12 are one more and one less than a multiple of three respectively, so there's no reason that there should be only finitely many such primes p. And so this is what the Prime Patterns Conjecture tells us. It basically states that unless there is some underlying reason why a prime "pattern" such as {p, p+a, p+b, p+c, ..., p+z} must all be prime for only finitely many primes p, then it will be prime for _infinitely_ many primes p. And so Bau proceeded by finding a prime pattern for not just two or three primes as in my examples above, but 48 primes instead -- 24 twin prime pairs. And the prime pattern he found is: {p, p+2, p+12, p+14, p+18, p+20, p+42, p+44, p+48, p+50, p+72, p+74, p+90, p+92, p+102, p+104, p+132, p+134 p+168, p+170, p+180, p+182, p+198, p+200, p+210, p+212, p+222, p+224, p+258, p+260, p+270, p+272, p+282, p+284, p+300, p+302, p+312, p+314, p+342, p+344, p+348, p+350, p+378, p+380, p+390, p+392, p+408, p+410} One can check to see that there's no reason for there to be only finitely many instances (which includes _zero_ instances) of this pattern among the primes. If p is odd then all of the numbers above are odd. If p is one less than a multiple of three, then all the numbers on the left are one less than a multiple of three, and all the numbers on the right are one more than a multiple of three. If p is one less than a multiple of five, then none of the numbers given above are multiples of 5, and so on. And so the Prime Patterns Conjecture states that this pattern must occur _infinitely_ often. It might be exceedingly _rare_ for all 48 of those numbers to be prime -- just as flipping a coin 48 straight times and obtaining heads is also extremely rare. But we get to flip the coin _infinitely_ many times, and when we do so, then the coin will eventually land heads 48 straight times -- the probability of this is 1. And so we consider _infinitely_ many primes, and this prime pattern _must_ occur eventually. The size of p might be enormous, but it will occur. And so Bau writes: twins((p-1) + 419) <= twins(p-1) + twins(419) So twins(p-1) is the cardinality of the set of prime pairs less than p. We don't knonw what it is, but it's some number, say M. Then twins((p-1) + 419) must be exactly M+24, since we add the 24 prime pairs that we found in the pattern above. As for twins(419), we can count it ourselves. The prime pairs less than 419 are: {3, 5, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421} And there are clearly only 22 prime pairs there. And so we have: twins((p-1) + 419) <= twins(p-1) + twins(419) M + 24 <= M + 22 Subtracting M from both sides gives: 24 <= 22 which is a blatant contradiction. Therefore, if the Prime Patterns Conjecture holds, then the conjecture given by tommy1729 must fail. QED This is an existence proof. The proof doesn't state how to find p (which is very large). It only states that (infinitely many such) p must exist.
From: Transfer Principle on 22 Apr 2010 14:51
On Apr 18, 5:07 am, master1729 <tommy1...(a)gmail.com> wrote: > > > Transfer Principle <lwal...(a)lausd.net> writes: > > of RF, TO, MR, tommy1729 was going to be easy. > RF = Ross Finleyson ? > TO = Tony Orlow i guess. > MR = ? Mitch Raemsch ??? i hope not !? Fine then. Since I'm not helping _any_ "cranks" by defending _every_ "crank," I'll drop MR from the list as being too "cranky" even for me to defend. > but , plz dont include musatov* or ultrafinitists. Even standard theorists find some ultrafinitists, such as Yessenin-Volpin, worth considering > * unemployable retarded son of a fat german [...] [snip rest of racist rant] |