for x,y > 7 twins(x+y) <= twins(x) + twins(y)
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From: Transfer Principle on 3 May 2010 20:09 On Apr 22, 12:43 pm, James Burns <burns...(a)osu.edu> wrote: > (Ask yourself who refers to the cranks most often > as cranks, though maybe they don't mind as long as > they are only called "cranks".) (For this question, I make the assumption that Burns intends to say that _I_ am the one who uses the "crank" label the most. If this assumption is false, then one can ignore the following section and skip down to the next quote from Burns.) When I first came up with the label "standard theorist," it was with the intention of using a similar-sounding label such as "nonstandard theorist" to describe the standard theorists' opponents. Thus, instead of writing "Many 'cranks' don't believe in uncountable sets," I'd write "Many nonstandard theorists don't believe in uncountable sets." But I decided to stick to "cranks," since that label is much more well-known, so it's immediately identifiable what I mean by "cranks" (rather than have to answer the question "What do you mean by nonstandard theorists?" over and over again). Since I don't believe that anyone should be called "cranks," I at least put the label in scare quotes, and sometimes even put "so-called" in front of the word "crank." But I only use the word "crank" to echo their opponents' use of the term. If it's true, as Burns hints at here, that I actually use the word "crank" more than their opponents (the "crank"-busters), then there is something that I can do to emphasize that I only use the word in response to their opponents' use of the word as follows: From today on, I shall not be the first poster in any thread to use the word "crank." I shall only use the word "crank" if an opponent has already used the word in that thread. The same goes for any other five-letter insult that commonly appears on sci.math. If the opponents avoid the word "crank," then they should be commended for not using the word -- I shouldn't try to stir up trouble by using the word myself. > I think it is fair to say that your Grand Project > has decayed to the point that you are only defending > /your/ right to decide for /everyone else/ who is > and is not a crank. What I'd rather do is defend my right to _convince_ everyone else who is and isn't a "crank." But to do this, I need to come up with a theory in which the alleged "crank"'s claims are provable. That would convince posters that there really is rigor behind the post so that the poster isn't a "crank." But is such a theory possible? Earlier in this thread, I attempted to write such a "crank"-defending theory, but Hughes proved that it is inconsistent. Certainly, inconsistent theories will convince no one that the poster isn't a "crank." An ad hoc theory in which the poster's claims are assumed as axioms won't stop the "crank" accusations either. The reason that I haven't posted in this thread for two weeks is that I was trying to fix the theory so that it's no longer inconsistent. Since the proof in ZF that every set has a transitive closure requires the Axiom of Infinity, I considered a theory in which we replace Infinity with the negation of that result: There is a set with no transitive closure. In other words: There exists a set a such that for every transitive superset x of a, there exists another transitive superset y of a such that x is not a subset of y. But even though this isn't obviously inconsistent, I haven't been able to tie this to the claims of any "crank" yet. What does this mean? Does it mean that there is _no_ reasonable theory that can prove a "crank" claim? Does this mean that my "project" really is "dead," as Burns implies in the next line: > Why not bury the corpse of the project [...] My answer to this question is in the next line: > [...] and move on? The reason I don't "move on" is that I find nothing appealing to "move on" to. If I admit that my project is dead, then I'm saying that Burns and his allies are right all along. Burns and his allies have a rigorous theory, ZFC, on which to stand, while their opponents post claims that contradict ZFC. If I accuse them of opposing all theories other than ZFC, all they have to do is ask for a rigorous theory in which their claims are provable, and I won't be able to come up with a rigorous theory quickly enough. So it's not that the standard theorists oppose all theories other than ZFC, but that there is _no_ rigorous theory that does what their opponents want them to do, and so the use of the word "crank" to describe them is justified. To me this is unacceptable. No matter what Burns says, I won't "move on" to side with him and his allies, to agree with him when he writes: > some posters just could not be reasoned with and that the posters who "can't be reasoned with" are his opponents, and that the only reasonable posters are Burns himself and those who agree with him. Give me something more desirable to "move on" to, and I'll consider "moving on" to it. Until then, I will keep considering posts of the opponents of Burns and search for rigorous theories in which their claims are provable no matter how long it takes.
From: Jesse F. Hughes on 3 May 2010 20:22 Transfer Principle <lwalke3(a)lausd.net> writes: > What I'd rather do is defend my right to _convince_ > everyone else who is and isn't a "crank." But to do > this, I need to come up with a theory in which the > alleged "crank"'s claims are provable. That would > convince posters that there really is rigor behind > the post so that the poster isn't a "crank." I wonder why you still don't get the obvious point: if *you* come up with a theory in which this or that claim is true, it doesn't imply that there was rigor behind the original post. But never mind. Have at it! You spend all this time talking about what you need to do, and remarkably little time doing it. -- Jesse F. Hughes "I just define real numbers to be all those on the number line, as they were defined before Dedekind and Cauchy." -- Ross Finlayson's simple definition.
From: Transfer Principle on 3 May 2010 22:26 On May 3, 5:22 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > What I'd rather do is defend my right to _convince_ > > everyone else who is and isn't a "crank." But to do > > this, I need to come up with a theory in which the > > alleged "crank"'s claims are provable. That would > > convince posters that there really is rigor behind > > the post so that the poster isn't a "crank." > You spend all this time talking about > what you need to do, and remarkably little time doing it. I've spent _lots_ of time thinking about theories! In fact, the reason that I avoided posting in this thread is that I wanted to come up with a theory to replace the one that Hughes already proved inconsistent. But as of yet, the theory that I have still isn't satisfactory. I don't claim that the following is a rigorous theory that proves a "crank" claim -- at least not yet. I only post the following in order to prove to Hughes that I have been spending more than "remarkably little time" trying to come up with a rigorous theory. Once again, let's begin with ZF, and once again, let's drop the least popular axiom among "cranks," Infinity, and change it to a new axiom. But unlike in the last failed attempt, I will add only a single axiom, not a schema. So what is a good axiom to add? The most obvious axiom would be ~Infinity, but I don't want ~Infinity. In another thread, Tim Little pointed out that Infinity differs from the other axioms of ZF in that it begins with an existential, rather than a universal, quantifier. It's the only axiom (other than Empty Set, which some consider redundant) that states that a certain set exists (a successor-inductive set, and eventually omega). Now the negation of an existential statement is essentially a universal statement, and so ~Infinity would be universal like all the other axioms. But I decided that I want my new axiom to be existential, just as the original Axiom of Infinity is. I notice that in ZF, the Axiom of Infinity is used to prove that every set has a transitive closure. Thus, I can use the Deduction Theorem to come up with: ZF |- "every set has a tr. cl." ZF-Infinity |- Infinity -> "every set has a tr. cl." ZF-Infinity |- "not every set has a tr. cl." -> ~Infinity ZF-Infinity+"not every set has a tr. cl." |- ~Infinity So I can replace the Axiom of Infinity with a new axiom stating that not every set has a transitive closure, and then I'd have a new theory in which ~Infinity is a theorem. And this axiom can be written as an existential axiom: There is a set with no transitive closure. This can be expanded as: There exists a set a such that for every transitive superset x of a, there exists another transitive superset y of a such that x is not a subset of y. Written in full symbolic notation with only the primitive "e" we have: Ea (Ax (AzAw ((zea v zewex) -> zex)) -> (Ay ((AsAt ((sea v setey) -> tey)) & Eu (uex & ~uey))) (Here I take advantage of the fact that "x is transitive," AzAw (zewex -> zex), and "x is a superset of a", Az (zea -> zex), can be combined together to form AzAw ((zea v zewex) -> zex).) Notice that many of zuhair's theories also dealt with transitive closures. Keeping this in mind, we can try to write the axiom in zuhair notation: ..a,x,zw;za|zwx,>zx;>,y;st;sa|sty,>ty;.u,ux~uy Since zuhair has changed his notation several times, I'm not sure whether this is the latest version of it. In particular, I believe I can change the last part to: ..a,x,zw;za|zwx,>zx;>,y;st;sa|sty,>ty;.ux~uy since Eu uex & ~uey can't mean (Eu uex) & ~uey -- the ~uey must be still in the scope of Eu. There might be other places where I can shorten the zuhair notation, based on how his notation exactly works. Now what can we prove with this theory? It's hard to look up already known theories about sets which don't have transitive closures, since in the standard set theory, every set does have a transitive closure. I believe that I can prove in ZF-Infinity that if every element of a set a has a transitive closure, then a has one as well. Proof: Using the Replacement Schema, replace each element of a with its transitive closure. Then using the Union axiom, take the unary union of the resulting set. Then take the Boolean union of this set with a. The resulting set is the transitive closure of a. QED (If this proof is in error, feel free to correct it, making sure not to use the Axiom of Infinity or assume that every set has a transitive closure.) The contrapositive of this theorem is that if a set a has no transitive closure, then neither does at least one of its elements. So the set a whose existence is guaranteed by the new axiom has an element, a_1, with no transitive closure, and then a_1 has an element a_2 without a transitive closure as well, and so on. This seems to produce an infinitely descending chain, which doesn't exist in ZF via Foundation/Regularity -- but note that the proof that no infinitely descending chain requires _Infinity_ as well as Foundation. Thus in ZF-Infinity, we can't prove that an infinitely descending chain exists -- and in fact, with the new axiom, we prove that such a chain does exist! All of this may be fine -- but so far, this doesn't seem to relate to any "crank" claim at all. So let's try to connect this to a "crank" claim. But which one should we try? Fortunately, Hughes, in his infamous ..sig, provides us with a "crank" claim right here: > "I just define real numbers to be all those on the number line, as > they were defined before Dedekind and Cauchy." > -- Ross Finlayson's simple definition. In this post, RF eschews the standard real numbers as defined via D-cuts or C-sequences. Instead, RF wants to define the reals via geometry -- the points that lie on a (number) line. RF is not alone in this thought -- AP has also favored a geometric definition of the reals as well. In particular, rather than have a "Ruler Postulate" that automatically assigns a real number to every point on the line, we want to describe the points on the line _first_, then determine what type of numbers best describe the points. In other threads, including his most recent thread, RF states that he wants adjacent points -- indeed, he wants the set of points on a line to be order-isomorphic to the set of natural numbers (which would require new definitions of both naturals and reals). The name of this isomorphism will be the Equivalency Function, to be defined using EF : N -> [0,1]. To define the points adjacent to 0, we can use H_2, the Zermelo naturals, such as: EF(0) = 0 EF({0}) = iota EF({{0}}) = 2iota EF({{{0}}}) = 3iota where iota is a nonzero infinitesimal. (This of course doesn't exist in standard analysis, but then again, this isn't standard analysis. Also, I refer to the elements of H_2, but notice that H_2 isn't a set in this theory.) To define the points adjacent to 1, we can use some of our new sets based on a, such as: EF(a) = 1 EF(a_1) = 1-iota EF(a_2) = 1-2iota EF(a_3) = 1-3iota This is as far as I've gone thus far. Of course, the theory is far from finished. We've only found the points in [0,1] such that either finitely or cofinitely many of them lie between 0 and that point, but we still have infinitely many points in between -- so what is the set x such that: EF(x) = 1/2 ? Also, what exactly is this set a? All we know about a is that it lacks a transitive closure and that one of its elements is a_1. But is a_1 the only element of a, or are there other elements of a as well? The standard Axiom of Infinity tells us that a successor inductive set exists, but set theorists don't deal with an arbitrary successor inductive set, but rather a particular set, omega. We don't know what particular set a is supposed to be. So I still have a long way to go. I know that the theory isn't finished, yet I post this in order to show Hughes that I've been spending more than "remarkably little time" on considering new theories. And I'll have to spend more than "remarkably little time" trying to finish it.
From: Jesse F. Hughes on 3 May 2010 22:38 Transfer Principle <lwalke3(a)lausd.net> writes: > There exists a set a such that for every transitive > superset x of a, there exists another transitive > superset y of a such that x is not a subset of y. Very minor nit: in text forums like this, it's better not to use "a" as a variable name, for readability's sake. -- Jesse F. Hughes "A factor is simply something that multiplies against another factor to produce a 'product'." -- James Harris offers a definition.
From: Jesse F. Hughes on 3 May 2010 22:42
Transfer Principle <lwalke3(a)lausd.net> writes: > This seems to produce an infinitely descending chain, > which doesn't exist in ZF via Foundation/Regularity -- > but note that the proof that no infinitely descending > chain requires _Infinity_ as well as Foundation. Thus > in ZF-Infinity, we can't prove that an infinitely > descending chain exists -- and in fact, with the new > axiom, we prove that such a chain does exist! I'm confused by your claim. Can you spell out the exact formula you think that this theory proves regarding infinite descending chains? -- "Just because you're ... in a Ph.d program it does not mean that you're up to the challenge of being a real mathematician. Only those who have a purity of mind and dedication to the truth as the highest ideal have a chance." --James Harris, as Sir Galahad the Pure. |