for x,y > 7 twins(x+y) <= twins(x) + twins(y)
in [Math]
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From: Jesse F. Hughes on 13 May 2010 18:00 Transfer Principle <lwalke3(a)lausd.net> writes: > (Note that brevity isn't necessarily better than verbosity -- > sometimes I wish that MR would be _more_ verbose!) Really? *Really?* -- Jesse F. Hughes "They don't want a choice, they want something other than Windows." -- billwg explains that refusing Windows isn't a choice.
From: Transfer Principle on 13 May 2010 18:08 On May 10, 10:09 am, James Burns <burns...(a)osu.edu> wrote: > [S]uppose that Poster Y asserts that 1 > 0.999... > and it is clear he means to use those symbols in the > standard way. What does Burns mean by "the standard way"? If by "the standard way," he means the definition in classical analysis by which it is exactly equal to 1, then Poster Y's assertion is equivalent to "1>1" (substitution). To me, "1>1" is a statement which even I won't defend or attempt to find a theory for. To me, simply by posting "1 > 0.999...," it is clear he _doesn't_ mean to use those symbols in the standard way at all. It's clear that Poster Y means to use the symbols in a nonstandard way -- a way in which 0.999... differs by 1 by a nonzero infinitesimal. And so it's up to me to _find_ a way to use it that's rigorous and not ad hoc. > People tell him he is mistaken. People > go on at great length explaining why he is mistaken, > all to no avail. Eventually, he convinces people that > he is not merely mistaken but a crank. Suppose that > you find another mathematical system, another > interpretation of "1 > 0.999..." in which it is true. > Your use of "1 > 0.999..." is, in an important sense, > not even the same thing Poster Y is saying. But what exactly is Poster Y saying? I doubt that he would be saying "1>1" -- which is what he would be saying if he means the standard definition of 0.999.... (An analogy: Poster Z writes, "The positive integral factors of 14 are 1,2,7, and 14." If I assume that here "integral" means "antiderivative," then the statement makes no sense, until I find a definition, such as the adjectival form of "integral," which does make sense, since I assume that Poster Z is rational.) I start from the assumption that Poster Y is rational, that the poster wouldn't write something as blatantly false as "1>1" and search for a definition that does make sense, such as "1 > 1-iota" for some nonzero (infinitesimal) iota. Burns, on the other hand, appears to start from the opposite assumption, namely that Poster Y is _irrational_, when he criticizes me for searching for an infinitesimal theory. (But I could be wrong about Burns's assumption here.) > Why would you expect your newly introduced system to > change anyone's mind about Poster Y's crankhood? Have any of my newly introduced systems changed anyone's mind about Poster Y's "crank"-hood yet? Not yet -- but that could be because I have yet to post any theory that is sufficently rigorous and non-ad hoc. (The closest that I have come is declaring that 0.999... is the surreal 1-1/omega, but even this causes problems as 1-1/omega doesn't work exactly as Poster Y wants it to.)
From: Transfer Principle on 13 May 2010 18:25 On May 13, 3:00 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > (Note that brevity isn't necessarily better than verbosity -- > > sometimes I wish that MR would be _more_ verbose!) > Really? MR is often criticized for having posts that are _too_ short -- so short that one can't tell exactly what he's talking about. Typical MR posts (from August 2009): "The infinitely small is the first quantity and is closest to zero." "There is a smallest quantity." ".9 repeating is the quantity closest to 1. Math begins with the infinitely small." And of course, typical criticism of MR posts include that MR won't _explain_ what he means by "the infinitely small," a "smallest quantity," "quantity closest to 1," and so on. And typical criticism of my defense of MR posts usually involve that there isn't enough content in MR's posts for me to make a rigorous theory out of, so it's foolish for me to try. Therefore, I no longer try to defend MR's posts. If Hughes is trying to imply that brevity is seldom a bad thing, and that my posts will be so much better if I could be briefer as MR is brief, then maybe the next time someone asks me to explain a theory, I'll give a brief MR-like response. In fact, I'll do exactly that in my next post. I will give an intentionally brief MR-like response to a question Hughes asked of me earlier in this thread. So Hughes won't be able to complain that this next post will be too verbose.
From: Transfer Principle on 13 May 2010 18:27 On May 3, 7:47 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > You have a set theory in which not every set has a transitive > closure. You also have a fairly random comment by RF regarding the > reals. Maybe I'm missing something, but it seems to me that RF's > claim is utterly unrelated to transitive closures. > I just don't see why you think that this theory is related to Ross's > opaque comment. 1-iota is the smallest quantity below one. The null axiom theory works because it works.
From: christian.bau on 13 May 2010 20:42
On May 13, 11:08 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > To me, simply by posting "1 > 0.999...," it is clear he > _doesn't_ mean to use those symbols in the standard way > at all. It's clear that Poster Y means to use the > symbols in a nonstandard way -- a way in which 0.999... > differs by 1 by a nonzero infinitesimal. And so it's up > to me to _find_ a way to use it that's rigorous and not > ad hoc. Let's give it a name. We call this mystical 0.999... by the name x. Now three simple questions: What is (1 + x) / 2? What is the square root of x? What is the difference between (1 + x) / 2 and the square root of x? These poor souls thinking deep thoughts about the nature of 0.999... don't realise that they have only advanced their knowledge by an infinitesimal amount of size 1 - 0.999... . They are so stuck on that particular way to write a number that they can't progress from there. Here is a very, very simple approach that actually gives you something useful: Assume there is a set S with the following properties: 1. On the set S four operations +, -, * and / and three relations <, = and > are defined in a way which follows the twelve axioms of arithmetic. 2. The set S contains the real numbers as a proper subset, that is every real number is an element of S, but S contains at least one element which is not a real number. 3. The operations +, -, * and / and the relations <, = and > are consistent with the operations of the same name in the real numbers, so if x and y are both real numbers, then x + y using the definition of S's "+" operator and using the definition of "+" in the real numbers gives the same result. The mystical 0.999... could be an example of an element of S which is not a real number, you just need to get the definitions for +,-,* and / right so that they work for this element as well. But now we get to the meat. Under the assumptions in (1) to (3) prove the following: 0. Prove that the complex numbers are _not_ an example for such a set S. 1. Prove that S has an element which is greater than every real number. We may call this element "infinity". 2. Prove that 1 / infinity is an element of S which is greater than zero, but less than every real number that is greater than zero. We may call this element "epsilon" 3. Prove that the axiom of completeness doesn't apply in S, in other words in S there are non-empty bounded sets that don't have a smallest upper bound. 4. Prove that infinity and epsilon are not real numbers. Prove that 2 * infinity > infinity. Prove that infinity - epsilon < infinity. Prove that infinity * epsilon = 1. Write down lots of other elements of S. 5. Find a simple model for S. In other words, find a simple set S that actually has all the properties from (1) to (3). The bit "Write down lots of other elements of S" should really give you an idea for this. It's not difficult. It is really quite easy, and it gets you lightyears ahead of the "1 > 0.999..." crowd. |