Cantor Confusion
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From: Han.deBruijn on 14 Oct 2006 15:09 Randy Poe schreef: > Han de Bruijn wrote: > > > I merely note that there is no requirement in the problem that > > > the limit be the value at noon. > > > > The limit at noon - iff it existed - would be the value at noon. > > Wrong. That is a flat out incorrect statement showing a > fundamental misunderstanding about what limits mean. > > A CONTINUOUS function at x0 has the property that the > limit of f(x) as x->x0 is f(x0). But not all functions are > continuous. Almost. Any function which is defined everywhere at an interval of real numbers is also continuous at the same interval. Or, with other words: For real valued functions, being defined is very much the same as being continuous. This fact is known as Brouwer's Continuity Theorem: http://www.andrew.cmu.edu/user/cebrown/notes/vonHeijenoort.html#Brouwer2 Brouwer's Continuity Theorem is cosmologically valid. Han de Bruijn
From: Han.deBruijn on 14 Oct 2006 15:25 stephen(a)nomail.com schreef: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > I'm not interested in the question whether set theory is mathematically > > inconsistent. What bothers me is whether it is _physically_ inconsistent > > and I think - worse: I know - that it is. > > What does "physically inconsistent" mean? Can you give an example > where set theory is "physically inconsistent"? Take e.g. the axiom of infinity. > The balls and > the vase problem is not such an example, as it is not > physically realizable. Denied. The balls in a vase problem is good as the approximation of a physical thought experiment for (non)times before noon. > Yes, set theory can model unphysical > things, but so can any mathematics. For example, suppose > you have an acceleration of 10m/s^2. To determine your > velocity after n seconds you calculate > > / n > | 10 dt > / 0 > > Of course this is wrong if n is 30000. Does that make > calculus physically inconsistent? Or is it just the case > that calculus can be used to describe unphysical situations? You forget that _all_ applied mathematics is an approximation. You can employ the above only in the domain where it is (always approximately) valid: Newtonian mechanics. But even relativistic mechanics is _never_ exact, though it may be beter with high velocities. Thus the keywords are approximation and uncertainity, everywhere where Applied rules the roast. And that's precisely the point where OUR "paradoxes" disappear. Han de Bruijn
From: Han.deBruijn on 14 Oct 2006 15:52 Dik T. Winter schreef: > In article <b2f47$452f51eb$82a1e228$2726(a)news2.tudelft.nl> > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > ... > > b(t) diverges at noon. Thus b(noon) is undefined. Thus t is not time. > > > > It's impossible in a renormalized mathematics that limits are different > > from the actual values of functions at that place. > > "Diverges at noon" is strange wording in my opinion. I would say > b(t) diverges when going to noon. And so the limit does not exist. Yes. That's a better wording perhaps. > But this is not in contradiction with b(noon) = 0. NO with contemporary mathematics. But with my renormalized mathematics: YES. The gist of renormalized mathematics is to add a tiny bit of error propagation to the exact outcome. (According to my matra: a little bit of Physics.) You know: empirical sciences are _never_ free of "errors", inexactness and uncertainity. Then it's easy to see that a limit "at noon" can not be different from a value "at noon". Quite according to common sense, which in this case is not different from sound physical reasoning. But the limit at noon does not exist. Thus a value at noon does not exist either. Natura non facit saltus. Nature does not jump. Leibniz. > However, let's have a look at the entier function. > lim(x from 0 -> 1) entier(x) = 0, but entier(1) = 1. > It seems that the limit is different from the actual value of the > function. If we convolute the entier function with a Gaussian function with very small spread sigma, then it becomes continuous (: kind of a smoothened staircase). And your argument evaporates accordingly. Yes, that is the power of renormalization. Take mathematics "as is" and add that little bit of Physics. For your eyes only. My "numerical differentiation schemes" work with the same Gaussians. This doesn't mean that they are better than yours. It only means that I find it important to have a uniform theory here. Han de Bruijn
From: Han.deBruijn on 14 Oct 2006 16:00 Dik T. Winter schreef: > In article <ddeb9$452e55fe$82a1e228$16456(a)news1.tudelft.nl> > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > > Set Theory is simply not very useful. > > Oh. So you think that Banach spaces are not very useful? You think that > a book like "The Algebraic Eigenvalue Problem" is not very useful? You > may note that both are heavily based on set theory. Think I could rewrite the relevant stuff in those books without using any set theory. I'm not actually going to do it, though. (BTW, I find Banach Spaces not very useful either) Han de Bruijn
From: Virgil on 14 Oct 2006 16:00
In article <1160813247.796685.225130(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Dik T. Winter wrote: > > > In article <1160649760.068206.172400(a)b28g2000cwb.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > To inform the set theorist about the possible existence of sets with > > > > finite cardinality but without a largest number. > > > > > > Interesting but in contradiction with the definition of the concept of > > > "finite set". So you are talking about something else than "finite > > > sets". > > > > It would seem he is. I don't understand why people use words in non- > > standard ways without explaining what they mean. They are guaranteeing > > that no one will understand them. > > A finite set is a set with a number of elements, which is smaller than > some natural number. As far as I know this notion is covered by the > standard meaning of words. A better definition is that a set is finite if and only if it has (exactly) a natural number of elements. But even this requires a pre-definition of the natural numbers. Given the axiom of choice, that definition is equivalent to the Dedekind definition: A set is finite if and only if there is no function injecting it into any of its proper subsets. |