Cantor Confusion
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From: David Marcus on 14 Oct 2006 19:04 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > The problem is that you believe in an actually infinite series of > > > finite numbers. This belief leads to intermingling digit position and > > > number of digits just according to the needs and, after all, it leads > > > to such self-contradictive statements as we have seen here with "the > > > vase". > > > > There is nothing self contradictory in "the vase" but there are > > contradictions between the axiom systems on which "the vase" is based > > and the assumptions of those who oppose those axiom systems. > > > > According to the ZFC system: The vase is empty at noon, because all > natural numbers left it before noon. > By means of the ZFC system we can formulate sequences and their limits > in mathematical language. From this it follows that lim {n-->oo} n > 1. > And from this it follows that the vase is not empty at noon. > > You will and must disagree, but the spectators will at least get to > know this inconsistency. ZFC does not talk about "vases". If you say that ZFC is inconsistent, please give a statement using the language of ZFC. -- David Marcus
From: David Marcus on 14 Oct 2006 19:08 Virgil wrote: > In article <1160815255.198845.303670(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > Han de Bruijn wrote: > > > > Dik T. Winter wrote: > > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > ... > > > > > > It is not > > > > > > contradictory to say that in a finite set of numbers there need not > > > > > > be > > > > > > a largest. > > > > > > > > > > It contradicts the definition of "finite set". But I know that you are > > > > > not interested in definitions. > > > > > > > > Set Theory is simply not very useful. The main problem being that finite > > > > sets in your axiom system are STATIC. They can not grow. Which is quite > > > > contrary to common sense. (I wouldn't imagine the situation that a table > > > > in a database would have to be redefined, every time when a new row has > > > > to be inserted, updated or deleted ...) > > > > > > Is your claim only that set theory is not useful or is contrary to > > > common sense? Or, are you claiming something more, e.g., that set theory > > > is mathematically inconsistent? > > > > It is not useful and contrary to common sense but above all it is > > mathematically inconsistent. > > Such outrageous claims require proofs which "Mueckenh" has not been able > to provide. That standard set theories may violate "Mueckenh"'s amour > propre does not constitute proof. A proof is certainly required, but as a first step, MP should provide a clear statement of the inconsistency. Discussing whether it is an inconsistency seems pointless until MP can provide such a statement using modern terminology. -- David Marcus
From: Randy Poe on 14 Oct 2006 19:09 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > 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. > > And you are in charge of determining which functions are continuous and > which are not? Where do you get this stuff from? How do you translate a statement that some functions are not continuous into "I am in charge of determining if some functions are continuous"? No, I am not in charge. Non-continuous functions are non-continuous now and forever. They were non-continuous before I existed, they will remain non-continuous after I'm gone. The number of balls in the vase is such a function. > The function f(t) = 9t is continuous, because the function 1/9t is > continuous. Yes, but that is not the number of balls in the vase. f(t) = sin(t) is continuous also. So what? What does that have to do with the vase? - Randy
From: Alan Morgan on 14 Oct 2006 21:29 In article <1160858900.410979.56520(a)m7g2000cwm.googlegroups.com>, <mueckenh(a)rz.fh-augsburg.de> wrote: > >Alan Morgan schrieb: > >> >But sqrt(-1) does not yield contradictions, as far as I know. >> >> It violates the heck out of my intuition. The objections to set >> theory seem to arise from someone's dislike for the conclusions >> or an inability to do mathematics correctly. > >Is Weyl this someone? "...classical logic was abstracted from the >mathematics of finite sets and their subsets...Forgetful of this >limited origin, one afterwards mistook that logic for something above >and prior to all mathematics, and finally applied it, without >justification, to the mathematics of infinite sets. This is the Fall >and original sin of [Cantor's] set theory ..." And David Hilbert said "No one will drive us from the paradise which Cantor created for us". Alan -- Defendit numerus
From: stephen on 14 Oct 2006 21:49
Han.deBruijn(a)dto.tudelft.nl wrote: > 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. Why is that physically inconsistent? Is the continuum physically inconsistent? Is a speed of 10^12 m/s physically inconsistent? >> 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. In the finite version every ball is also removed before noon. Given n balls, each ball #i is added at time -(1/2)^(floor(i/10)) and removed at time -(1/2)^i. No matter how many balls there are, all balls will be removed 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 What does this have to do with what "physically inconsistent" means? You seem to never actually define this cute phrases you insist on using. Stephen |