Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Virgil on 14 Oct 2006 18:30 In article <1160857717.822360.61130(a)i42g2000cwa.googlegroups.com>, 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? > The function f(t) = 9t is continuous, because the function 1/9t is > continuous. And for what value of t does one have noon? > > Regards, WM
From: Virgil on 14 Oct 2006 18:34 In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil schreef: > > > In article <267fc$452f5def$82a1e228$15540(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > Virgil wrote: > > > > > > > In article <b8869$452f4a39$82a1e228$32738(a)news2.tudelft.nl>, > > > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > >> > > > >>How about: I'm not interested in "the end". I don't know where the end > > > >>is. And I don't care as well. As long as the end is somewhere where it > > > >>causes a uncertainity which is acceptable for my purpose. > > > > > > > > But what if "the end" isn't anywhere because there isn't one? > > > > > > > > As soon as you posit an end, you run into problems. You would be much > > > > better off saying that all such questions about an end to the naturals > > > > are unanswerable, and stick to what you can explicitly construct. > > > > > > Both approaches run into problems. Either you accept infinities, either > > > you accept a little bit of Physics: uncertainity and inexactness. Guess > > > you know what my choice is. Guess I know what your choice is. > > > > I do not object to the constraints of the mathematics of physics when > > doing physics, but why should I be so constrained when not doing physics? > > Because (empirical) physics is an absolute guarantee for consistency? It doesn't guarantee anything unless many assumptions about "reality" are valid, such as the assumption that what we have observed in the past we will continue to observe in the future, i.e., that the "laws" of physics do not change over time.
From: Virgil on 14 Oct 2006 18:35 In article <1160857922.568001.291390(a)h48g2000cwc.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > > Han de Bruijn 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? Wouldn't your comments be > > better posted to sci.physics? Most people in sci.math are (or at least > > think they are) discussing mathematics. > > ONE world or NO world. > > Han de Bruijn Thanks be that HdB has not the power to enforce his Procrustean notions.
From: William Hughes on 14 Oct 2006 18:36 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > >> > > But there > > is a last N iff the number of digit positions in 0.111.... > > is finite. > > Exactly. But two posts back you said Why should the number of finite positions be of any influence, if we know that the theorem is true for any finite position? I guess you changed your mind. >Either the number of digit positions is finite No one has denied that claiming the number of integers is finite will lead to the results you claim. On the other hand, no one has claimed that claiming the number of integers is finite is a common sense solution. >or there are some positions undefined Nope. At this point we are looking for sufficient conditions for a single M to exist. Saying there are some postitions undefined will not do this. > or an infinite set does not actually exist, Nope. At this point we are looking for sufficient conditions for a single M to exist. Saying that an infinite set does not actually exist will not do this. You need to add that there is an upper bound to the integers (i.e. you must deny potential as well as actual infinity). > meaning that the number of digits is finite though unbounded. There is no such thing as a "finite though unbounded" integer. So what is this "number" in "number of digits". To get the result you want (there exists a single M) you have to claim that the number of integers is finite (not actually infinite, not potentially infinite, finite). Knock youself out. - William Hughes
From: Virgil on 14 Oct 2006 18:37
In article <1160857922.727908.74990(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1160675932.010700.124010(a)m7g2000cwm.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > 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. > > > > > > We know that a set of numbers consisting altogether of 100 bits cannot > > > contain more than 100 numbers. Therefore the set is finite. The largest > > > number of such a set cannot be determined, as far as I know. > > > > That set is indeterminate. Just use Ascii notation. The string > > "Graham's number" fits in 100 bits. > > > > > Could you determine it? Or would you prefer to define that such ideas > > > do not belong to mathematics? Then I would not be interested in that > > > definition. > > > > It is an indeterminate set. > > The set of bits is determined: exactly 100. What you can build from 100 > bits belongs to the power set of this set. It is probably large but > certainly not infinite. if one is allowed to use arbitrary bases, then it IS infinite. |