Cantor Confusion
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From: stephen on 11 Dec 2006 08:26 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han.deBruijn(a)dto.tudelft.nl wrote: >> >>>step...(a)nomail.com schreef: >> >>>>Do you or do you not wish to abolish any mathematics >>>>that involves infinity? If you are perfectly content >>>>to let others freely explore whatever they wish, then >>>>why are you so aggressive? >> >>>_What_ infinity. That's the question. Mainstream mathematics has mixed >>>up infinity so much that it's not a sensible notion anymore. >> >> What is mixed up about infinity? Care to cite an example >> where mainstream mathematics has mixed up infinity? > See the many threads in this group, e.g. this one. I see a handful of confused, and ill informed people. >> And >> shouldn't people be free to explore whatever infinities >> they wish? You were the one complaining that about >> your freedoms being restricted, yet you seem perfectly >> content to deny those freedoms to others. > People should be free to explore whatever nonsense they wish, > but please: don't call it mathematics. > Han de Bruijn So you do want to restrict the freedoms of others, despite your protestations that set theory was restricting your freedom. So, as I said earlier, you are a hypocrite. Stephen
From: William Hughes on 11 Dec 2006 08:31 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > The reversal given was > > > > > > > > For every natural number n there exists a line L(n), such that > > > > every natural number m <= n is an element of L(n) > > > > > > > > There exists a line L, such that for every natural number n, > > > > every natural number m<=n, is contained in L. > > > > > > > > Note the movement of the phrase "every natural number". > > > > > > > > Please provide an alternate formulation that does not > > > > involve the set of natural numbers. > > > > > > The set N can be involved and the quatifier can be changed as noted > > > above as long as it is asured that: > > > 1) every line has a finite number of elements > > > 2) there is no element of the diagonal outside of every line. > > > > > > If you disagree please provide a counter example (with a finite line). > > > > The problem is not making a statment for each > > finite line. The problem is combining all these statments > > (one for each natural number n) into a single statement. > > > Why should there be a problem as long as *all* lines are finite? The problem is not with any one line, it is with the attempt to combine all lines into one. The fact that *all* lines are finite does not mean there is a last line. > Do you have a counter example where quantifier reversal in any finite > line / set was prohibited? No. For any natural number n the statement For every m<=n, there exists a line L(m) such that m is in L(m) can be reversed to form There exists a line L(n) such that for every m <=n, m in in L(n) However the L(n) is different in every pair. You can combine all lines into There exists a line L such that for every natural number n n is in L if and only if there is a last line. - William Hughes
From: stephen on 11 Dec 2006 08:32 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han.deBruijn(a)dto.tudelft.nl wrote: >> >>>Virgil schreef: >> >>>>In article <1165695789.274304.75780(a)80g2000cwy.googlegroups.com>, >>>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>> >>>>>stephen(a)nomail.com schreef: >>>>> >>>>>>functions, etc. as all of those things can be modelled with set theory. >>>>> >>>>>The topic of functions has been handled separately on my web page: >>>>> >>>>>http://hdebruijn.soo.dto.tudelft.nl/www/grondig/natural.htm#fd >>>>> >>>>>In a nutshell: the mainstream definition is narrow-minded because the >>>>>whole notion of _TIME_ is lacking. >>>> >>>>Position, velocity and acceleration are specifically expressed as >>>>functions of time, so I have no idea of what HdB is talking about. >> >>>Huh, no. Let f(x) = 2.x then time is involved with multiplying x by 2. >> >> That is your idea of time? And you think that cannot be modelled >> in set theory? You are becoming increasingly irrational. > Okay. Let's "model" it in set theory. Let's have your "cost" associated > with f(x) = 2.x and call it C(?) = whatever. Then you have to calculate > that cost and every time you do it there will be time involved with such > a cost calculation. So you have to define a cost function for the cost > function as well. And a cost function for the cost function of the cost > function. Don't you see that this is an _explosion_ instead of a model? So what is the cost function for f(x)=2.x. Do you have to calculate that in order to calculate 2.x? If not, then do you have to calculate the cost function for the cost function for f(x)=2.x? If so, then how do you calculate anything at all? You are the one claiming this viscious circle exists. How do you avoid it? If you can calculate f(x)=2.x without calculating the cost function, then why do you think that cannot be modelled in set theory? > There is only one way to avoid a vicious circle of the kind and that is > to accept that time is not a set. > Han de Bruijn Nobody has said that time is a set. But time can be modelled using sets. Apples are not sets either, nor are they numbers, but you can reason about apples using sets and numbers. Your arguments are becoming increasingly irrational. You really do not appear to have any rational opposition to set theory, just emotional ones, and your arguments boil down to Communist style propaganda slogans. Stephen
From: Han de Bruijn on 11 Dec 2006 09:09 stephen(a)nomail.com wrote: > Nobody has said that time is a set. But time can be modelled using > sets. Modelling time as implied with functions results in a vicious circle, as I have demonstrated in a previous poster. > Apples are not sets either, nor are they numbers, but you > can reason about apples using sets and numbers. Your arguments are > becoming increasingly irrational. You really do not appear to have > any rational opposition to set theory, just emotional ones, > and your arguments boil down to Communist style propaganda slogans. Well, you're quite close, after all. But I can also say that _your_ arguments boil down to Capitalist style propaganda slogans. Who is the hypocrite here? At least _I_ will not deny that my ideas _are_ influenced by the world outside. While I don't expect _you_ to admit that Set Theory is only "fundamental" because of its setting within the Capitalist Economic System. You think you are objective and free of emotion, but you are not. Your propaganda is as bad as mine. Han de Bruijn
From: mueckenh on 11 Dec 2006 09:32
William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > Recall this post from Dec 1 > > > > > > We extend this to potentially infinite sets: > > > > > > A function from the potentially infinite set A to the > > > potentially infinite set B is a potentially infinite set of > > > ordered pairs (a,b) such that a is an element of A and b is > > > an element of B. > > > > A function, according to modern mathematics, is a set, actually fixed > > and complete > > Yes, but according to modern mathematics the natural numbers > are a set, actually fixed and complete. You cannot pick and choose the > bits of modern mathematics you want to use. But I can point to those bits which are selfcontradictory and refuse to accept them. Your assumption of an actually potentially infinite set is such a piece. Regards, WM Regards, WM |