Cantor Confusion
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From: Han.deBruijn on 9 Dec 2006 16:51 David Marcus schreef: > No. I'm just looking for any mathematics. This is sci.math, after all. Really? Then you could have joined me in those _lonely_ threads about Chebyshev and stuff, and solve some real puzzles. There are plenty of the kind. Han de Bruijn
From: Virgil on 9 Dec 2006 16:56 In article <MPG.1fe4d29bd9a1ecdd989a25(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Bob Kolker wrote: > > David Marcus wrote: > > > > > Nonsense. If you use a word, you must define it. Please define > > > "countable number". > > > > Not every word can be (verbally) defined. This would lead to > > circularity. Some words can only be defined by ostention or example. > > Or, by giving axioms that the word satisfies. However, something must be > given or we are only discussing poetry. Mathematicians have to start with an imprecise common language which one can only hope is understood by all its users in sufficiently similar ways, and gradually try to eliminate ambiguities by a succession of careful definitions until what remains is a considerably limited language but is, as far as possible, without ambiguity.
From: Virgil on 9 Dec 2006 17:03 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. > > Absolute rigour is a phantom, even in mathematics. Absolute lack of rigor is chaos, especially in mathematics.
From: David Marcus on 9 Dec 2006 17:21 Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > To those who are unable to see it, there is no evidence that there is > > > any sort of mathematics not describable by set theory. But this is a > > > vicious circle. Mainstream mathematics simply DOES NOT ALLOW any sort > > > of mathematics that violates set theory. > > > > Quite a silly thing to say. Do you have any evidence for such an absurd > > statement? [ ... snipped "counter evidence" with Category Theory ... ] > > Ah, now don't act as if you didn't have those heated debates with some > manifest opponents of set theory, i.e. Wolfgang Mueckenheim. If you've been reading the threads, then you should know that WM has so far failed to present any mathematics at all. All he does is present incorrect arguments that he insists follow from the standard axioms and then proclaim: "Behold, standard mathematics is inconsistent." Big deal. Anyone can prove 2 = 1, if they bend a rule here and there. > I'm not > talking about any possibilities to replace set theory by look-alikes. > > I'm talking about rejecting any monolithic foundation for mathematics, > any "foundation" that narrows down my freedom of thinking. I hate any > form of NewSpeak, whether it is called Set Theory, Category Theory or > Object Oriented Programming. I've seen too many of these. Fine. Give some evidence that it "narrows your thinking". I.e., present some mathematics that can't be done using ZFC as a foundation. -- David Marcus
From: stephen on 9 Dec 2006 17:21
Han.deBruijn(a)dto.tudelft.nl wrote: > MoeBlee schreef: >> Han de Bruijn wrote: >> > It's quite simple. Set Theory can not be the foundation for mathematics, >> > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but >> > it's not a set. >> >> What pure mathematical calculation cannot be represented as a proof in >> set theory? (And a proof, being a sequence of formulas, is, in Z set >> theory as a meta-theory, a set.) > Any calculation runs in TIME. And time is not a set. But time can be modelled as a function, and a function can be modelled as a set. You seem rather hung up on calculations. I am not even sure what you mean by a "calculation". We can calculate 2*3, but most of use do not actually have to calculate anything, as we have memorized the answer. One way to talk about calculation is to look at Turing machines, or any of the other equivalent models of computation. This gives us a very precise definition of what a "calculation" is. So here is a Turing machine that calculates x*y, where x and y are represented in unary: b0 # R # b1 b1 1 R # b2 b1 # R # e0 b2 1 R 1 b2 b2 # R # c0 c0 1 R B c1 c0 # L # d0 c1 1 R 1 c1 c1 # R # c3 c3 1 R 1 c3 c3 # L 1 c4 c4 1 L 1 c4 c4 # L # c5 c5 1 L 1 c5 c5 B R B c0 d0 B L 1 d0 d0 # L # d1 d0 # L # d1 d1 1 L 1 d1 d1 # R # b1 e0 1 R # e0 e0 # R # e1 e1 # L # h e1 1 L 1 h b0 is the start state, h is the halt state. The states are grouped according to the function they are part of. The Turing machine consists of a set (yes a set) of states and transitions, that operate on a set of symbols, and on a tape, which is just a sequence of symbols. Here is the calculation of 2*3. Each pair of lines shows the state of the tape, and location of the head, and the state of the machine. "Time" is simply the progression from one state to another. There are 98 steps in this calculation. So the calculation is simply a finite sequence of steps, and a sequence is simply a function from a {1, ... n } for some n, and a function of course can be modelled as a set. #11#111################################# ^ b0 #11#111################################# ^ b1 ##1#111################################# ^ b2 ##1#111################################# ^ b2 ##1#111################################# ^ c0 ##1#B11################################# ^ c1 ##1#B11################################# ^ c1 ##1#B11################################# ^ c1 ##1#B11################################# ^ c3 ##1#B11#1############################### ^ c4 ##1#B11#1############################### ^ c5 ##1#B11#1############################### ^ c5 ##1#B11#1############################### ^ c5 ##1#B11#1############################### ^ c0 ##1#BB1#1############################### ^ c1 ##1#BB1#1############################### ^ c1 ##1#BB1#1############################### ^ c3 ##1#BB1#1############################### ^ c3 ##1#BB1#11############################## ^ c4 ##1#BB1#11############################## ^ c4 ##1#BB1#11############################## ^ c5 ##1#BB1#11############################## ^ c5 ##1#BB1#11############################## ^ c0 ##1#BBB#11############################## ^ c1 ##1#BBB#11############################## ^ c3 ##1#BBB#11############################## ^ c3 ##1#BBB#11############################## ^ c3 ##1#BBB#111############################# ^ c4 ##1#BBB#111############################# ^ c4 ##1#BBB#111############################# ^ c4 ##1#BBB#111############################# ^ c5 ##1#BBB#111############################# ^ c0 ##1#BBB#111############################# ^ d0 ##1#BB1#111############################# ^ d0 ##1#B11#111############################# ^ d0 ##1#111#111############################# ^ d0 ##1#111#111############################# ^ d1 ##1#111#111############################# ^ d1 ##1#111#111############################# ^ b1 ####111#111############################# ^ b2 ####111#111############################# ^ c0 ####B11#111############################# ^ c1 ####B11#111############################# ^ c1 ####B11#111############################# ^ c1 ####B11#111############################# ^ c3 ####B11#111############################# ^ c3 ####B11#111############################# ^ c3 ####B11#111############################# ^ c3 ####B11#1111############################ ^ c4 ####B11#1111############################ ^ c4 ####B11#1111############################ ^ c4 ####B11#1111############################ ^ c4 ####B11#1111############################ ^ c5 ####B11#1111############################ ^ c5 ####B11#1111############################ ^ c5 ####B11#1111############################ ^ c0 ####BB1#1111############################ ^ c1 ####BB1#1111############################ ^ c1 ####BB1#1111############################ ^ c3 ####BB1#1111############################ ^ c3 ####BB1#1111############################ ^ c3 ####BB1#1111############################ ^ c3 ####BB1#1111############################ ^ c3 ####BB1#11111########################### ^ c4 ####BB1#11111########################### ^ c4 ####BB1#11111########################### ^ c4 ####BB1#11111########################### ^ c4 ####BB1#11111########################### ^ c4 ####BB1#11111########################### ^ c5 ####BB1#11111########################### ^ c5 ####BB1#11111########################### ^ c0 ####BBB#11111########################### ^ c1 ####BBB#11111########################### ^ c3 ####BBB#11111########################### ^ c3 ####BBB#11111########################### ^ c3 ####BBB#11111########################### ^ c3 ####BBB#11111########################### ^ c3 ####BBB#11111########################### ^ c3 ####BBB#111111########################## ^ c4 ####BBB#111111########################## ^ c4 ####BBB#111111########################## ^ c4 ####BBB#111111########################## ^ c4 ####BBB#111111########################## ^ c4 ####BBB#111111########################## ^ c4 ####BBB#111111########################## ^ c5 ####BBB#111111########################## ^ c0 ####BBB#111111########################## ^ d0 ####BB1#111111########################## ^ d0 ####B11#111111########################## ^ d0 ####111#111111########################## ^ d0 ####111#111111########################## ^ d1 ####111#111111########################## ^ b1 ####111#111111########################## ^ e0 #####11#111111########################## ^ e0 ######1#111111########################## ^ e0 ########111111########################## ^ e0 ########111111########################## ^ e1 ########111111########################## ^ h This whole calculation, including time, can be modelled in set theory. Stephen |