Cantor Confusion
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From: Virgil on 8 Dec 2006 03:15 In article <1165564379.213843.105070(a)j72g2000cwa.googlegroups.com>, cbrown(a)cbrownsystems.com wrote: > Virgil wrote: > > In article <1165552164.399063.145330(a)f1g2000cwa.googlegroups.com>, > > cbrown(a)cbrownsystems.com wrote: > > > > > Virgil wrote: > > > > In article <MPG.1fe27d2a5a4ae60a989a01(a)news.rcn.com>, > > > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > I do nothing. The tree cares that even in the limit the number of > > > > > > paths > > > > > > cannot become uncountable. 2^n remains the cardinal number of a > > > > > > countale set, even in the limit n --> oo. That's why I devised the > > > > > > tree! > > > > > > > > > > The number of paths in a tree of height n is 2^n. Are you saying that > > > > > the number of paths in the infinite tree is lim_{n -> oo} 2^n? > > > > > > > > Actually, if one interprets 2^n as the number of functions from > > > > {0,1,2,...,n-1} to {0,1} in NBG, or. equivalently. as the number of > > > > functions from {1,2,3,...,n} to {1,2}, then 2^n -> 2^N as n increases > > > > without limit and 2^N, interpreted that same way, is an uncountable > > > > set. > > > > > > What do you mean by "the number" of functions? > > > > I mean the cardinality of the set of all such functions, of course. > > > > > > > In what sense does the > > > sequence of natural numbers (1,2,4,8, ..., 2^n, ...), approach "the > > > number" of the set of functions N->{0,1}? > > > > For each finite n in NBG, n = {0,1,2,...,n-1}, so the set 2^n consists > > of all functions from n to {0,1} = 2, and is of cardinality equal to the > > number of such functions. > > > > So as n --> N, 2^n --> 2^N. > > Ah! So, because (1,2,4, ..., 2^n, ...) is a subsequence of (1,2,3, ..., > n, ...), then as n->N, n->2^N. That is neither what I said nor what I meant. In set notation (rather than number notation) give non-empty sets A and B, the shorthand notation for the set of all functions from B to A is A^B. A slightly less compact notation, used in the category of sets and functions, would be Hom(A,B). Thus Card(A^B) = Card(A)^Card(B) for all finite sets A and B. And Card({0,1})^{0,1,...,n-1}) = Card({0,1})^Card({0,1,...,n-1}) = 2^n for all natural numbers, Card({0,1,...,n-1}) = n in N. Then as {0,1,...,n-1} --> N, {0,1}^{0,1,...,n-1} --> {0,1}^N and 2^n --> 2^N
From: Han de Bruijn on 8 Dec 2006 03:19 Lester Zick wrote: > On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >>If you use ZFC (or something similar) as your foundation for >>mathematics, then everything is a set. Of course, while solid >>foundations are good to have, if you are living on an upper floor, you >>may prefer to ignore what is going on in the basement. > > So you're saying that set "theory" is all of mathematics? Of course > since what you say isn't necessarily true that's not exactly a ringing > endorsement of set "theory". 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. Set theory may be of limited use, but it's supremacy is complete nonsense, and will be overruled in time. Han de Bruijn
From: Virgil on 8 Dec 2006 03:47 In article <59dc9$45791d79$82a1e228$8084(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Bob Kolker wrote: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> You cannot imagine the integer [pi*10^10^100]. > > > > That is not an integer, dummkopf. It is an irrational real number. > > Dummkopf? Who? Doesn't "[ .. ]" stand for the "floor" function? > > Han de Bruijn Not always. Sometimes it is just a grouping symbol like parentheses, (), or brackets, {}. Sometimes it indicates a closed interval, as in [0,1]. To be less ambiguous, WM could have written "Floor(pi*10^10^100)", but even then, there are many people who would not know what he meant.
From: Virgil on 8 Dec 2006 03:53 In article <68588$45791ff3$82a1e228$8581(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Lester Zick wrote: > > > On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > >>If you use ZFC (or something similar) as your foundation for > >>mathematics, then everything is a set. Of course, while solid > >>foundations are good to have, if you are living on an upper floor, you > >>may prefer to ignore what is going on in the basement. > > > > So you're saying that set "theory" is all of mathematics? Of course > > since what you say isn't necessarily true that's not exactly a ringing > > endorsement of set "theory". > > 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 A calculation is an application of mathematics, but may actually be physics, or chemistry, or merely commerce. Accounting involves calculations, and while its cacluations involved may be in a sense mathematical, they are not Mathematics.
From: Han de Bruijn on 8 Dec 2006 04:05
David Marcus wrote: > That makes sense. And, most people have no understanding of the function > concept, so they don't see its utility or pervasiveness. Huh? Most people, especially the people who work with their hands, have a much better understanding of the function concept than you may think! An example. What is the best way to disassemble a device and assemble it again? (For example if you need to clean the parts or repair something) Take a big towel. Take the parts out of the device and put them on the towel in the same order as you have taken them out. After you have done your thing, put the parts back into the device, but now in the _reverse_ order as you have taken them out: last out, first in. The above is quite common practice among i.e. metal workers. The mathematical background of this is the formula for the inverse of a composed function: (A.B.C.D.E.F.G.H.K.L)^(-1) = L^(-1).K^(-1).H^(-1).G^(-1).F^(-1).E^(-1).D^(-1).C^(-1).B^(-1).A^(-1) Maybe they don't know it with their minds, but they _do_ it with their hands. My own conclusion has been: show some respect for "most people" ! Han de Bruijn |