Cantor Confusion
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From: Virgil on 14 Dec 2006 16:10 In article <2aa3c$45810b32$82a1e228$31266(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <1166011032.914983.204230(a)79g2000cws.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >>Han de Bruijn schrieb: > >> > >>>William Hughes wrote: > >>> > >>>>Tony Orlow wrote: > >>>> > >>>>>Well, the proof is simple. Any finite number of subdivisions of any > >>>>>finite interval will only identify a finite number of real midpoints in > >>>>>that interval, between any two of which will remain more real midpoints. > >>>>>Therefore, there are more than any finite number of real points in the > >>>>>interval. > >>>> > >>>>This just shows that the number of real points is unbounded. > >>>>It does not show it is infinite (unless of course you use the > >>>>fact that any unbounded set of natural numbers is infinite). > >>> > >>>Isn't unbounded the same as infinite, i.e. = not finite = unlimited = > >>>without a limit? > >> > >>Unbounded is potentially infinite but it is not necessarily actually > >>infinite. > > > > AS in honest mathematics, the one implies the other, in honest math one > > has neither or one has both. In ZFC and NBG, both. > > Let's repeat the question. Does there exist more than _one_ concept of > infinity? Isn't unbounded the same as infinite = not finite = unlimited > = without a limit? Please clarify to us what your "honest" thoughts are. > > Han de Bruijn Unless one is working in some system in which the axioms are set out in advance, one is operating in a fog. Without some assumptions (other than the rules of logic) one cannot deduce anything about anything. So one must start with SOME assumptions. It is sensible to require all those assumptions to be explicitly stated, so that one can be sure of precisely what is being assumed. Such collections of statements are called axiom systems. And it is plain that no sound mathematics can be developed unless based on some axiom system as its solid foundation. The only issue then is what system(s) of axioms. For arithmetic and analysis, ZFC and NBG have been found useful. For geometry, Hilbert's refinement of the Euclidean axioms has been found useful. It would be of great advantage to find a single axiom system sufficient to cover all of mathematics, but that has not yet been shown to be possible. However ZFC and NBG have each been shown to be sufficient for a great deal of mathematics, which is why so much attention is paid to them. There are some other possibilities along the lines of Category Theory or Topos theory that I understand are in the running as well, but are as well developed.
From: Virgil on 14 Dec 2006 16:11 In article <d1914$45811514$82a1e228$2825(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > William Hughes wrote: > > > Therefore, as there are at least as many digit positions in 0.111... > > as there are primes, there are an infinite number > > of digit positions in 0.111... > > Binary or decimal? Irrelevant!
From: Bob Kolker on 14 Dec 2006 16:18 Virgil wrote: > > And it is plain that no sound mathematics can be developed unless based > on some axiom system as its solid foundation. Arithmetic was around long before it was axiomatized and people were proving theorems about integers. For example Gauss and Euler. Bob Kolker
From: Lester Zick on 14 Dec 2006 17:13 On Thu, 14 Dec 2006 16:18:04 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Virgil wrote: >> >> And it is plain that no sound mathematics can be developed unless based >> on some axiom system as its solid foundation. > >Arithmetic was around long before it was axiomatized and people were >proving theorems about integers. For example Gauss and Euler. And apparently "-" was around long before "+". ~v~~
From: Jonathan Hoyle on 14 Dec 2006 17:43
Bob Kolker wrote: > Virgil wrote: > > > > And it is plain that no sound mathematics can be developed unless based > > on some axiom system as its solid foundation. > > Arithmetic was around long before it was axiomatized and people were > proving theorems about integers. For example Gauss and Euler. > > Bob Kolker True, but you are ignoring Virgi's adjective "sound". Calculus existed way back during the time of Newton and Leibniz, but you could hardly call their use of the infinite and infinitessimals at all "sound" by today's standards. It wasn't until Bolzano and Weierstrass made things truly rigorous in the 19th century was Calculus anywhere near sound. It is in fact their essential treatments that we are taught Real Analysis today, not Newton's. (Newton's work would be barely recognizable today with its "fluxions" and "fluents".) Bolzano and Weierstrass gave way to more rigor in numbers by Cantor, and then rigor in Set Theory by Zermelo and Fraenkel. Then with the wonderful contributions of Hilbert, Lebesgue, Godel, and others, mathemaatics today is far more rigorous than it was over a century ago. Even infinitessimals were consistently defined by Robinson. With the exception of Aristotle's Logic and Euclid's Geometry, much of mathematics would not be considered acceptable by today's standards. Even Guass and Euler played a bit fast and loose (although they were considered impeccably precise in their day.) In ancient times, arithmetic was discovered in much the same way physical laws were. "Hey, notice that when we do this, that always happens..." As centuries of very hard work, mathematicians have boiled arithmetic assumptions down to some basic axioms, and all of the remaining theorems flow forth. |