Cantor Confusion
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From: Eckard Blumschein on 21 Nov 2006 08:43 On 11/9/2006 12:17 AM, Virgil wrote: > As infinitely many paths pass through each edge no path can "own" more > than an infinitesimal share of that edge according to that "equal > sharing" rule. And unless we are in something like Robinson's > non-standard analysis, less than infinitesimal means zero. I wonder. I have to agree with Virgil. Is there something wrong?
From: William Hughes on 21 Nov 2006 08:45 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Every element except the first is "outside" at least one line. > > > > While it is true that there is no diagonal element that is "outside" of > > EVERY line, that is quite a different issue. > > > > > No, just this is the decisive issue. > > > > In ZF, statements with counterexamples are not theorems. > > One counter example contradicts ZFC: > There is not one single element of the diagonal which is not contained > in a line. This line contains this and all preceding elements. Quantifier dyslexia again. For every n there exists an line L(n) such that L(n) contains all elements m <=n Does not imply There exists a line L such that for every n, L contains all elements m <=n -William Hughes
From: Eckard Blumschein on 21 Nov 2006 08:52 On 11/12/2006 8:28 PM, William Hughes wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > As you stated "Goes on forever" is a property of the natural > numbers. The set {1,2,3,...} is just the natural numbers. So > this set must go on forever. Being admittedly not very familiar with set theory, I nonetheless wonder if sets are considered like something going on forever.
From: Eckard Blumschein on 21 Nov 2006 09:00 On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: > Every set of natural numbers has a superset of natural numbers which is > finite. Every! I am only aware of the unique natural numbers. If one imagines them altogether like a set, then this bag may also be the only lonly one.
From: William Hughes on 21 Nov 2006 09:07
Eckard Blumschein wrote: > On 10/30/2006 12:07 AM, William Hughes wrote: > > > But note, to define an integer, the definition must be > > communicated. We have only a finite number of bits > > to use for this communication (this includes comunicating > > the details of any compression scheme). So there are > > only a limited number of integers that it is possible to define. > > Among these is the largest integer that it is possible to > > define. > > > > If we claim that the only integers that exist are the ones > > that have been or will be defined, > > then there is a largest possible integer. > > > > - William Hughes > > Having looked around and realized a lot of impolite and opiniated > persons, I consider you, William Hughes, like a perhaps rare positive > exception. > > Hopefully you will not mistake me if I do no longer hesitate to > nonetheless comment on your reasoning. > While I do not question your last sentence being a correct conclusion if > the premise holds, I cannot not see any possibility to reasonably and > concisely quantify a largest possible number. > > Do we really have to define numbers? I tend to be satisfied when I have > an executable rule which may create as many natural numbers as I like, > as does Archimede's axiom. Executable means effectively about the same > as communicable. It excludes postulated a priori existence. > > Do not get me wrong. I consider the properties executable and > communicable like belonging to the abstract mathematical idea, not to > any physical limitation to it. The process of counting indefinitely is > never completely executable. Accordingly all created numbers are finite, > and there is no principial reason for a largest possible number to be > found. This is of course the standard position. It is not WM's position. On the larger question of whether the set of all natural numbers exists. Your "executable rule" defines what is commonly referred to as "potential infinity". The problem is the difference between saying The set of all natural numbers exists and We can generate an arbitrarily large set of natural numbers (i.e. the difference between actual and potential infinity) is mostly one of terminology. There is no element of the set of natural numbers that does not exist as an element of some arbitrarily large set. So there is absolutely no difference between the elements of "the infinite set of natural numbers" and the elements of "the potentially infinite set of natural numbers". About the only thing you can say about the "potentially infinite set of natural numbers" is that it is not a set. So what? You still have this potential, this thing that allows you to get new natural numbers whenever you want. This potential has all the properties of a set but the name. - William Hughes |