An uncountable countable set
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From: Han de Bruijn on 20 Sep 2006 03:38 David R Tribble wrote: > mueckenh wrote: > >>>Nothing has changed. There is no complete set of natural numbers. Any >>>set that can be established is a finite set. Hence, the probability to >>>select a number divisible by 3 is 1/3 or very very close to 1/3. > > Virgil wrote: > >>That presumes that the allegedly finite set of naturals that can be >>constructed is nearly uniform with respect to divisibility by 3 at >>least, and probably by other numbers as well. What is the justification >>for this assumption? Wolfgang says litteraly: "_or_ very very close to 1/3". > It also presumes that this allegedly finite set of naturals contains > exactly 3n elements, in order that the distribution is an exactly > uniform distribution. Likewise, this set must contain 2n elements > so that exactly 1/2 of the naturals are even; and this set must > contain 5n naturals so that exactly 1/5 of them are multiples of 5; > and so forth. > > This presumption leads us to conclude that the number of naturals > (in the finite set of naturals) is 2 x 3 x 5 x 7 x 11 x 13 x ..., i.e., > equal to the largest composite integer. Which is also presumed > to be finite, of course. I find this rather hard to accept. A more precise treatment (with limits) is in: http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962?hl=en& The gist of the argument being that for large n and a << n the error between floor(a/n)/n and 1/a becomes neglectable. (Take a = 3). Han de Bruijn
From: Han.deBruijn on 20 Sep 2006 04:53 Mike Kelly wrote: > The limit of a sequence need not have a property that each of its > elements has. The limit of the sequence 1,1,1,1,1,1,1,1, ... ,1, ... is 1. Han de Bruijn
From: Tony Orlow on 20 Sep 2006 09:51 MoeBlee wrote: > Tony Orlow wrote: >> Given N is the standard naturals: >> >> Actually infinite(s) = E seS A neN index(s)>n >> Potentially infinite = A seS index(s)eN ^ ~E neN index(s)<>n > > First, that doesn't even make sense as a definition, since you have a > free variable 'S' (upper case 'S') that is in the definiens but not in > the definiendum. Sorry, hastily written. The 's' in the definiendum should be an 'S', obviously, and the second rule isn't quite right, on second thought. >> Actually infinite(S) <-> E seS A neN index(s)>n >> Potentially infinite(S) <-> A seS index(s)eN ^ A neN E seS index(s)=n. > > Next, what is the logistic system? First order logic What are the non-logical primitives? e > index > What are the non-logical axioms? These are definitions of potential vs. actual infinity. Consider them axiomatically stated if you wish. What is the definition of 'index'? The count of elements up to and including a given element. The number of steps between and element and the first element in a linear order, plus 1. > What is the definition of '>' (if it's different from the standard > 'greater than' relation on natural numbers)? What is the definition of > '<>'? Forget '<>' for now - I removed it. '>' is the standard order relation on real quantities such that a>b ^ b>c -> a>c. > > But please do not give me yet more definitions that don't eventually > lead to an endpoint with primitives only. > > Anyway, please, please, please, why don't you just learn something > about mathematical logic so that you'd know how to formulate a > definition. A definition does not have a free variable in the definiens > that is not in the definiendum. You shouldn't even need mathematical > logic for that; you instincts alone should warn you. It was a typo. Whoops. Obviously, though, I wasn't defining potentially infinite elements of sets, but potentially infinite sets. > > MoeBlee > ToeKnee
From: Tony Orlow on 20 Sep 2006 10:02 Han de Bruijn wrote: > Tony Orlow wrote: > >> Han de Bruijn wrote: > >>> Precisely! Mathematicians get confused by the idea of a "bijection", >>> which is an Equivalence Relation, which in turn is a "generalization" >>> of "common equality" (yes: the one in a = b). But the funny thing is >>> that EQUALITY HAS NEVER BEEN DEFINED. So there is actually nothing to >>> "generalize". Equivalence relations are a "generalization" of nothing. >>> >>> But, fortunately, reality is more simple than this. Every equality is >>> an equivalence relation. And every equivalence relation is an equality. > [ ... snip ... ] >> >> I agree with that last statement, but would disagree that equality is >> not definable. It depends on difference, most basically, and where >> none is detected, two things can be said to be equal. > > Panta rhei, ouden menei (= everything flows, nothing remains the same). > There is no such thing as "difference, [...] where none is detected" in > nature (and culture). All equality is "in some sense" and relative. But > a picture says more than a thousand words: > > http://hdebruijn.soo.dto.tudelft.nl/fototjes/gezocht.htm > > Han de Bruijn > Consider the equally spaced staircase from (0,0) to (1,1), as the number of steps increases from 1 without bound. Is it the same as the diagonal line? Inductively we can prove that the length of the staircase is 2 at every step. Does it really suddenly become sqrt(2) in the infinite case? By the measures of point set topology, all points in the staircase become indistinguishable in location from the those of the diagonal, so by this thinking, all difference has disappeared, and the two objects are equal. However, using a segment-sequence topology, staircase n is the concatenation of n pairs of segments, denoted by x and y offset, described by {0,1/n} {1/n,0}, whereas the corresponding segments of the diagonal, between the points on the diagonal where perpendicular lines pass through the vertices of the staircase, are of the form {sqrt(2)/2n,sqrt(2)/2n}. The fact that the directions of the two curves are different at every point explains the difference in length, but this distinction cannot be detected by looking at pointwise location alone.
From: Tony Orlow on 20 Sep 2006 10:07
mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: >> Yes, I can see now that these are all finite sets. >> >> And which are proper subsets of infinite sets. The set of all naturals >> that have been written now, for example. Obviously it's an ever >> growing set as time goes on, and will never contain the entire set >> of naturals that are possible. So it's simply a finite subset of N, >> and always will be. >> >> Somehow you are using this fact to "prove" that N can't exist, perhaps >> employing some marvelous mathematical logic that has not been >> tainted by mainstream teachings. You show several finite sets. >> How do they prove anything about infinite sets? > > "a reasonable way to make this conform to a platonistic point of view > is to look at the universe of all sets not as a fixed entity but as an > entity capable of 'growing', i.e. we are able to 'produce' bigger and > bigger sets" [A.A. FRAENKEL, Y. BAR-HILLEL, A. Levy: Foundations of Set > Theory, 2nd ed., North Holland, Amsterdam (1984) p. 118]. > > Why should simple infinite sets exist in another way? Just because > there is an axiom which cannot be satisfied like the axiom that there > be a straight bent line? > > Regards, WM > You mean like the circumference of an infinite circle? :D Tony |