An uncountable countable set
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From: Tony Orlow on 20 Aug 2006 12:36 David R Tribble wrote: > Tony Orlwo writes: >>> If the formula applies to an infinite number of finites, then does the >>> sum of the aleph_0 finite naturals equal (aleph_0^2+aleph_0)/2? > > Dik T. Winter wrote: >> I think there is a misunderstanding. The formula >> sum{i = 1 .. n} i = n * (n + 1) / 2 >> holds for every finite number n, so it holds for infinitely many finite >> numers n (as there are infinitely many finite numbers n). But we can >> not switch to an infinite sum (that is something different). > > Not without proof that the formula works for infinite n, anyway. > But we're still waiting for Tony to provide that proof. > David, this is not a matter of proof given standard definitions, but a redefinition of the principle of induction such that it applies to the infinite case. When an inductive proof demonstrates that f(x)>g(x) for all x greater than some y, all infinite values falls into that category. So, it simply amounts to removing the restriction that induction only applies to finite n. Once this restriction is lifted, then we can "prove" that, for instance, for all n>2, n^2>2n, and therefore this also holds for all infinite n. The implications of this assumption are a very nice system of ordering infinities which goes far beyond what cardinality can even hope to attain. If you want to see that it applies to infinite n visually, picture the unary enumeration in question, which forms a staircase, from an infinite distance. You have some sort of square divided diagonally. whatever infinite n you have chosen, you have an nxn square divided in half. That's your n^2/2. The remaining n/2 is a linear measure, being 1 dimensional, and does not add more than infinitesimally to the area occupied by the set of units. > > Tony Orlwo writes: >>> In standard theory, would this not equal aleph_0, and if so, does it make >>> sense that sum(n=1->aleph_0: 1) = sum(n=1->aleph_0: n), when n>1 for all >>> n>1? The scond would appear to be clearly a larger sum. > > Dik T. Winter wrote: >> No that does not make sense. sum{n = 1 .. oo} 1 is not defined, the same >> holds for sum{n = 1 .. oo} n. If you want to use them you have to >> provide a definition for them. > > But Tony thinks he has provided a definition, based on his "Big'un" > number. The problem, of course, is that he simply assumes that > arithmetic operations on Big'un work "es expected" without providing > any proof of that whatsoever. > > It's one thing to provide a definition, it's quite another to prove > that it is a consistent definition. > I think the responsibility lies with you to point out an inconsistency that arises from my assumptions. Set theory is not "proven true". It cannot prove itself consistent. It took years of trying, with some succcess in detecting failures, to refine set theory so that it was somehow actually consistent. But, it cannot "prove" itself so. Please show where I am being inconsistent, not with set theory, but within my own assumptions. Remember, I don't claim to believe in transfinite set theory, and don't intend to be consistent with it. What does work is the combination of IFR, N=S^L, infinite-case induction, and Big'un. There is no contradiction between those. Thanks, TO
From: David R Tribble on 20 Aug 2006 12:36 Tony Orlow wrote: > ... it's inductively provable likewise that the size of a successor > ordinal is one greater than its max element. That would make aleph_0 1 > greater than the max finite natural. Except that Aleph_0 is not a successor ordinal. It's not even an ordinal. And the fact that there is no maximum finite natural. So your "inductive proof" has a couple of holes in it. > The only way around this is the declaration of the limit ordinals, > but that's just a philosophical monkey wrench. Ordinals and cardinals are necessities if we want to talk about set "order" and "size" in any kind of logical, well-defined way. What do you call the least ordinal that is greater than all finite ordinals? You don't have a name for it, do you? What do you call the "size" of a countable set with no end? You don't have a name for it, do you? What do you call the "size" of an uncountable set? You don't have a name for it, do you? If you are going to keep taking this political approach to mathematics, condemning what's already established and functional but not providing any workable alternatives, you might want to consider running for office instead.
From: Virgil on 20 Aug 2006 12:36 In article <1156082279.755828.80850(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > It is fairly easy to see that if there is no injection from the set of > > naturals to any proper subset, then there must be a largest natural. > > > > And if there is such an injection, then the set is Dedekind infinite. > > There is an injection from the number of natural numbers into the sizes > of natural numbers. That is not the issue. Is there an injection from N to a proper subset of N? I claim that the mapping s: N -> N : n |-> n \/ {n} is such an injection, so that N is Dedekind infinite.
From: Virgil on 20 Aug 2006 12:39 In article <ec9pqd$6iu$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > Here is my problem. Given normal combinatorics But by eschewing any axiom system, TO cannot educe "normal combinatorics". And even if he could, his intuitive assumptions would make the whole structure inconsistent.
From: Virgil on 20 Aug 2006 12:43
In article <1156082414.663394.94630(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > There is no question that only countable sets of numbers can be > > > constructed in the mathematical sense. > > > > That depends on what construction methods are allowed. > > No. That "Mueckenh" denies it does not make it false. > > > Inherent in the > > axioms of ZF and beyond are the existences of uncountable sets. > > Therefore ZF and beyond is inherently false. By what standard? Where does "Mueckenh" get his "truths" from? What are his basic truths, his axioms? Until "Mueckenh" can specify them, he is only expressing opinions and has no truth at all. |