An uncountable countable set
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From: Tony Orlow on 5 Sep 2006 20:21 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> MoeBlee wrote: >>>>> Tony Orlow wrote: >>>>>> Yes, and the universe is consistent by definition, so math should be >>>>>> consistently overall as well. >>>>> Unless the universe is a set of sentences, the notion of consistency >>>>> does not even apply. >>>> The universe is governed by the properties of the elements within it, >>>> which properties are statements true about those elements. >>> You are taking properties to BE statements. That's fine if you have >>> some coherent philosophy to support it. > >> Actually, I am taking properties to be truth values applied to >> statements about a given object. "It is red" is a statement with a truth >> value close to 1 for most fire engines, and a truth value near 0 for >> most newspapers (except in old jokes). Fire engines have the property >> that that statement is usually true for them. So, a property is a truth >> value associated with a statement. Better? > > So, by directly plugging in your definition of 'properties', your > previous statement becomes: "The universe is governed by the truth > values applied to statements about given objects of the elements within > it, which truth values applied to statements about given objects are > statements true about those elements. > > Whatever that means, it is difficult to see that it implies that > consistency is a property of the universe as opposed to a property of > sets of sentences. > > MoeBlee > What it means becomes much more apparent when you realize, as I would think would be obvious, that the "given objects" are "those elements". Then we can remove the redundancies in your formula above and get: "The universe is governed by the truth values applied to statements about the elements within it, which truth values applied to those statements are true about those elements." A better way to view the universe of sets is as a relation between the objects within it and 1-place predicates concerning objects, where each object has a truth value associated with each predicate, between 0 and 1 inclusive. Thus, these predicates define subsets of the universe, eh? :) Tony
From: David R Tribble on 5 Sep 2006 20:23 mueckenh wrote: >> All we can attach to it is the number of elements known >> or existing. Disregarding physical constraints ... > David R Tribble schrieb: >> I was not aware that abstract mathematical concepts (e.g., sets) >> had any physical constraints. > mueckenh wrote: > Then you should learn it. It you are unable to physically (i.e. in > written form or in your mind) distinguish all the elements of a set, > then the set does not exist. Nope, that still does not explain why abstract mental concepts are limited by physical constraints. I don't believe it. mueckenh wrote: >> ... we can assume that the >> elements of the set of natural numbers are counted by the largest >> natural number temporarily known. > David R Tribble schrieb: >> So how many numbers are in this sequence?: >> 1, 2, 3, ..., 10^100, 10^100+1 > mueckenh wrote: > How many can you write down? To give you a hint: Can you determine > whether the first 10^100 digits of pi make up a naturall number? I can prove that they do. > Exchange the last igit of this number P by 6 and find out whether the > new number is larger than P or not. If I'm allowed to use base-16 notation, I can do that rather easily. (You're probably not aware that there is an formula for finding any hexadecimal digit of pi.) David R Tribble schrieb: >> What exactly is it about your set theory that physically constrains >> your sets to being less than some arbitrarily chosen maximum size? > mueckenh wrote: > Not arbitrarily cosen. There is no means in the universe to surpass > this amount of information. How do you know? You have proof?
From: Tony Orlow on 5 Sep 2006 20:29 MoeBlee wrote: > Tony Orlow wrote: >> I find this last sentence vague because of the word "induction". If you >> refer to inductive proof, then I am familiar with the Peano axioms and >> the underlying logical construction which makes inductive proof valid. > > You have only a faint idea. If you actually read a book on the subject, > you'd find how much deeper, richer, and rigorous this is. > >> If you refer to inductive logic, the formulation of rules from instances >> of fact, then there are statistical methods coupled with feedback that >> make it possible. Which were you talking about? > > Mathematical induction. Inductive sets, et al. As in mathematical logic > and set theory, which is deductive. Not inductive logic as in the other > sense of inductive - empirical based inference (or however you want to > define it). > > MoeBlee > So, what is it you think I DON'T get about inductive proof, sets, and recursion? As far as I can tell, very few dare to question the predefined rules as set forth, but I have yet to see any valid counterexample to my rules regarding inductive proof in the infinite case. Where an equality between expressions is proven for all n, it is valid for infinite n. Where an inequality is proven for all n greater than some finite m, and the difference upon which the inequality is based does not have a limit of 0 as n->oo, it holds also in the infinite case. The staircase in the limit vs. the diagonal line was a valiant effort at a counterexample, but obviously flawed in its reliance on the limitations of point set topology, and the only other attempt was based on a half hidden limit of 0 for the nested inequality as n->oo, based on a function discontinuity, so was dismissible without any fanfare. So, if you have what you think is a valid counterexample which shows how little I know about inductively defined structures and how they suddenly change character when n=oo, well, bring it on. :) Tony
From: Tony Orlow on 5 Sep 2006 20:32 MoeBlee wrote: > Tony Orlow wrote: >> And yet, you are saying it's not technically correct. So, 1 is a natural >> but not a rational or real? If rigorous formulations come to that >> conclusion, then rigor does not ensure correctness. > > You're hopeless. You didn't understand a thing I said, which may be my > fault for not providing adequate explanation in the context of your > ignorance of the subject; but it is no my fault that you won't even > look at a book on mathematics, such as even a introductory text in real > analysis. > > MoeBlee > Your whole point here is ludicrous. You are arguing that the naturals are not a subset of the rationals, which are not a subset of the reals. While each superset may require a more complex construction than the subset, all elements of the subset are covered by the superconstruction. As points on the real line, they are all real numbers. What you're saying amounts to what I said above, which is indeed absurd. So, no, I don't understand a thing you're saying, when you say naturals aren't reals. Sorry. Tony
From: Tony Orlow on 5 Sep 2006 20:36
Virgil wrote: > In article <1157471262.974912.294510(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Virgil schrieb: >> >>> In article <1157367209.318653.75760(a)p79g2000cwp.googlegroups.com>, >>> mueckenh(a)rz.fh-augsburg.de wrote: >>> >>> >>>> Counting is the most primitive version of addition. >>> Counting is not addition at all. >>> >> So you cannot even count, in your infinite unphysical mind? > > Counting precedes adding, in that one may count when unable to add. > > But how is one able to add without being able to count? > > A primitive version of counting, from which the word "calculus" > allegedly derives, is Greek shepherds in classical times counting their > flocks by pairing off their sheep with pebbles (calculi). Yes, counting does precede addition. Addition is repeated increment, multiplication is repeated addition, exponentiation is repeated multiplication, tetration is repeated exponentiation... Increment is fundamental. It's not just successor, but successor with measure, and this is what generates the natural numbers. :) Tony |