An uncountable countable set
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From: Virgil on 19 Sep 2006 16:55 In article <451035d2(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Randy Poe wrote: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> Mike Kelly wrote: > >> > >>> Infinite natural numbers. Tish and tosh. Good luck explaining that idea > >>> to schoolkids. > >> Look who is talking. Good luck explaining alpha_0 to schoolkids. > > > > I think I was 10 when I saw the proof that the rationals are > > countable, and first saw the notation "aleph_0". I don't remember > > having a problem with it. > > > > - Randy > > > > Perhaps you blocked it out. I know that, as soon as I was presented with > the "proof" that there are as many evens as all naturals, even though > that's only half of them, I detected a logical contradiction. TO's alleged "logical contradiction" is just his mental indigestion over an idea to subtle for him to grasp. The infiniteness is different from finiteness. The > "equivalence" between the dense set of rationals and the sparse set of > naturals, on the real line, only clinched it for me. It's simply > unconscionable. The theory is wrong. Does TO deny the existence of a bijection between the set of naturals ad the set of rationals? What TO cannot stomach is the notion that the SET of rationals has no necessary order built into it, but can be ordered in any way one likes. That is not the same as for the ordered field of rationals, where the order is built into the structure.
From: Virgil on 19 Sep 2006 17:00 In article <45103c66(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > A natural number is a whole number. That may include both positive and > negative whole numbers, because we can count equally well up or down. What does TO include as "whole numbers"? Does he mean the set of integers? If so, why not use the standard term, and if not what does TO mean? If TO chooses to include as "whole numbers" anything that cannot be expressed with a sign and a finite binary string, then he is outside of mathematics. > That means we take one unit at a time. We can treat 1 as 0 and vice > versa, if we wish. They're only symbols. So, we can define increment and > decrement as "find the rightmost this and invert rightward form > there", or the "rightmost that". Or, m and w, if you recall. Math is all > about the relationship between measure and symbolic representation. Can > we easily decrement ...1111? Uh, ...1110. Increment? TO is off in his delusionary world again, and well outside of any mathematical world.
From: Virgil on 19 Sep 2006 17:02 In article <45104f87(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > I'm well aware of the notions of 'potential infinity' and 'actual > > infinity' that go back through at least a couple thousand years of > > philosophy and philosophy of mathematics. > > So, you ask for axioms defining the terms, then cannot gives any > "explication" yourself? I gave you axioms a little while ago. Nothing that would fit into any other axiom system, and not enough to form a system by themselves.
From: MoeBlee on 19 Sep 2006 17:13 Tony Orlow wrote: > So, you ask for axioms defining the terms, then cannot gives any > "explication" yourself? 1. I didn't say I couldn't give any explication. I said that I cannot give an explication that is any better than those usually found in general writeups on the subject. 2. I am not the one whose mathematical explanations depend on those terms. YOU are the one who is using those terms to propose to explain how a set can be unbounded but finite. So, I am not the one who needs to explain those terms, since they're NOT terms that I rely upon, but rather YOU are the one who needes to explain those terms since they are terms that you DO rely upon. > I gave you axioms a little while ago. Those were supossed to be axioms or defintions or what? They fail as definitions for the reason I stated. As axioms, they use yet more undefined terminology and you don't give any indictation as to what the primitives, remaining axioms, and defintions are supposed to be part of your system.. MoeBlee
From: David R Tribble on 19 Sep 2006 19:33
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? 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. |