Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 27 Jul 2005 23:00 In article <MPG.1d51a1178def0152989fb9(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d504db641c57d4a989f97(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > Virgil said: > > > Since I was at all times dealing with a maximal binary tree, and as all > > such are tree isomorphic, there is no way that I could have switched > > trees. > You switched interpretations of the tree, using a tree where the paths were > digital numbers to prove the nodes were countable, and using a tree where the > paths were subsets of the naturals to prove the paths were uncountable, based > on the notion that a power set of the naturals is not "countable", which is > nonsense anyway. If the base was the same tree, that is all that matters. The fact that I used different proofs to prove different things hardly constitutes a valid objection. What TO really is objecting to is the validity of both proofs. I really did prove that the set of branches (or the set of nodes) of a maximal binary tree are denumerable but that the set ofpaths is not. THAT is what TO objects to. > > > > > > If TO prefers it, I could rephrase the proof so that each branch is > > > > a finite string of binary bits and each path is an infinite string > > > > of binary bits. The same conclusion holds in either form, that the > > > > set of branches (or nodes) is countably infinite set but the set > > > > of paths is not. > > > > > Why don't you try it with a SINGLE tree, rather than using two > > > different trees? Try it with a branch being a bit and a path being a > > > bitstring, or a branch denoting membership of an element in a path > > > representing a subset. Just don't chnage trees in the middle of your > > > proof, and you may get a correct result. There are precisely half as > > > many paths as branches in any complete binary tree, finite or > > > infinite. It's this kind of nonsensical conclusion that demonstrates > > > the inconsistencies of set theory. > > > > Theorem: In a maximal binary tree, the set of nodes and the set of > > branches are both counably infinite but the set of paths is uncountably > > infinite. > An INFINITE maximal binary tree, I assume you mean. > > > > Proof: > > In a maximal binary tree, every node except the root is the terminus of > > exactly one branch, and the position of every node except the root node > > can be indicated by a finite sequence of L's for left branches and R's > > for right branches connecting the root node to that given node. This > > same string identifies the branch terminating in that given node. > So you have the same number of branches as nodes, minus 1. You say > each path is finite, but that is not the case in an infinite binary > tree. What I said, for those who can read better than TO, is that the sequence of branches and nodes between the root node and any other node is finite. Since in a maximal binary tree no complete path ends in a node (or ends at all) This finiteness does not apply to any such path. > > > > This gives a 1 to 1 correspondence between the finite strings of L's and > > R's and the branches (or the non-root nodes). Such sets of finite > > strings can be countable. > Did you just draw a bijection between the branches and the paths? No! Try reading what I actually say, TO, rather than what you want me to have said. > > > > We do the same thing, with the same tree, for paths, except that since > > no path has an ending node or a last branch, each path-string is of > > infinite length. > Um, wait a minute. Why? Just because TO can't read is no reason for anybody else to stop. Did you not say "every node except the root node can be > indicated by a FINITE sequence of L's for left branches and R's for right > branches connecting the root node to that given node"? Aren't those the paths > in the tree NO! A path in a maximal binary tree does not have any terminal or leaf node, it goes on forever, just like the tree of which is is a part. > > > > The set of finite branch-strings is countably infinite. > > The set of infinite path-strings is infinite but not countable. > So strings of branches are different from paths? Interesting. Care to explain > the difference? A finite string of branches has an ending node, but a path does not. > > > > Q.E.D. > > >It's a dishonest > > > mathematical trick, and typical in this area, whether conscious or > > > not. > > > > It is mathematical, and might be called a trick, since it is so easy, > > but it is straightforward, and not in the least dishonest, and it does > > precisely what I said it would do, show that the set of branches (or, > > equivalently, the set of nodes) is countably infinite but the set of > > paths is infinite but uncountable. > Sure, if the paths are finite, and then suddenly infinite. I have been very careful not to speak of paths when I meant finite chains of branches and nodes. That TO still sees "path"s where none exist may be due to his excessive self-medication.
From: Poker Joker on 27 Jul 2005 23:01 "Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote in message news:dc9f0t03cj(a)drn.newsguy.com... >>> >Robert Low said: >>> > >>> >> OK, so how many elements are there in the set of all finite >>> >> natural numbers? > > Tony replied. > >>>>Some finite, indeterminate number. > > That is an out-and-out contradiction. Let FN be the > collection of all finite natural numbers. You say that > FN is finite. You say that that means that its size is > equal to some finite natural number. He never said that. He said "Some finite, indeterminate number." He didn't say "Some finite, indeterminate natural number." But even if he did, mathematicians have their own meaning of words and therefore his natural numbers might be different than mathematicians. > So call that number > L. If L is finite, then it must be an element of FN, > because FN is the collection of *all* finite natural > numbers. But that means that FN contains at least L+1 > elements: 0, 1, 2, ..., L. That contradicts the claim > that FN contains exactly L elements. Too bad he didn't imply that stuff. > Your theory is self-contradictory. Not that *you* would > ever notice the contradiction, because you are just making > things up as you go. You are just playing, not caring whether > what you're saying makes sense or not. I think you are the one that is trying to put nonsense into his post. > -- > Daryl McCullough > Ithaca, NY
From: Virgil on 27 Jul 2005 23:02 In article <MPG.1d51a192a95c0c4f989fba(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > Yes, that's the appropriate model. > > > > >and infinite funtime, it could conceivably produce infinite results. > > > > HOW?? No matter how long it runs, EVERY printed value is finite. > > WHEN EXACTLY do you think it would start producing "infinite" values > > (whatever those might look like). > > > > > At the point that the runtime became infinite, which is obviously not an > identifiable point. Never! But the prinout will, in finite time exceed any preassigned finite value, which is precisely what is meant by unbounded.
From: Daryl McCullough on 27 Jul 2005 22:48 Tony Orlow (aeo6) wrote: >> Who cares about S_oo? That's not one of the S_n. >You mean there aren't an infinite number of n in N? Okay, maybe this will make it clear to you: There is exactly 1 natural number that is less than 1: namely 0. There is exactly 2 natural numbers that are less than 2: namely 0 and 1. There are exactly 3 natural numbers that are less than 3: 0, 1, and 2. You see the pattern? For any n, the number of naturals less than n is equal to n. Now, how many natural numbers are less than infinity? -- Daryl McCullough Ithaca, NY
From: Poker Joker on 27 Jul 2005 23:07
"Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote in message news:dc9c6502t8a(a)drn.newsguy.com... > Basically, for any particular statement S, Tony will > sometimes claim S, and sometimes claim the negation. > Whichever one is most convenient for his argument. So > sometimes there is a biggest finite natural, sometimes > there isn't. Sometimes mathematical induction is valid, > sometimes it isn't. Sometimes you use bijections to > prove that sets are the same size. Sometimes you don't. Tony makes a model Cantorian. > -- > Daryl McCullough > Ithaca, NY > |