An uncountable countable set
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From: MoeBlee on 27 Aug 2006 17:59 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Logic determines truth. Induction is more than just a form of proof. > > > > Logic determines VALIDITY. And induciton is indeed more than just a > > form of proof. But your concept of induction is uninformed, > > superficial, and confused. > > > > MoeBlee > > > > That second sentence refers to inductive logic, as opposed to deductive > logic, of which "inductive" proof is a form. My second sentence referred to induction in deductive mathematics. If you were referring to, inductive logic, then of course, my remarks would have to be adjusted in that light. > When you say my concept of induction is "uninformed, superficial, and > confused", I would disagree. In my opinion, those that think it just > "springs from the axioms" without a sound logical basis are the confused > and uninformed ones. That is again a clear demonstration of your ignorance. If you read just the least bit on this subject, then you'd find how induction is sourced in mathematical logic. MoeBlee
From: MoeBlee on 27 Aug 2006 18:10 Tony Orlow wrote: > All I have > suggeted as a change to inductive proof is that we consider any infinite > value to be greater than a finite one, and consider equalities between > expressions proven inductively true for al x greater than some finite y > to remain true even when x is infinite, since x is still greater than > that y. Give me your logistic system, axioms, primitives, and definitions. Then we'll talk. In the meantime, all of what you just said is Tonyspeak babble, given that you have not defined your terms nor said from what axioms and by what logic your claims are supposed to come from. MoeBlee
From: MoeBlee on 27 Aug 2006 18:27 Tony Orlow wrote: > So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I > never thought so. It just occurred to me that even with the naturals isomorphically embedded and not a subset of the reals, your demand that the naturals are a subset of the reals can be met just by renaming. Call the finite ordinals 'first-stage-naturals', and each for step in the construction, keep nicknaming the members of range of the embedding as 'second-stage-naturals', etc. (the construction of each system is like a "staging area or staging exercise" for the next constructed system). Then nickname the particular subset of the reals as 'naturals'. It's all trivial and irrelevent anyway, since these are but nicknames while the formal theory has only predicate and operation symbols to assign. We assign, say, the 32nd 1-place operation symbol to the particular set of concern here that is a subset of the set of reals. Whether we nickname that as 'the naturals' or as 'the embedded set from the naturals' is merely a question of informal nicknaming practice and has little mathematical importance. You learned, as we all did, in about the fifth grade, that the naturals are a subset of the integers are a subset of the rationals are a subset of the reals. However, when we get into the real serious stuff, we see that with certain constructions, that fifth grade notion of subset is not precise and is sharpened into a rigorous formulation in set theory. MoeBlee
From: Dik T. Winter on 27 Aug 2006 20:31 In article <1156585516.788671.21150(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > And similar holds for the width. So either you state that nor the height, > > nor the width are 1, or you state that both are 1. That depends entirely > > on your viewpoint. But as the height and width of my staircase at step n > > is 1 - 1/2^n, I would say that the height and width of the completed > > staircase is the limit of that number. Apparently you disagree. > > > > > There is no asymptote. Hence your > > > comparison of 1 and oo fails. > > > > Yes, I wanted to simplify the picture to show that what holds for the > > height also holds for the width. > > > And that is just the deviation from the natural numbers. Here the 1 in > width aleph_0 is asserted to exist for the set of natural numbers > *without* a number omega. What TO and I are arguing is that either none > of them do exist or that both exist. Yes, you are arguing, but not providing a proof. > So aleph_0 is the cardinality of > the set including the *number* omega. Right. > Less numbers are not sufficient > to obtain aleph_0. Let us remove omega from that set. What is the resulting cardinality? As there is a bijection between that set and the set including omega, their cardinalities are the same. > Your example shows very clear that either both LUBs have to be included > - or none. In my opinion we choose none. But whatever you choose, you > cannot avoid to see the symmetry, can you? I see the symmetry, I also choose none. And I think I have stated that already. > > > As long as we are in the naturals we know: Each natural is the sum of > > > all preceding differences. Infinitely many differences require an > > > infinite sum. > > > > Wrong. In mathematics the concept of "infinite sum" is not defined, > > But if there are infinitely many naturals then the infinite sum is > necessary. Not at all. Why do we need it? > > as I > > have told you already *many* times. We can talk losely and state that > > the infinite sum of all those numbers is indeed aleph-0, but note that > > that is talking losely. All infinite sums as in sum{n = 1 .. oo} > > used in mathematics are defined by limits. > > You told me that set theory does not involve limits: "As set theory > does not know about limits, I fail to see that". Indeed. Set theory does not define limits. For the kind of limits I quoted above, you need ordering, concepts of measuring things and whatever. Outside the field of set theory. The limit I give is *not* a set theoretical limit. It involves quite a bit more. > > > If you insist that there are infinitely many naturals and > > > if infinity is a number aleph_0, then there is also a magnitude > > > aleph_0. > > > > Makes no sense, again. What do you mean with the term "magnitude"? > > Just the same a size. OK, I kind of understand. I do not state that infinity is a number aleph_0. There exists a host of infinities. And aleph_0 is the infinity that gives the equivalence class of sets that are in bijection with the set of natural numbers; the canonical set with cardinality infinity. > > The only connection I know of is where it is used in connection > > with real or complex numbers. And it can also denote some norm > > (in general Euclidean norm) in vector spaces. And as the finite > > cardinals equate to the non-negative integers, that equate to their > > magnitude, you might consider a cardinal number to represent its own > > magnitude. Do you mean that? > > Exactly. And in addition, as long as the numbers are finite, ordinal > and cardinal is the same size. That is just why the cardinal cannot be > infinite as long as the ordinal is finite (as expressed by "the > infinite number of finite numbers"). But indeed. If the ordinality of a set if infinite (w or larger), then so is its cardinality (aleph-0 or larger). On the other hand, if the cardinality of a set is infinite (aleph-0 or larger), and if it can be well-ordered, so is its ordinality after well-ordering (w or larger). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 27 Aug 2006 20:46
In article <1156585631.775313.153510(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > 1/3 is that path which turns left, right, left, right, left, right, ... > > > and never ceases to change its direction. > > > Which edge do you miss? Which dge could not be enumerated? > > > > As in your tree, by definition of the tree, each edge terminates at a node, > > what is the number of the edge that terminates at 1/3? What is the > > number of the edge that terminates at 1/5? > > I don't understand these questions. 1/3 and 1/5 are infinite paths > consisting of infinitely many edges. You state that you can number the edges. Each edge terminates at a node, and if 1/3 and 1/5 are in your tree, there must be edges that terminate at those nodes. In that case you should be able to tell me what the numbers of those edges are. Or do you now claim that there is *no* edge that terminates at those nodes? If so, where do the nodes come from? Out of thin air? > > As the set of all edges is countable, you > > should be able to state that. The problem with your counting method of > > edges (count first level 1, next count level 2, etc.) you will already > > have exhausted all natural numbers by counting all edges that are at the > > end of finite paths. > > There are no finite paths and there are no ends at all in my tree. During your construction process all paths are finite, and there is an edge that terminates at a node terminating the path. I meant the paths during the intermediate process of construction. > If > you think that 0.1 would be represented by a finite path, what should > then be represented by the path 0.1000...? The same. > By your arguing you would never be able to count the digits of 1/3 > because "you will already have exhausted all natural numbers by counting > all edges that are at the end of finite" sequences. Wrong. To count digits we need only one number for each level. > > > Do you believe that the set of edges of 1/3 is uncountable? > > > > That is not the set of all edges in the tree. > > No. But would you say that the set of all edges of 1/3 and of 1/5 and > of 1/7 were uncountable? No. Assuming of course that 1/3, 1/5 and 1/7 are in your tree. > How many paths are required to obtain an > uncountable set of edges, according to your opinion? Depends quite a bit on a host of questions. In your tree with terminated edges, also the paths are terminated, so neither becomes uncountable. But also 1/3 is not in that tree. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |