An uncountable countable set
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From: Virgil on 21 Aug 2006 16:51 In article <1156189552.184903.323170(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > But often some expression like 0.111... is ivolved where not all digit > positions can be indexed by natural numbers and n is understood to > approach something I do not know. Your ignorance is not an acceptable excuse to those of us who have no trouble indexing every digit of "0.111..." by a natural number. > > The cardinal numbers do not > > form a field, only a ring (and not even a division ring). So > > arbitrary division is not defined, definition is not even possible. > > Division was possible and was practised in fact long before rings and > fields were known. Not outside of what later became known as rings. > > Do you agree that aleph_0 = aleph_0 ? > > Do you agree that division by n is a process which leads to a partition > of a number (of units) into n equal shares (and possibly a remaining)? Only when the cardinality of the set of units being divided is a natural and the divisor is a positive natural. Division of rationals by rationals or reals by reals is quite different. So we have no reason to suppose that divisions involving other than naturals need behave like division of naturals, or even be possible without specific definitions of what it is and how it works.. > > One must have a very restricted mind to believe that without fields and > rings division was impossible. The set of positive naturals is not even a ring, but it allows at least two forms of division, and this has been recognized by most of us from well before "Mueckenh".
From: Virgil on 21 Aug 2006 17:16 In article <1156189715.669425.236390(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > TO fails to note that induction does not apply. > > The form of induction is: > > Let F(x) be a predicate whose argument, x, is a member of a minimal > > inductive set S.* > > If F(x) is true for the first member of S > > and > > if whenever F(x) is true then F(successor(x)) is also true. > > then > > the principle of induction says that F(x) is true for each member of S. > > In particular induction says that each member of the set of natural > numbers is a member of a finite set, i.e. is covered by a finite > sequence. Then "Mueckenh" is using a different form of induction than I am. My form defines an inductive set as a set S together with an injective function s:S -> S such that (1) there is a unique member of S, call it o, not in the image of s. (2) If T is a subset of S such that (a) o \in T and (b) if t \in T then s(t) \in T then T = S. As it happens, von Neumann's N, with s:N ->N : n \-> n \/ {N}, and {} as unique non-successor, is an inductive set. > > It does not say anything about what is true for the set S itself, only > > for its members. > > > The set itself is "its members". While a set is entirely determined by its members, '"Mueckenh"s reading would require that {{}} be regarded an an empty set even though it has a member. > The set of natural numbers is "all natural numbers" and nothing else. Only if one mistranslates "all" as "the set of". > Or what is the difference between "all natural numbers" and the set? All natural numbers are finite but the set of natural numbers is not. All natural numbers are either even or odd, but the set of natural numbers is neither. All natural numbers have natural number successors, but the set of natural numbers does not. All natural numbers have squares which are natural numbers , but the set of natural numbers does not. And so on, ad infinitum.
From: Virgil on 21 Aug 2006 17:22 In article <1156189788.777422.231630(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Except that ordinal numbers cannot work that way, > > as it would require each ordinal to be a member of itself, > > which is not possible for ordinals. > > For the set of finite positive whole numbers ordinal and size are > equal. > The first element has value 1. > > Any problem about that? As any definition of "whole number" depends on one's definition of natural number, so why "whole number' instead of the more natural "natural number"? > The set does not exist? It does in ZF, ZFC and NBG. > > Regards, WM Declaring that non-sets have members is non-sense.
From: Virgil on 21 Aug 2006 18:02 In article <1156189929.275271.20710(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <ec8jpl$t2j$1(a)ruby.cit.cornell.edu>, > > > For every edge in an infinite binary tree, there are uncountably many > > non-termnating paths passing through (or including) that edge. > > That is an assertion without justification and without logic. > > > The number of terminating paths is irrelevant. > > > > WE already have a satisfactory proof that there are injections but no > > surjections from any set to its power set, so that the cardinality of > > any set is strictly less that the cardinality of its power set. > > > Wrong. It would be the case if set theory was undisputedly consistent. Within ZF and ZFC and NBG it is consistent enough so that not even "Mueckenh" can prove it otherwise without invoking additional assumptions. > But as we just investigate consistency, you cannot presuppose it. With > your attitude it is impossible to find any inconsistency even in an > inconsistent theory. Deplorably you are too simple to recognize that. If "Mueckenh" can deduce from any axiom system both a statement within the system and its negation, "Mueckenh" will have found his inconsistency. If it should transpire that this never occurs then, at least as far as anyone can tell, the system is consistent. Goedel showed something like: for any system capable of producing standard arithmetic, there can never be 'proof' of consistency without inconsistency. So there can be no more proof of the consistency of "Mueckenh"'s system that of ZF r ZFC or NBG.
From: David R Tribble on 21 Aug 2006 18:49
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. > David R Tribble wrote: >> 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. > Tony Orlow wrote: > Apologies. I mean omega. The concept with the limit ordinals is that > they are the first after the last of something that doesn't end, the > next biggest thing. But, when all successor ordinals are exactly one > larger than their max element, then limit ordinals are something else > entirely. The declaration of omega as that thing right after all the > finites itself leads to silliness. omega is defined as the least ordinal greater than every finite ordinal. It has well-defined properties, e.g., it is an ordinal, it obeys ordinal arithmetic including ordering (<) and set arithmetic (+, x, ^). In contrast, your Big'un does not have any such well-defined properties. You define it as "the number of reals in [0,1]", which of course is c, but then you tack on "and it operates arithmetically just like a finite value" without proof or elucidation. Which is just silly. Tony Orlow wrote: >> The only way around this is the declaration of the limit ordinals, >> but that's just a philosophical monkey wrench. > David R Tribble wrote: Tony Orlow wrote: >> Ordinals and cardinals are necessities if we want to talk about >> set "order" and "size" in any kind of logical, well-defined way. > Tony Orlow wrote: > Not limit ordinals and transfinite cardinalities, and in the finite > case, a count's a count. You don't need to call 1 an ordinal sometimes > and a cardinal at others. It's just a natural, a count. 1 is: - a natural - an integer - a rational - a real - an ordinal - a cardinal - a multiplicative identity - not a prime, not a composite etc. Why exclude a couple of properties you don't like? David R Tribble wrote: >> What do you call the least ordinal that is greater than all finite >> ordinals? You don't have a name for it, do you? > Tony Orlow wrote: > obviously, that's omega, if you entertain such a useless idea. I don't. Proving my point. > There is no least infinity, any more than there's a greatest finite, > where addition or subtraction changes the value. It always should. In standard arithmetic and standard set theory? Then prove it. I you mean in your theory, you still have to prove it. But first you have to define your system consistently. David R Tribble wrote: >> What do you call the "size" of a countable set with no end? >> You don't have a name for it, do you? > Tony Orlow wrote: > No, I find focused concentration on the Twilight Zone, the "boundary" > between finite and infinite, to be a rather fruitless exercise. There is > no such distinct boundary. Correct. The finite and the infinite ordinals do not "meet" at any point. And yet they are still ordered with respect to each other. David R Tribble wrote: >> What do you call the "size" of an uncountable set? You don't >> have a name for it, do you? > Tony Orlow wrote: > Uh, yeah. Infinite. Infinite sets can be finely ordered formulaically. Do you have different names for those different formulaically ordered infinite sets? Or do you agree with Ross that all infinite sets are the same, so one name is enough? David R Tribble wrote: >> 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. > Tony Orlow wrote: > It's not functional, and I have provided functional alternatives. But not consistently defined alternatives. > But, maybe I should run for office anyway. Maybe we can outlaw this thing.... > ;) haha Indiana might have a special docket for that kind of legislature, right next to their tabled proposal for their exclusive official value of pi. |