The Definition of Points
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From: Virgil on 12 Apr 2007 16:32 In article <461e879d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > On Mar 31, 5:18 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> Virgil wrote: > >>> In article <1175275431.897052.225...(a)y80g2000hsf.googlegroups.com>, > >>> "MoeBlee" <jazzm...(a)hotmail.com> wrote: > >>>> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote: > >>>>> They > >>>>> introduce the von Neumann ordinals defined solely by set inclusion, > >>>> By membership, not inclusion. > >>> By both. Every vN natural is simultaneously a member of and subset of > >>> all succeeding naturals. > >> Yes, you're both right. Each of the vN ordinals includes as a subset > >> each previous ordinal, and is a member of the set of all ordinals. > > > > In the more usual theories, there is no set of all ordinals. > > > > Right. Ordinals are...ordered. Sets aren't. Ordinals have a unique ordering by reason of their being ordinals. Sets in general have all sorts of orderings, but none which is as inherent in their being sets as the ordinal order is in sets being ordinals. > >> > >> Anyway, my point is that the recursive nature of the definition of the > >> "set" > > > > What recursive definition of what set? > > > > Oh c'mon! N. ala Peano? (sigh) What kind of question is that? Does TO seem to thing that N is the only set defineable recursively or that "successor" is the only recursively defineable operations on sets? > > > >> Order is defined by x<y ^ y<z -> x<z. > > > > Transitivity is one of the properties of most of the orderings we're > > talking about. But transitivity is not the only property that defines > > such things as 'partial order', 'linear order', 'well order'. > > > > It defines order, in general. Only to TO. For everyone else, other properties are required. For example, in addition to transitivity, ((x>y) and (y>x)) -> x = y is a necessary property /every/ ordering. Also there are lots of transitive relations which are not orderings, at least as usually understood. E.g., universal relations, which hold true for all x and y in the relevant set. So that TO's notion of an ordering does not necessarily order anything. > > >> I suppose > >> this is one reason why I think a proper subset should ALWAYS be > >> considered a lesser set than its proper superset. It's less than the > >> superset by the very mechanics of what "less than" means. The mechanics of "less than" depends on what standard of measurement one is using, so claiming that one measure measures all is a procrustean fallacy. > There can always be a 1-1 correspondence defined between a set > with no end and its proper subset with no end, even if that > correspondence is so complicated so as to defy all attempts to define > it. Trivially false. Neither the set of reals nor the set of rationals has an end, and the rationals are a proper subset of the reals, but there is no bijection between them. And, given the axiom of choice, any well ordered uncountable set even has well ordered countable subsets with which it does not biject.
From: MoeBlee on 12 Apr 2007 17:04 On Apr 12, 11:16 am, Tony Orlow <t...(a)lightlink.com> wrote: > > And what is your definition of "infinite"? > "greater than any finite" Define 'finite' and 'greater than'. Nevermind, you have no primitives anyway to which ANY of your definitions ultimately revert. MoeBlee
From: MoeBlee on 12 Apr 2007 17:07 On Apr 12, 11:30 am, Tony Orlow <t...(a)lightlink.com> wrote: > Lester Zick wrote: > > What grammar did you have in mind exactly, Tony? > > Some commonly understood mapping between strings and meaning, > basically. Grammar is syntax, not meaning, which is semantics. What you just described, an intrepative mapping from strings to meanings of the strings is semantics, not grammar. MoeBlee
From: K_h on 12 Apr 2007 18:15 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:461e8864(a)news2.lightlink.com... > Virgil wrote: >> In article <4611182b(a)news2.lightlink.com>, >> Tony Orlow <tony(a)lightlink.com> wrote: >> >> >>> It's true that the set of consecutive naturals starting at 1 with size x >>> has largest element x. >> Not unless x is less than or equal to some natural. > > Prove it, without assuming it. Virgil's assumption is a natural one since w is not a member of itself.
From: MoeBlee on 12 Apr 2007 17:36
On Apr 12, 12:25 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > In the more usual theories, there is no set of all ordinals. > > Right. Ordinals are...ordered. Sets aren't. That is not what I said. And what you said I would put as: For all x, if x is an ordinal, then the membership relation on x is a well ordering. But it is not the case that, for all x, the membership relation on x is a well ordering. And that is not the reason I say that there is no set of which all ordinals are members. > >> In > >> this sense, they are defined solely by the "element of" operator, or as > >> MoeBlee puts it, "membership". Members are included in the set. Or, > >> shall we call it a "club"? :) > > >> Anyway, my point is that the recursive nature of the definition of the > >> "set" > > > What recursive definition of what set? > > Oh c'mon! N. ala Peano? (sigh) What kind of question is that? You've never understood what Peano arithmetic is. Let's be specific, and start with first order And, since you're talking about RECURSIVE definitions, then there is no recursive definition stated as a formula of PA that is a definition of 'the set of natural numbers' or 'N' or anything like that. So, if you have some real mathematical notion here, you'll just have to say what it is. > > Transitivity is one of the properties of most of the orderings we're > > talking about. But transitivity is not the only property that defines > > such things as 'partial order', 'linear order', 'well order'. > > It defines order, in general. No it doesn't. Different KINDS of orderings are defined by having properties of which transitivity is only one of them. > > It is not always the case though that a set is not 1-1 > > with some proper subset of itself. In the finite, the two aspects > > coincide, but not in the infinite. That's just the way it is in the > > more usual set theories. That does not stop you from formulating a > > different theory though. > Right. There can always be a 1-1 correspondence defined between a set > with no end and its proper subset with no end, I can't comment unless you define 'no end'. > even if that > correspondence is so complicated so as to defy all attempts to define > it. I have no idea what MATHEMATICAL notion of 'complicated' and 'defy attempts' you have. In some cases, for a particular set, we show a particular bijection between the set and a proper subset of itself. In other instances, if the axiom of choice (or a weaker variation) is used, then we infer the existence of a bijection but wihtout showing a particular one. As to that, no one denies that it is non-constructive. If it is constructivity that you demand, then no one will deny that ZFC is not the theory for you. On the other hand, if you require constructivity but also you keep repeating that non-standard analysis is an alternative for you, then you only show yourself to be an ignoramus about the subject, since non-standard analysis depends on the axiom of choice (or at least weaker variants). > Perhaps that is the motivation behind Choice. Zermelo's motivation was to prove that every set is well ordered. > But, where that > violates the "lesserness" of the proper subset, it cannot be considered > a comprehensive comparison of "size". Another level of analysis needs to > be applied to get "measurable" results. Good that you use "measurable" in scare quotes, since you don't use it in any precise mathematical sense. And, again, for the thousandth time, that mathematicians use words such as 'cardinality' or 'equinumerosity' doesn't preclude anyone from investigating other properties either in set theory or in an alternative theory. So there's just no MATHEMATICAL import to all your whining about this. MoeBlee |