The Definition of Points
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From: Tony Orlow on 12 Apr 2007 14:54 Mike Kelly wrote: > On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 1 Apr, 04:44, Tony Orlow <t...(a)lightlink.com> wrote: >>>> cbr...(a)cbrownsystems.com wrote: >>>>> On Mar 31, 5:45 pm, Tony Orlow <t...(a)lightlink.com> wrote:> >>>>>> Yes, NeN, as Ross says. I understand what he means, but you don't. >>>>> What I don't understand is what name you would like to give to the set >>>>> {n : n e N and n <> N}. M? >>>>> Cheers - Chas >>>> N-1? Why do I need to define that uselessness? I don't want to give a >>>> size to the set of finite naturals because defining the size of that set >>>> is inherently self-contradictory, >>> So.. you accept that the set of naturals exists? But you don't accept >>> that it can have a "size". Is it acceptable for it to have a >>> "bijectibility class"? Or is that taboo in your mind, too? If nobody >>> ever refered to cardinality as "size" but always said "bijectibility >>> class" (or just "cardinality"..) would all your objections disappear? >> Yes, but my desire for a good way of measuring infinite sets wouldn't go >> away. > > You seem to be implying that the existence and acceptance of > cardinality as one way of measuring infinite sets precludes the > invention of any other. This is patently false. There is an entire > branch of mathematics called "measure theory" which, roughly speaking, > examines various ways to measure and compare infinite sets. Measure > theory builds upon set theory. Set theory doesn't preclude mesure > theory. > > Of course, if *your* ideas were to be formalised then first of all > you'd have to pull your head out of.. the sand, accept that you've > made numerous egregiously erroneous statements about standard > mathematics, learn how to communicate mathematically and learn how to > formalise mathematical ideas precisely. Look at NSA and the Surreal > numbers if you need evidence that non-standard ideas can be expressed > clearly and coherently within an existing framework of mathematical > expression. > > You may be a lost cause though. You've spent, what, three years > blathering on Usenet and your mathematical understanding and maturity > hasn't improved a jot. It seems like you genuinely don't want to > learn. Is ranting incoherently just your way of blowing off steam? > You're not entirely wrong, Mike. I mean, you've been a jerk through all of this, but you have a point. In order to supplant what is currently the overly axiomatic bent of the field of mathematics and return to a balance between the deductive and the inductive sides of logic requires that one delve into the very foundations of logic itself, and form a new basis for determining what constitutes evidence in the field of mathematics, and how this evidence should be fed as deductively derived input into the inductive process of choosing axioms from which to build theorems. I am working on how to balance this, but it's not easy, and life's not easy, and I have other things to do. But, I'm doing this, too. I wish I had more time to research, but when I find the time to do this here, it's occasional. I promise to try harder in the future. My threads do get attention, because I raise some valid issues. You want a solution? Help me out. :) >>>> given the fact that its size must be equal to the largest element, >>> That isn't a fact. It's true that the size of a set of naturals of the >>> form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? >> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. > > No. This is not true if the set is not finite (if it does not have a > largest element). Prove it, formally, please, from your axioms. > > It is true that the set of consecutive naturals starting at 1 with > largest element x has cardinality x. Forget cardinality. Can a set of naturals starting with 1 and with size X possibly have any other maximum value besides X? This is inductively impossible. > > It is not true that the set of consecutive naturals starting at 1 with > cardinality x has largest element x. A set of consecutive naturals > starting at 1 need not have a largest element at all. Given the definition of the naturals, given any starting point 0, a set of consecutive naturals of size y has maximum element x+y. Does the set of naturals have size aleph_0? If so, then aleph_0 is the maximal natural. > > Do you see that changing the order of words in a statement can change > the meaning or that statement? Do you see that one statement can be > true, and another statement with the same words in a different order > can be false? > This is not quantifier dyslexia, and I am not interested in entertaining that nonsense, thanx. >> Is N of that form? > > N is a set of consecutive naturals starting at 1. It doesn't have a > largest element. It has cardinality aleph_0. If aleph_0 is the size, then aleph_0 is the maximal element. aleph_0 e N. Or, as Ross likes to say, NeN. > > -- > mike. > tony.
From: Tony Orlow on 12 Apr 2007 15:10 stephen(a)nomail.com wrote: > In sci.math Mike Kelly <mikekellyuk(a)googlemail.com> wrote: >> On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: >>> Mike Kelly wrote: >>> >>>> That isn't a fact. It's true that the size of a set of naturals of the >>>> form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? >>> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. > >> No. This is not true if the set is not finite (if it does not have a >> largest element). > >> It is true that the set of consecutive naturals starting at 1 with >> largest element x has cardinality x. > >> It is not true that the set of consecutive naturals starting at 1 with >> cardinality x has largest element x. A set of consecutive naturals >> starting at 1 need not have a largest element at all. > > To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define > "size" such that set of consecutive naturals starting at 1 with size x has a > largest element x, he can, but an immediate consequence of that definition > is that N does not have a size. > > Stephen > Stephen!! :) <3 Exactly! (smooch - sorry if that got toooo personal) That's one of the points I've been "ranting" about for years. The choice of axioms determines the conclusions allowed. If one assumes that adding or subtracting a nonzero quantity changes the value of any quantity, based on a the linear model, then there is no smallest infinity. If one assumes that equalities proven inductively hold in the infinite case, and inequalities where the limit as x->oo is not 0, then a host of different conclusions result, including IFR. I have been accused of not being able to follow assumptions to their deductive conclusions, but that's not the problem. I am trying to analyze which starting assumptions, or combinations thereof, lead to these "erroneous" conclusions regarding infinity. It's not that they don't follow. It's that they're led by the cart. So, thanks for being fair. I think we should all hope for such fairness in this type of arena de calculi. I'm not trying to be ignorant, but I'm not drinking the water, either. Peace. Tony
From: Tony Orlow on 12 Apr 2007 15:17 Mike Kelly wrote: > On 2 Apr, 17:12, step...(a)nomail.com wrote: >> In sci.math Mike Kelly <mikekell...(a)googlemail.com> wrote: >> >>> On 2 Apr, 15:49, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>>> That isn't a fact. It's true that the size of a set of naturals of the >>>>> form {1,2,3,...,n} is n. But N isn't a set of that form. Is it? >>>> It's true that the set of consecutive naturals starting at 1 with size x has largest element x. >>> No. This is not true if the set is not finite (if it does not have a >>> largest element). >>> It is true that the set of consecutive naturals starting at 1 with >>> largest element x has cardinality x. >>> It is not true that the set of consecutive naturals starting at 1 with >>> cardinality x has largest element x. A set of consecutive naturals >>> starting at 1 need not have a largest element at all. >> To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define >> "size" such that set of consecutive naturals starting at 1 with size x has a >> largest element x, he can, but an immediate consequence of that definition >> is that N does not have a size. >> >> Stephen > > Well, yes. But Tony wants to use this line of reasoning to then say > "and, therefore, if N has size aleph_0 then aleph_0 is the largest > element, which is clearly bunk". This is where his claims that > "aleph_0 is a phantom" come from. But, obviously, this line of > reasoning doesn't apply to all notions of "size". mike, what I am saying is that, assuming inductive proof works for all equalities, this equality holds: |N|=max(N). If this is the case, then aleph_0 cannot be infinite, while also being a member of N, and therefore is self-contradictory, and cannot exist any more than a largest finite. > > This isn't quite quantifier dyslexia, but it's related I guess. You may guess that, and I've heard it a lot. but this should serve as a lesson. Deduction is not the only kind of logic, any more than animals are the only kind of life forms. One is built upon the other, and requires it. Axioms are still in development, and not delivered on stone tablets. Tony > describes one notion of size where N doesn't have a size. Then he > wants to point to cardinality, which is another notion of size, and > triumphantly exclaim "Look, cardinality gives a size to N! Cardinality > is bunk! Aleph_0 is a phantom!". Who he thinks he is fooling is beyond > me. > > -- > mike. > mike, I'm trying to point out that inductive proof should not be limited to the finite case, any more than sequences should be limited to the countable. There are ways out of the morass. Watch out for that sinkhole...keep your eye on the horizon. ;) tony.
From: Tony Orlow on 12 Apr 2007 15:25 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. >> 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? >> introduces a notion of order which is not present in the mere idea >> of membership. > >> 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. >> This is generally >> interpreted as pertaining to real numbers or some subset thereof, but if >> you interpret '<' as "subset of", then the same rule holds. > > Yes, the subset relation on any set is a transitive relation. > Yes. It's related. >> 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. > > A proper subset is less than a proper superset of it, in the sense > that the proper superset has all members of the proper subset plus at > least one more. Yes. It's true. 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. > > MoeBlee > Right. 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. Perhaps that is the motivation behind Choice. 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. ToeKnee
From: Tony Orlow on 12 Apr 2007 15:27
MoeBlee wrote: > On Mar 31, 5:39 am, Tony Orlow <t...(a)lightlink.com> wrote: > >> In order to support the notion of aleph_0, one has to discard the basic >> notion of subtraction in the infinite case. That seems like an undue >> sacrifice to me, for the sake of nonsense. Sorry. > > For the sake of a formal axiomatization of the theorems of ordinary > mathematics in analysis, algebra, topology, etc. > > But please do let us know when you have such a formal axiomatization > but one that does have cardinal subtraction working in the infinite > case just as it works in the finite case. > > MoeBlee > Sorry, MoeBlee, but when I produce any final product in this area, cardinality will be a footnote, and not central to the theory. As I work on other things, so do I work on this. Take care. ToeKnee |