The Definition of Points
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From: stephen on 18 Apr 2007 17:51 In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>> Mike Kelly wrote: >>>> The point that it DOESN'T MATTER whther you take cardinality to mean >>>> "size". It's ludicrous to respond to that point with "but I don't take >>>> cardinality to mean 'size'"! >>>> >>>> -- >>>> mike. >>>> >> >>> You may laugh as you like, but numbers represent measure, and measure is >>> built on "size" or "count". >> >> What "measure", "size" or "count" does the imaginary number i represent? Is i a number? >> The word "number" is used to describe things that do not represent any sort of "size". >> >> Stephen > Start with zero: E 0 > Define the naturals: Ex -> Ex+1 > Define the integers: Ex -> Ex-1 > Define imaginary integers: Ex -> sqrt(x) > i=sqrt(0-(0+1)), so it's built from 0 and 1, using three operators. It's > compounded from the naturals. That does not answer the question of what "measure", "size" or "count" i represents. And it is wrong on other levels as well. You just pulled "sqrt" out of the air. You did not define it. Claiming that it is a primitive operator seems a bit like cheating. And if I understand your odd notation, the sqrt(2) is an imaginary integer according to you? And sqrt(4) is also an imaginary integer? You also have to be careful about about claiming that i=sqrt(-1). It is much safer to say that i*i=-1. If you do not see the difference, maybe you should explore the implications of i=sqrt(-1). So what is wrong with Start with zero: E 0 Define the naturals: Ex -> Ex+1 Define omega: Ax I did that using only one operator. > A nice picture of i is the length of the leg of a triangle with a > hypotenuse of 1 and a leg of sqrt(2), if that makes any sense. It's kind > of like the difference between a duck. :) That does not make any sense. There is no point in giving a nonsensical answer, unless you are aiming to emulate Lester. Stephen
From: MoeBlee on 18 Apr 2007 18:17 P.S. This looks like a good article stressing some of the history: http://citeseer.ist.psu.edu/cache/papers/cs/21901/http:zSzzSzwww.math.ucla.eduzSz~aslzSzbslzSz0303zSz0303-001.pdf/kanamori97mathematical.pdf MoeBlee
From: MoeBlee on 18 Apr 2007 18:44 On Apr 18, 3:17 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: P.P.S. Tony, in view of the fact that the link I gave is more concerned with historical aspects, I want to emphasize that I strongly believe that the way to understand set theory is to study one or two systematic and modern textbook treatments. The idea is to know a particular treatment that is the culmination of the historical process but is not bound to any informality, vagueness or problems that might be found in late 19th and early 20th century developments that have since been supplanted by systems that can be made fully rigorous. Especially, Cantor's own treatments are not at issue since his pre- formal theory and even Zermelo's own initial pre-formal theories have been supplanted by Z and various set theories that can be read in a way that we can see them as being formalizable with perfect rigor. Also, note that when talking about well ordering, sometimes when it gets down to the real technical nitty gritty, you have to be careful to distinguish between a well ordering that is reflexive (I call that a 'weak well ordering') and a well ordering that is irreflexive (I call that a 'strict well ordering' or just a 'well ordering') since different authors use 'well ordering' in those two different ways, and if you're fail to note which sense the author means, then you can be led to some erroneous conclusions. MoeBlee
From: Lester Zick on 18 Apr 2007 18:52 On Wed, 18 Apr 2007 12:40:56 -0700, Bob Cain <arcane(a)arcanemethods.com> wrote: >Alan Smaill wrote: >> Phil Carmody <thefatphil_demunged(a)yahoo.co.uk> writes: >>> >>> Can I have a coffee without milk? >>> I'm sorry, we don't have any milk. >>> Ah, OK, how about a coffee without cream? >> >> like the silent "k" in "frying pan" >> > >Note to Lester. The above contains demonstrations of wit. Perhaps, >by example, you can learn something both about wit and about >demonstration. Whereas from you I can learn something about being witless. ~v~~
From: Lester Zick on 18 Apr 2007 19:42
On Wed, 18 Apr 2007 14:31:35 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 17 Apr 2007 12:20:01 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> What question? You seem to think there is a question apart from >>>> whether a statement is true or false. All your classifications rely on >>>> that presumption. But you can't tell me what it means to be true or >>>> false so I don't know how to answer the question in terms that will >>>> satisfy you. >>>> >>>> ~v~~ >>> A logical statement can be classified as true or false? True or false? >> >> A logical statement as opposed to what, Tony? > >As opposed to, say, an arithmetic formula. So arithmetic formulas are not logical? >>> In other words, is there a third option, for this or any other statement? >> >> Hard to tell without seeing the statement. >> >> ~v~~ > >No, it's up to you. Good. The logical statement "it is black" is true. > A logical statement is one that has some measure of >truth, from false to true. One can consider just false and true, or one >can consider a multilevel logic like a scale from 1 to 10, or even a >probabilistic logic with all real values from 0 through 1. Since you >only speak of truth versus falsity, I imagine you are considering the >first type, or Boolean binary logic. So "black is crow" is either true or false? Or is not a logical statement? 'Fraid you'll just have to help me out here, Tony. Just tell me what you want me to say. ~v~~ |