The Definition of Points
in [Math]
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From: Tony Orlow on 24 Apr 2007 19:26 Lester Zick wrote: > On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>> If you want to talk about the truth values of individual facts used in >>>> deduction, by all means, go for it. >>> I don't; I never did. All I ever asked was how people who assume the >>> truth of their assumptions compute the truth value of the assumptions. >>> >>> ~v~~ >> By measuring the logical implications of their assumptions. > > Or perhaps by measuring the truth of their assumptions instead. > > ~v~~ How do you propose to do that? 01oo
From: stephen on 24 Apr 2007 20:05 In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > Well, if the axiom systems we develop produce the results we expect > mathematically, then we can be satisfied with them as starting > assumptions upon which to build. My issue with transfinite set theory is > that it produces a notion of infinite "size" which I find > unsatisfactory. I accept that bijection alone can define equivalence > classes of sets, but I do not accept that this is anything like an > infinite "number". So, that's why I question the axioms of set theory. Why would you question the axioms? The axioms do not contain the words "number", "size", or "infinite". As you have been told over and over again, the results do not depend on the names we use. Stephen
From: Ben newsam on 24 Apr 2007 20:31 On Tue, 24 Apr 2007 19:08:46 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> And I believe it obvious that the one "mechanism" has to be the >>>> process of "alternation" itself or there is no way to produce anything >>>> other than our initial assumption. A chain is no stronger than its >>>> weakest link and if you make dualistic apriori assumptions neither of >>>> which is demonstrably true of the other in mechanically exhaustive >>>> terms you already have the weakest link right there at the foundation. >>>> >>> Again, please comment on my use of "not" to define the relationship >>> between true() and false(). >> >> Okay that's an improvement. But one of the things I don't see is how >> you produce and "truth values" between 0 and 1 using not. >> >> ~v~~ > >If you already have truth values between 0 and 1, as in a probabilistic >model, then not(x) is defined as 1-x, which is between 0 and 1 for x >between 0 and 1. In that kind of system, not(x) is arithmetic. For >uncorrelated x and y, and(x,y)=x*y, and or(x,y)=not(and(not(x),not(y))), >or 1-(1-x)(1-y), or x+y-x*y. Where there is a correlation between x and >y, it becomes hairier. I think that's what you're trying to address with >your large and/or green apples? You're wasting your time. Trust me.
From: Ben newsam on 24 Apr 2007 20:36 On Tue, 24 Apr 2007 15:12:19 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >wrote: > >>>> If I say 3x+3=3(x+1) is that true? >>>> Yes, it's true for all x. >>> >>> How about for x=3/0? >>> >> >>Division by pure 0 is proscribed because it produces an unmeasurable oo. >>If x is any specific real, or hyperreal, or infinitesimal, then that >>statement is true for all x. 3/0 is not a specific number. > >In other words the statement isn't true for all x. <Sigh> Tap * Dance = Porridge
From: Ben newsam on 24 Apr 2007 20:39
On Tue, 24 Apr 2007 15:15:45 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >wrote: > >>>> If I say a or not a, that's true for all a. a and b are >>>> variables, which may each assume the value true or false. >>> >>> Except you don't assign them the value true or false; you assign them >>> the value 1 or 0 and don't bother to demonstrate the "truth" of either >>> 1 or 0. >>> >> >>1 is true, 0 is false. If a is 0 or 1, then we have "0 or 1", or "1 or >>0", respectively. Since or(a,b) is true whenever a is true or b is true, >>or both, or(1,0) and or(0,1), the only possible values for the >>statement, are both true. So, or(a,not(a)) is always true, in boolean >>logic, or probability. >> >>Intuitively, if a is a subset of the universe, and not(a) is everything >>else, then the sum of a and not(a) is very simply the universe, which is >>true. > >Yeah but you still haven't proven that 1 is true and 0 false or what >either of these terms has to mean in mechanically exhaustive terms. Try this: "1" is everything or anything. "0" is whatever "1" is not. You may assign the terms "true" and "false" if you wish. Or vice versa. |