Galileo's Paradox
in [Math]
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From: Bob Kolker on 12 Dec 2006 14:09 Eckard Blumschein wrote: > > Cantor's first diagonal argument (stolen from Cauchy) does not show > uncountability of the reals but merely countability of the rationals. Wrong! It show the assumption that the reals can be written down as an ennumerable list leads to a contradiction. But if the set of reals were indeed enumerable there would be such a list. But such a list cannot exist, hence the set of reals is uncountably infinite. Is there something about a proof by contradiction that bothers you? Bob Kolker
From: Bob Kolker on 12 Dec 2006 14:10 Eckard Blumschein wrote: > > > I cannot imagine any reason why an intelligent person may reiterate this. That is the definition of coutably infinite. Bob Kolker
From: cbrown on 12 Dec 2006 14:19 Tony Orlow wrote: > This still makes no sense. x and y are reals, or at least reside along > the same metric from comparison. It's like asking whether a baseball is > more or less than a washcloth. triangles are not quantities, but > geometrical objects. I agree. So when you say infinite(x) <-> A y in R, x > y I can only assume you mean x is a real number being compared to y in "the usual way"; because that is the usual meaning of "x > y" when y is a real number. And there is no such real number x such that infinite(x). That's my complaint about your definition - ">" is defined for real numbers already, but you are "secretly" using the same symbol (">") to mean a /different thing/ - to compare real numbers with "a quantity". Which is fine and a common thing to do, /if/ you have defined what you mean by ">" and "a quantity"; otherwise it's just something floating around in your head that you haven't stated explicitly. We're not mind readers! Your verbal definition of "a quantity" seems to be limited to: something that you claim can be compared to a real number using the symbol ">". For example, if a triangle can be compared to a real number using the symbol ">", then a triangle is a "quantity"; otherwise it's not. That does not include a host of other things that I know you want, but have not said. I assume you will later claim that a "quantity" is not simply a thing that can be compared to a real number using the symbol ">"; but also has the property that it can be compared to every other "quantity" in such a way that the symbol ">" is a total order on "quantities" and real numbers, that preserves the usual meaning of "x > y" when both x and y are real numbers. But until you actually /state/ your definitions, axioms and rules, this is only guessing. For example, here's a sketch of what I think you want: "Let the set of quantities be some superset of R, with a total order ">=" which extends the usual ordering ">=" of R. Then if x is a quantity (is in the set of quantities), then infinite(x) <-> for all y in R, x >= y". This of course assumes that we mean the usual things by "R", "set", "superset", "extends", "is in the set", and "total order". Note that we still /cannot/ prove from this definition that the triangle T is a quantity; nor can we prove that it is /not/ a quantity. For all we know from the above definition, the set of quantities could possibly be exactly the same as the set R union {T}; with ">=" defined so that x >= T for all x in R. But we /can/ deduce things like "if infinite(x) and infinite(y), then exactly one of x > y, x = y, or x < y is true" or "if infinite(x) then x > 1.72". Which I think is the type of thing you want to prove. Cheers - Chas
From: cbrown on 12 Dec 2006 14:32 Eckard Blumschein wrote: > I will perhaps no longer reply to nonsensical replies. > In exchange for this, could you perhaps no longer state nonsensical statements? Thanks in advance! Cheers - Chas
From: Tonico on 12 Dec 2006 15:14
Eckard Blumschein ha escrito: ..........`...................................................ยด........................... Unfortunately, I will be hindered for a while to continue our discussion. > Who could grasp my insights and suggestions? > Robert Kolker is out of anything. > TO does not even understand Cantor. > Virgil will not be able to surrender. > Lester Zick? ??? > Maybe someone else. ************************************************** Well, there's a challenge! It won't though be easy to fill Eckie's shoes: cranks his size, his haughtiness, stupidity and density are not easy to find. Somebody jumping to take the job...? tonio |