Galileo's Paradox
in [Math]
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From: Virgil on 5 Dec 2006 17:37 In article <45759016.3020802(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/4/2006 11:52 PM, Virgil wrote: > > >> I am an electrical engineer > > > > Shocking! > > Why? It's what electricity does!
From: Virgil on 5 Dec 2006 17:41 In article <457593EB.9030809(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > Rationals are p/q. This system cannot be improved by adding genuine > (i.e. rational) numbers. One can merely move to the fictitious > continuous alternative. All numbers are equally genuine in any meaning of "genuine" other than EB's improper meaning of "irrational".
From: Virgil on 5 Dec 2006 17:48 In article <457596BC.3040307(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/4/2006 10:47 PM, David Marcus wrote: > >> Standard mathematics may lack solid fundamentals. At least it is > >> understandable to me. > > > > If it is understandable to you, then convince us you understand it: > > Please tell us the standard definitions of "countable" and > > "uncountable". > > I do not like such unnecessary examination. Then stop trying to examine others. > Countably infinite refers to bijection. > Uncountable is a correcting translation of the German nonsense word > ueberabzaehlbar = more than countable. I understood that "ueber" could also bean "beyond", and "beyond counting", at least in the English meaning of "counting", is precisely what uncountable means. > Meant is: there is no bijection to the naturals. Actually, that "meaning" would make finite sets uncountable. A better statement would be that there is no surjection from the naturals. That excludes finite sets.
From: Lester Zick on 5 Dec 2006 17:50 On Tue, 05 Dec 2006 11:53:13 -0500, Tony Orlow <tony(a)lightlink.com> wrote: [. . .] You know, Tony, I got to thinking last night there may be a way to avoid this whole paradoxical situation. Let's say on the one hand we have what I would call cardinal algebra by which I just mean the conventional algebra dealing with finites such as r+y=z and so on. Then we wish to ascertain the nature and properties of such expressions as r+dr. And I have maintained that the addition of infinitesimals such as dr doesn't alter the size of finites such as r. Now there is actually a precedent in conventional mathematics for this situation. With complex numbers you actually have two component numbers: one conventional algebraic and one imaginary. Thus we can't say that r+ni is actually larger than r unless n is even. Now what I propose is something I'll call phase array algebra where in addition to conventional imaginary components we have other phases as well. We also have infinitesimal components such as dr. In other words to express a number such as c+ni+kdr we have not only the classical finite cardinal algebraic phase c and the neoclassical imaginary phase ni, we also have an infinitesimal phase kdr. So in effect we have no way to say c+ni+kdr is larger than c alone unless ni or kdr contribute something further to the magnitude of c. This is despite the presence of the "+" sign because "+" may not mean exactly what it means in the context of classical finite cardinal algebra alone. The interesting thing about phase array algebra is that you can make up phase array components all day long but unless we can phase one component into the c phase there is no impact on the c phase. These I'll just call phase transitions or rules for converting one phase into another. Now we're all aware of phase transition rules for imaginary numbers. c+ni is no greater or less than c alone unless n is even in which case there is a transition from the i phase to the c phase and c+ni becomes larger than c alone. The same is true of infinitesimal numbers such as kdr only the phase transition rules are different. As long as k is finite there is no way to say that c+kdr is larger than c alone in finite terms. However once definite integral calculus is invoked there is a transition between the infinitesimal phase and the finite cardinal algebraic phase which allows us to state that c+kdr is greater than c alone. But only then. In theory I suppose every number and numerical concept has a number of concomitant but otherwise unrelated phases associated with it linked to the c phase only through phase change transitions rules which relate any one phase to the c phase in which conventional algebra is done. At least that's the way I read the situation. Thus when I say that c+kdr is no larger than c alone because k is finite I'm just saying kdr is not in the same phase as c. Now what you can recognize in all this is the misinterpretation and even perhaps the misrepresentation of the "+" sign when someone says that 1+dr=1 or 00+1=00 for that matter. It doesn't mean the same as the "+" sign in classical finite algebra any more than it would mean the same with complex numbers until some phase transition occurs between phases which renders the result of one phase consistent with classical c phase algebra. In fact recalling a suggestion I offered you a couple of weeks back there is an interesting parallel. Then I suggested that rather than trying to cram finites and infinites onto one real number line you might consider putting finites on one line and infinites on another dimensional line which is exactly the way imaginaries are conceived. So in effect we have a c algebraic phase concentrated on one line and other kdr infinitesimal phases concentrated on other lines normal to it with the only real difference being phase transition rules between the two. At least I hope this clears up the situation for you in terms of classical algebraic arithmetic relations between infinites and finites. I think this analysis is pretty much definitive but who knows? We may yet have to try again if this doesn't work. ~v~~
From: Virgil on 5 Dec 2006 17:50
In article <45759AD4.3040505(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/4/2006 9:58 PM, Bob Kolker wrote: > > Eckard Blumschein wrote: > > > >> No wonder. Exactly this selfdelusion was the intention of dedekind. > > > > 'Twas no delusion. > > > > The Dedikind cut defining the square root of 2 is just as well defined > > as the successor to the integer 2. > > > > Bob Kolker > > Dedekind's cuts did not create any new number. The square root of two is > still what it was before. The cuts provided a construction of sets having the properties of those numbers, i.e., a model of the reals within set theory. |