Galileo's Paradox
in [Math]
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From: Virgil on 11 Dec 2006 00:07 In article <457cbd4b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457c48ef(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Odd powers of i equal i > > > > Or -i. > > > > For integers n, > > > > i^(4*n + 1) = 1 > > > > but > > > > i^(4*n - 1) = -1 > > > > Try learning some arithmetic before pontificating. > > Well, if that wasn't about as out of context as it could be. You must be > really bored. I have had a great deal of experience finding and correcting arithmetical errors in boring papers.
From: Virgil on 11 Dec 2006 00:16 In article <457cbff3(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <457c1fa0(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <457b8ccf(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <457b5606(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> David Marcus wrote: > >>>>>>> Tony Orlow wrote: > >>>>>>>> David Marcus wrote: > >>>>>>>>> Tony Orlow wrote: > >>>>>>>>>> There is nothing wrong with saying E and N have the same > >>>>>>>>>> cardinality. > >>>>>>>>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse > >>>>>>>>>> Function Rule, which indeed can parametrically compare sets mapped > >>>>>>>>>> onto > >>>>>>>>>> the real line, using real valued functions, as a generalization to > >>>>>>>>>> set > >>>>>>>>>> density. It works for finite and infinite sets. So, what is wrong > >>>>>>>>>> with > >>>>>>>>>> trying to form a more cohesive theory of infinite set size, which > >>>>>>>>>> distinguishes set sizes that cardinality cannot? There certainly > >>>>>>>>>> seems > >>>>>>>>>> to be intuitive impetus for such a theory. > >>>>>>>>> Fine. Please state your rule. Let's take a look. > >>>>>>>> I've been over this a lot, but hey, what's one more time. Practice > >>>>>>>> makes > >>>>>>>> perfect. > >>>>>>>> > >>>>>>>> We start with the notion of infinite-case induction, such that a > >>>>>>>> equation proven true for all n greater than some finite k holds also > >>>>>>>> for > >>>>>>>> any positive infinite n. > >>>>>>> What is a "positive infinite n"? > >>>>>>> > >>>>>> A value greater than any finite value. > >>>>> But if "n", is, as usual, reserved for indicating natural numbers, > >>>>> those > >>>>> which are members of every inductive set, there is no such thing. > >>>>> > >>>> What, now mathematics has declared what the single letter n means? It's > >>>> a variable. > >>> Every variable has a domain of definition. The variable 'n' is quite > >>> commonly required to have the set of finite naturals as its domain. > >>> > >>> I merely remarked that when that is the case, there are no values for n > >>> other than finite ones. > >>> > >>>>>> Why do you have a problem with the mere suggestion of an infinite > >>>>>> value? > >>>>> It is arithmetical operations with infinite values absent any > >>>>> definition of what those operations mean or what properties they have, > >>>>> to which we make legitimate objections. > >>>>> > >>>> Not really. > >>> Really. > >>> > >> No. Reread the following: > >> > >>>> If the expressions used can themselves be ordered using > >>>> infinite-case induction, then we can say that one is greater than the > >>>> other, even if we may not be able to add or multiply them. Of course, > >>>> most such arithmetic expressions can be very easily added or multiplied > >>>> with most others. Can you think of two expressions on n which cannot be > >>>> added or multiplied? > >>> I can think of legitimate operations for integer operations that cannot > >>> be performed for infinites, such as omega - 1. > >> Omega is illegitimate schlock. Read Robinson and see what happens when > >> omega-1<omega. > > > > I have read Robinson. On what page of what book does he refer to omega - > > 1 in comparison to omega? I do not find any such reference. > > He uses the assumption that any infinite number can have a finite number > subtracted, and become smaller, like any number except 0, so there is no > smallest infinite, just like you do with the endless finites. > Non-Standard Analysis, Section 3.1.1: > > "There is no smallest infinite number. For if a is infinite then a<>0, > hence a=b+1 (the corresponding fact being true in N). But b cannot be > finite, for then a would be finite. Hence, there exists an infinite > numbers [sic] which is smaller than a." Does TO claim that the infinite numbers of Robinson's non-standard analysis are in any way connected to the transfinite cardinals and ordinals of Cantor's analyses? Pray tell what ultrafilter generates standard cardinals and ordinals in the way that ultrafilters are needed to construct Robinson's non-standard reals from standard reals. > > Of course, he has no for omega. It's illegitimate schlock, like I said. There is a lot of schlock about but it is all of TO's generation. > >> If you don't know what "positive" or "infinite" mean, that's not my fault. > > > > My uncertainty about what TO means by a term is directly TO's fault. > > I know what mathematics usually means by many terms, but as TO > > frequently does not mean the same thing by any mathematical term as > > mathematics means, I cannot say that I know what TO means by them. > > Oh.
From: Virgil on 11 Dec 2006 00:18 In article <457cc0ce(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > I suppose now you're going to tell me I'm using nonstandard language.... Almost always, as far as the meaning of standard mathematical terms goes.
From: Virgil on 11 Dec 2006 00:21 In article <457cc17f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Axioms declaring existence of sets from nothing seem rather pretentious > to me. Isn't that precisely what TO's various alternate number systems do, declare something out of nothing?
From: Virgil on 11 Dec 2006 00:23
In article <457cc531(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > What do you mean by "exist"? We can postulate the existence of such a > set, and derive conclusions about it. It seems to me that it exists by > virtue of being referenced. So that every time a child references Santa Claus or the tooth fairy they come into existence? |