Galileo's Paradox
in [Math]
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From: Virgil on 10 Dec 2006 04:19 In article <457b8a2f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Language is a set of strings from an alphabet. Like Chinese and Japanese? Languages are acoustic at first, and are still learnt that way. Ideogramatics comes later.
From: Virgil on 10 Dec 2006 04:29 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. > 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. > > >> Surely you must have guessed enough what I meant to follow the > >> paragraph? (sigh) > > > > Guessing is not a reliable way of finding things out. > > Of course, reading is a little more reliable, but when one gets stuck on > such words as "positive" and "infinite", maybe one needs to do a little > guessing. If TO's descriptive powers are so insubstantial as to leave us perpetually guessing what he means, he will never be able to convey anything of mathematical interest or significance.
From: Bob Kolker on 10 Dec 2006 06:37 Virgil wrote: > > > Like Chinese and Japanese? Mildly non-linear. Think of latin characters printed on top of each other. And don't forget Korean Hongul. Bob Kolker
From: Bob Kolker on 10 Dec 2006 06:43 Virgil wrote: > In article <457b8a2f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >>Language is a set of strings from an alphabet. > > > Like Chinese and Japanese? > > Languages are acoustic at first, and are still learnt that way. > Ideogramatics comes later. All languages (used by humans in general discourse) are acoustic because they are primarily spoken and heard. Writing of any kind is a very recent development (in the last 6000 years or so). You are refering to the writing systems. It turns out that Japanese has two phonetic alphabets (Katakana and Hirugana) but the Chinese style ideographs (Kanjii) are preferred for traditional reasons. If the Japanese wished to, they could use a totally phonetic graphology. They just do not want to. The Chinese have used ideograms for much longer, but there is a phonetic system for representing the Mandarin pronounciation. The problem with Chinese is there are so many dialects which are mutually incompatable the only common written language is ideographic. The Japanese are under no such constraint. The alphabet derived from the Phonecian graphology was originally ideographic. The letters were pictures (or highly stylized representations) of things that eventually took on conventional sound values. Bob Kolker
From: Tony Orlow on 10 Dec 2006 09:00
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: > >>>> Why do you have a problem with the mere suggestion of an infinite value? >>> I don't have a problem with infinite values. However, you have to do >>> more than merely say "positive infinite n". Assuming your positive >>> infinite things aren't the same as something that we already know about >>> (in which case, you should just say so), you either need to give a >>> construction of these positive infinite things or you need to specify >>> their properties. You haven't done either. For example, how many of >>> these things are there? How do they relate to each other? How do they >>> interact with the natural numbers? Are the operations of addition and >>> multiplication defined for them? >> Well, I have been through much of that regarding such specific language >> approaches as the T-riffic digital numbers, but that's not necessary for >> this purpose. It suffices to say that, if a statement is proved true for >> all n greater than some finite k, that that also includes any postulated >> infinite values of n, since they are greater than any finite k. I don't >> need to construct these numbers. Consider them axiomatically declared. > > Then list the axioms for them. > (sigh) infinite(x) <-> A yeR x>y |