Galileo's Paradox
in [Math]
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From: Tony Orlow on 10 Dec 2006 21:06 Virgil wrote: > In article <457c658c$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: > >>>>>> 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 >>> Is that your only axiom? If so, then state your first theorem about >>> them and give the proof. >>> >> That's the only one necessary for what defining a positive infinite >> n. > > It is a definition, not an axiom. And as it does not imply existence of > anything, one has no more that other axioms will supply. > > > Since it is a statement of implication, it's a rule, and an axiom. That it serves as the definition of infinite doesn't make it not a rule for determining what values are infinite, once finites are defined. >> A whole array of theorems pop forth from infinite-case induction >> and IFR, such as that the size of the even naturals is half that of >> the naturals. > > Not unless you declare them as axioms, as neither holds without being > assumed. If n>k -> f(n) (inductively proven), and infinite(n) -> n>k, then infinite(n) -> f(n), and the property can be said to hold in the infinite case. I suppose what you want is an exact statement of IFR so that we can determine which inductive proofs are allowable? Or, what?
From: Tony Orlow on 10 Dec 2006 21:06 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.
From: stephen on 10 Dec 2006 21:11 Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: >> Tony Orlow wrote: >>> 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 >> >> Is that your only axiom? If so, then state your first theorem about them >> and give the proof. >> > That's the only one necessary for what defining a positive infinite n. A > whole array of theorems pop forth from infinite-case induction and IFR, > such as that the size of the even naturals is half that of the naturals. > That's a no-brainer. Go back to where I first answered your question at > length, and read again, at length. It's not transfinitology, but it's > also not nonsense. > Allow me to add another: > |{ x| yeR and 0<(x-y)<=1}|=Big'un. > That's the unit infinity. What is the definition of | | ? Stephen
From: Tony Orlow on 10 Dec 2006 21:18 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." Of course, he has no for omega. It's illegitimate schlock, like I said. >>>>>> 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. >> 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. Is it your dog's fault when he doesn't understand when you explain how the microwave works? > 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: Tony Orlow on 10 Dec 2006 21:21
Virgil wrote: > In article <457c1eb6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Bob Kolker wrote: >>> 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 >>> >> You guys both need to take your meds, and take note of a few facts. >> >> 1) While a symbol is generally considered to be a visual pattern, it is >> most generally any uniquely distinguishable sensory input. It could be >> an auditory sound, a latin letter, arabic numberal, hieroglyph, or asian >> pictogram. It can be a series of beeps and clicks, like Morse code. >> >> 2) While Western languages have a small alphabet, and words with an >> averga of a large number of characters, one can have a large alphabet, >> the size of the entire vocabulary, where each word is a single >> character. So, Asian pictographic languages is not excluded by that >> definition. >> >> 3) I have specifically been talking about digital number systems, which >> are a special case of languages, complete, and with a unique >> quantitative interpretation for any finite string. It doesn't help to >> wander off into sociological and psychological realms when discussing >> digital number system. > > Digital number representations are not languages, as they consist of > only proper nouns, with no verbs, adjectives adverbs pronouns, etc. A formal language is a set of strings: http://en.wikipedia.org/wiki/Formal_language I suppose now you're going to tell me I'm using nonstandard language.... |