Galileo's Paradox
in [Math]
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From: Tony Orlow on 1 Dec 2006 10:56 Eckard Blumschein wrote: > On 11/29/2006 7:36 PM, Tony Orlow wrote: >> Eckard Blumschein wrote: >>> On 11/29/2006 3:58 PM, Tony Orlow wrote: >>>> where one set contains all the >>>> elements of another, plus more, it can rightfully be considered a larger >>>> set. >>> All of oo? >>> >>> >> Yes. All of the naturals are integers. Only half of all the integers are >> naturals. >> >> All of the points in (0,1] are in (0,2], but only half of all of the >> points in (0,2] are in (0,1]. > > You are equating two quite different notions: smaller and half as large. Smaller means less large, meaning less than 1 times as large, as in a fraction in [0,1), such as 1/2. What is different about smaller and half as large? > In case of two finite heaps of size a and b of numbers, a=b/2 implies a<b. Generalize where possible. Why is this not true in the infinite case? > In case of a=oo and b=oo, we may have a=b/2 while a is not smaller than > b but simply not comparable: oo = oo/2. So, you adhere, then, to the tenets of imaginary alephs and the creed of transfinitology? What you say is true for the standard generalization from the finite using the *existence* of a bijection, called cardinality, but to claim that there is no valid justification for seeking a formulaic method that produces more intuitive results is an empty statement. To say that there are not twice as infinitely many point in (0,2] and in (0,1] is equivalent to claiming that there are more points in one or the other of (0,1] and (1,2], if there is any correlation at all between this infinite count, and measure, which there clearly can be. Tony
From: Tony Orlow on 1 Dec 2006 11:16 Eckard Blumschein wrote: > On 11/29/2006 7:40 PM, Tony Orlow wrote: >> Eckard Blumschein wrote: > >>> Let's rather say in Cantor's illusion of allegedly being able to count >>> the uncountable. >>> >> Uncountable simply means requiring infinite strings to index the >> elements of the set. That doesn't mean the set is not linearly ordered, >> or that there exist any such strings which do not have a successor. > > Infinite strings are a reasonable fiction. Requiring this fiction is > effectively the same as requiring something impossible. We are guessing > that in a line any point has to have a successor. Doing so, we see the > piture of points with space in between. Weyl went a side-step further. > He imagined the continuum a sauce with embedded single points. There are a bunch of ways to look at it. I was talking about the symbolic representation of the elements, since half of math is language. We can easily express such strings, where they have repeating digits, such as in the p-adics. Then, whatever maximum value such a string may have, the countable set of possible strings you can produce represent the rational portions of that number. If you allow uncountably long strings, you can even include infinite values, and digital represent such numbers as 3 oo^2 + 2 oo + log2(oo) + 23. You just have to specify the infinite digit locations for each countable neighborhood within the uncountable string, and the rational repeating patterns that connect them. That's the T-riffic numbers. The other half of math is what we express with the language: measure. That's where the continuum comes in. We can imagine successive points int he continuum, infinitesimal segments, as it were. We can view them as nilpotent or infinitely divisible. When we view points as infinitesimal segments, then we can preserve measure. > > Actually the continuum is a concept that complements the concept of > discrete numbers and complements it at a time. A genuine continuum > cannot at all be resolved into single points. Oh, I don't know about that. It depends on whether you allow actual infinity. Take the H-riffic number generator: E 1 E x -> E 2^x ^ E 2^-x This produces all positive real numbers. For any real point on the line, there are two other points it has as successors. Of course, it's true, most will require non-repeating infinite strings that can never be produced. > > What about existence, I got aware of an appealing idea: Existence means > common propereties. Reals, as indirectly defined with DA2, differ from > genuine numbers in being uncountable. So they only exist like a fiction. > > What's DA2 again? Anyway, ala Leibniz, each object IS the set of properties which it possesses, so any two objects with the exact same set of properties are the same object. I'm not sure you can say one or the other is fictitious based on difference alone. Apparently, you apply the term to all the numbers that can never be completely expressed in a standard digital format. It's very easy for me to express sqrt(1/2) in an H-riffic binary string of a few bits. So, perhaps it depends on the language you're using. Is it, perhaps, more a matter of constructibility? Tony
From: Tony Orlow on 1 Dec 2006 11:36 Lester Zick wrote: > On Thu, 30 Nov 2006 09:52:49 +0100, Eckard Blumschein > <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/29/2006 6:37 PM, Bob Kolker wrote: >>> Tony Orlow wrote: >>>> It has the same cardinality perhaps, but where one set contains all the >>>> elements of another, plus more, it can rightfully be considered a larger >>>> set. > > Tony, you know we've been over this previously. All "infinite" means > is lack of definition for a particular predicate such as numerical > size. And when you add numerical finites to numerical infinites the > result is still infinite. When you add anything to anything, you have more than you had, eh? That's pretty basic. Let's try to keep that in mind. > > This problem mainly arises I suspect because mathematikers insist on > portraying infinites as larger than naturals and somehow coming beyond > the range of naturals such as George Gamow's famous 1, 2, 3, . . . 00. > Then mathematkers try to establish certain numerical properties for > infinities by comparative numerical analysis and mapping with > numerically defined finites. However one cannot do comparative > numerical analysis and numerical analysis with numerically undefined > infinites anymore than one can do arithmetic. Infinites are neither > large nor small; they're just numerically undefined. Uh, what if you define them, and even work out a language for expressing them, and arithmetic that be performed on them, and they produce intuitive results that include measure, as well as count? Why do you claim that's impossible, because you don't like the idea? > >>> Not necessarily so, if it is an infinite set. >>> >>> Bob Kolker >> This time I agree with Bob. >> >> > > ~v~~
From: Tony Orlow on 1 Dec 2006 11:39 Lester Zick wrote: > On Thu, 30 Nov 2006 12:08:03 +0100, Eckard Blumschein > <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/29/2006 8:13 PM, Bob Kolker wrote: >>> Tony Orlow wrote: >>>> Uncountable simply means requiring infinite strings to index the >>>> elements of the set. That doesn't mean the set is not linearly ordered, >>>> or that there exist any such strings which do not have a successor. >>> Uncountable means infinite but not of the same cardinality as the >>> integers. For example the set of real numbers. It is an infinite set, >>> but it cannot be put into one to one correspondence with the set of >>> integers. >> Uncountable means: Counting is impossible. This property obviously >> belongs to fictitious elements of continuum. There is simply too much of >> them. So counting is not feasible. As long as one looks at a finite, >> just potentially infinite heap of single integers, one has to do with >> individuals. The set of all integers is something else. It is a fiction. >> It is to be thought constituted of an uncountable amount of >> non-elementary elements. Well this looks nonsensical. There is indeed a >> selfcontradiction within the notion of an infinite set. >> Non-elementary means not having a distinct numerical address. Element >> means "exactly defined by an impossible task". > > You make the same mistake of assuming "infinite" means "larger than" > when it only means numerically undefined. Infinites are neither large > nor small; they're only undefined. Consequently there are no numerical > relations or operations possible between them and finites. The reason > counting is not possible is not because infinites are huge or because > they form a continuum but because there is no numeric metric defined > for them and counting as well as every other arithmetic relation and > operation requires some kind of numeric definitional metric. > > ~v~~ Huh! So, what happens if I declare a number, Big'un, and say that that is the number of reals in (0,1]? What if I say the real line is homogeneous, so every unit interval contains the same number of points? And then, what if I say the positive number line is going to include Big'un such unit intervals, so it has Big'un^2 reals up to Big'un? Does the universe collapse, or all tautologies suddenly become false? 01oo
From: Tony Orlow on 1 Dec 2006 11:44
Bob Kolker wrote: > Tony Orlow wrote: > >> >> >> To establish an explicit bijection between infinite sets does require >> an ordering on the sets, at least in general. > > Not true. One can map the disk of radius one one onto the disk of radius > two without ordering points in either disk. Hint: Use a cone. Or if you > like vectors map the vector V of unit length into 2*V which has length > 2. No ordering in sight. > > So in genaral one does not require an ordering. > > Bob Kolker Are you saying you can explicitly state the mapping mathematically without any reference to the ordered coordinates which identify each point? Not that you have to explicitly address each point individually, but you do need to make reference to the coordinates in your mapping formula, no, and those coordinates are each elements of an ordered set, no? |