Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 30 Nov 2006 03:50 On 11/29/2006 6:12 PM, stephen(a)nomail.com wrote: > Six wrote: >>>>>>>>stephen(a)nomail.com wrote: >>>>>>>I was addressing your claim that there was "no extension from the >>>>>>>finite case". In the finite case, two sets have the same number >>>>>>>of elements if and only if there exists a one to one correspondence >>>>>>>between them. This very simple idea has been extended to the >>>>>>>infinite case. Incidentally, can you point me please to convincing evidence for all pertaining conclusions to be justified? >>>>>> OK. The idea of a 1:1 corresondence is indeed a simple idea. The idea of >>>>>> infinity is not. I see it the other way round. >>>>>That depends on what 'idea of infinity' of you are talking about. >>>>>The mathematical definition of 'infinite' is as simple as the >>>>>idea of a 1:1 correspondence. >>> >>>> The mathematical definition of infinity may be simple, but is it >>>> unproblematic? It seems to me that infinity is a sublte and difficult >>>> concept. It is indeed a fiction. >>> >>>What concept of infinity? Note, I said 'infinite', not 'infinity'. >>>You have been talking about Cantor and one-to-one correspondences, >>>so you have been talking about set theory. The word 'infinity' >>>is generally not used in set theory. It has been implicitely used if we directly or indirectly say there is an (infinite) set. The axiom of extensionality claims: A set is unambiguously set by its elements. In other words: There are infinitly many elements being united into an allegedly existing set. The infinite set is a fiction. So it is questionable whether or not it actually "is" in the sense one can attribute the same properties to it like to a finite set. It has no formal definition. Archimede's definition of a number is the same as the definition of potential infinity. What about actual infinity, one could write oo=a/0. Do not worry. Division by zero is strictly forbidden, yes. However, belief in the existence of an infinite set is pretty much the same. >>>'infinite' is used to describe sets, Systems of numbers are potentially infinite. If the original meaning of a set was not somewhat contrasting, there would not be any reason to deal with sets instead of opem ended series of numbers. The peculiarity of a set is: It has already been set. In other words, it is something perfect, something static. An infinite set is an intentional selfcontradiction. >>>and it has a very simple definition. What definition do you refer to? Cantor's definition of an (infinite) set has been proven untennable and withdrawn without substitute by Fraenkel. > >> I'm talking about mathematical meaning. Specifically I'm talking >> about "How many?", more or less etc.. > > "How many" is not a technical term. Ironically, Cantorian mathematics cannot accept a consequent distinction between countable (many) and uncountable (much) although it was Cantor himself who demonstrated that rationals are contable while reals are uncountable. The reason for this stunning fact is Cantor's misinterpretation of uncountable as "more than countable" (ueberabzaehlbar). > Cardinality corresponds to our > notion of "how many" in the finite case, In the finite case there is no reason to ask how much. Numbers are countable. According to Dedekind, ... >>>A set is infinite if there exists a bijection between the set and >>>a proper subset of itself. Because an infinite set is a fiction, it is questionable whether or not it is justified to attribute any subset to it. An infinite set would have indefinitely much of subsets, definitely too much as to focus on a single one. That is what mathematicians mean when >>>they say a set is infinite. There are other equivalent definitions. > >> I know already. I too, and I understood they are imprecise altogether. > So what are you asking? That is the definition of 'infinite set'. > It means mathematically exactly what it says. It is based on lacking understanding of what it means to have no limit. It ignores that an infinite set is the same as the fiction "actual infinity". Dedekind's definition contradicts Euclid who stated that the whole is always larger than its parts. Such contradiction is only allowed within the realm of mathematical fiction. >>>Do you have the same problem with prime numbers? Or even numbers? >>>The words 'prime' and 'even' have meanings outside of mathematics. >>>Do you feel obligated to drag those meanings into a discussion >>>of prime or even numbers? Telltale terminology unveils which errors led to the present mess in an allegedly fundamental part of mathematics. >>>> I accept that. The contradiction comes about if the one notion >>>> suggests equality of size and the other notion suggests inequality. Which >>>> they do, so there is a prima facie paradox. >>> >>>The problem is that you are using a word 'size' that you have >>>not defined. > >> True. I took it that people knew what I meant. And I think they do. > > No. I do not know what it means when applied to a set. Does > it mean "cardinality"? Fraenkel indeed explained cardinality like "generalized" size. > If so then we would not be having this discussion. > If it does not mean "cardinality", what does it mean? Can you give > me a mathematical definition of "size"? The size of a number is its value. If there are no infinite (no transfinite) numbers, then there is no reason to distinguish cardinal and ordinal nonsense. >> Certainly I write things in the heat of the moment which I later >> regret. But this wasn't meant as a cheap jibe. I've already conceded that >> following Cantor might in some deep way be right, Deep? You might even be prone to join scientology. Cantor himself claimed immediate contact to Him. Cantor's "theory" has been called "naive" for decades. It has been replaced by an aximatic method being not really correct but sucessfully designed in a slick manner. Look into the book Zahlen by Ebbinghaus et al. in order to find that "an obvious error led to something valuable". if it comes down to >> following productive branches and forsaking dead ends. I is overdue to forsake the dead end set theory. Ongoing big fuss about CH and AC, R*, etc. did not led to anything tangible. There is not even a single application of aleph_2 or higher alephs. Nobody needs aleph_0 and aleph_1 because the two qualities infinite and uncountable do not need a misinterpretation. >> There is an intuition that there are less squares (even numbers, >> primes, whatever) than naturals. This is not an intuition but correct reasoning for any given number of numbers. It merely lacks justification for open ended strings. >> We are talking here precisely of intuitions about infinite sets. Ebbinghaus called Cantor's way of thinking highly intuitive. What did he mean? He meant trusting in the guess of analogy. What is valid for numbers should be valid for transfinite numbers, too. >> It is not good enough to say: You're >> getting mixed up with finite sets, or: You can't rely on common sense >> intuitions in maths. Ironically, it's the proponents of what I call the Dedekind Cantor illusion who rely on half-religious half-common sense intuitions in maths. It was Galileo Galilei who resolved the paradox in a purely rational manner of thinking. >> So if there are less squares than naturals, squares as well as naturals do not have an upper limit. Therefore, we may conclude that there are neiter more nor equally many nor less. They are simply not comparable with each other. > Again, your problem is insisting that cardinality match some vague notion of 'how many' > that you have not defined. The basic problem is: He lacks the insight that cardinality is a cardinal mistake, something that has proven unfounded as well as useless. Eckard Blumschein
From: Eckard Blumschein on 30 Nov 2006 03:52 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. > > Not necessarily so, if it is an infinite set. > > Bob Kolker This time I agree with Bob.
From: Eckard Blumschein on 30 Nov 2006 04:09 On 11/29/2006 7:19 PM, Lester Zick wrote: I mean if you ask "how much gas" and get the answer "two > gallons" you've certainly measured the gas. Or if you ask "how much > space" and get the answer "two inches" you've certainly measured the > space. Common sense terminology tends to be nearly as logical as the mathematical one but more subtle. The question "how much gas" refers to the insight that the amount of gas is not directly countable. Nowadays, almost any continuous measurement yields in the end a numerical result because processing and transmission of digital signals is superior to processing and transmission of analogous signals. Eckard Blumschein
From: Eckard Blumschein on 30 Nov 2006 04:23 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. In case of two finite heaps of size a and b of numbers, a=b/2 implies a<b. 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.
From: Eckard Blumschein on 30 Nov 2006 04:40
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. 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. 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. |