Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 11 Dec 2006 08:14 On 12/6/2006 12:18 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> >> >> A subset inside the reals is comparable to a piece of sugar within tea. >> > > No. The elements of a piece of sugar are not tea. > > A is a subset of B if and only if every element in A is an element in B. > > Why do you make a simple concept more difficult with inept and inapt > analogies, when a straightforward definition is at hand? > > Bob Kolker You seems to like straigtforward definitions instead of thinking. Definitions are not always reasonable. Do you know what 1/2 and 1/4 definition caused some people to remove a signboard Sarah Heydrich and replace it by S. Heydrich? My metaphor sugar in tea refers to the really reals, those which were assumed for DA2 and those which resulted like the power set of the naturals.
From: stephen on 11 Dec 2006 08:36 Six wrote: > On Thu, 7 Dec 2006 17:25:40 +0000 (UTC), stephen(a)nomail.com wrote: >>Six wrote: >>> On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284(a)bris.ac.uk> wrote: >> >>>> >>>>Six wrote: >>>>> >>>>> I am very grateful to you for expanding on this. While I'm almost >>>>> certain I'm missing something, I'm afraid I still don't get it. >>>>> >>>>> How exactly does claiming that a 1:1 C is not necessarily >>>>> indicative of equality of size with infinite sets presuppose an inability >>>>> to map (eg) the binary and decimal representations of integers? >>>>> >>>>> There is still a 1:1 C between the two sets. It is still true that >>>>> for any finite sets a 1:1C implies equality of size. Moreover it's still >>>>> reasonable to suppose that a 1:1C implies equality of size in the infinite >>>>> case unless there are other, 'functional' reasons to the contrary. (Vague, >>>>> I know. Roughly, 1:1 C is a necessary but not sufficient condition for >>>>> equality of size.) >>>>> >>>>> The idea is that the naturals (in any base) form a paradigm or >>>>> norm, a standard against which other sets can be measured. >>>> >>>>The set of finite binary strings is a subset of the set of finite >>>>decimal strings. >> >>> I confess I hadn't fully appreciated this simple point, that >>> together with the fact that the strings just are, so to speak, the natural >>> numbers (in a given base). >> >>>>Then b) precludes them being the same size. >>>> >>>>They are also both the same size as the set of natural numbers. >>>> >>>>Thus they are the same size as each other. >>>> >>>>Contradiction. >> >>> One is driven to the conclusion that there is no base-independent >>> size for the natural numbers. >> >>How can the size be base dependent? The natural numbers are not base dependent. >>Any natural number can be expressed in any base. There is no natural number >>expressible in base 16 that is not expressible in base 10, or base 9, or base 2. >> >>I suppose you could claim that there is a set of decimal numbers, and a set >>of base 2 numbers, and a set of hexadecimal numbers, and that they are all >>different, and all have different sizes. But it is a strange notion of >>"different size" given that all the sets represent the same thing. > I agree it would be a strange notion of size. My views have > shifted, but I'm afraid you might still find them strange. I do not think > we can say those sets are different sizes, but I do not think we can say > they are the same size either. > Thanks, Six Letters Well if they do not have the same size, and they do not have different sizes, then the most sensible conclusion is that they simply do not have any size at all. That is not at all unreasonable. If your definition of "size" requires the size to be a natural number, then the set of natural numbers does not have a size. This of course gets back to the main point, which is you have to decide what "size" means in this context. If you do not have a definition of "size", it is rather pointless to talk about it. Stephen
From: Eckard Blumschein on 11 Dec 2006 08:47 On 11/28/2006 4:17 PM, stephen(a)nomail.com wrote: > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> On 11/27/2006 8:47 PM, stephen(a)nomail.com wrote: >>> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >>>> On 11/27/2006 3:21 AM, stephen(a)nomail.com wrote: >>> >>>>> There is no need to resolve the paradox. There exists a >>>>> one-to-correspondence between the natural numbers and the >>>>> perfect squares. The perfect squares are also a proper >>>>> subset of the natural numbers. This is not a contradiction. >>> >>>> What is better? Being simply correct as was Galilei or being more than >>>> wrong? (Ueberfalsch) >>> >>> Do you deny that there exists a one-to-one correspondence between >>> the natural numbers and the perfect squares? > >> I just learned that mathematical existence means common properties. I do >> not have any problem with the imagination of bijection between n and >> n^2. I merely agree with Galilei that this bijection cannot serve as a >> fundamental for ascribing a number to all n or all n^2. Both n and n^2 >> just have two properties in common: They are countable because they are >> genuine numbers, and they do not have an upper limit. > > So it is not 'more than wrong' to say that there exists a one-to-one > correspondence between the natural numbers and the perfect squares. It is correct to say there is an infinite one-to-one correspondence between the natural numbers and the perfect squares. > >>> Or do you deny that the perfect squares are a proper subset of the >>> naturals? > >> First of all, I do not consider the naturals and their squares identical >> with the belonging sets. While the naturals are potentially infinite, >> the term set is ambiguous in so far it claims to comprehend the naturals >> looked at number by number, as well as _all_ naturals. > > So you do not think the perfect squares are a proper subset of the > naturals? "M is a subset of N" means: All elements of M are also elements of N. Proper means additionally: M is not identical with N. > In which case you think there is a perfect square that > is not a natural? I think this is a ridiculous fallacy. > >> The latter ist something quite different and relates to actual infinity. >> I consider any infinite set a fiction if one claims to include _all_ of >> its elements. The natural numbers 1...n are potentially infinite, i.e. >> they are no fiction, while the set of all natural numbers as a whole is >> necessarily a fiction. > > That seems like a rather limited approach. What do you mean by limited? > > Stephen >
From: Eckard Blumschein on 11 Dec 2006 08:56 On 12/6/2006 8:43 PM, Virgil wrote: > In article <4576E9AC.2020006(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 11:36 PM, Virgil wrote: >> >> >> One has to be pretty miseducated in order to need the fancy aleph_1. >> > >> > That depends on whether one accepts or rejects the continuum hypothesis. >> >> Not at all. > > EB again demonstrates that mathematically he does not know what he is > talking about To my knowledge, it was uncertain for a while how to attribute the alephs to the continuum. Nonetheless, I guess it has been settled that aleph_1 corresponds to the uncountable continuum while aleph_0 denotes the countables. The CH confusion was unnecessary if all alephs are fancy. Uncountables are not countable of course. In principle, there is nothing between countable and uncountable.
From: Eckard Blumschein on 11 Dec 2006 09:11
On 11/28/2006 9:30 PM, Tony Orlow wrote: >> The relations smaller, equally large, and larger are invalid for >> infinite quantities. >> > > Galileo's conclusions notwithstanding, there are certainly relationships > among many countably and uncountably infinite sets which indicate > unequal relative measures. I certainly consider 1 inch to be twice as > infinitely many points Twice as infinitely many is Cantorian nonsense. Cantor was able to show himself that all natural and rational numbers do not have a different "size". Infinity is not a number. It has been understood like something which cannot be enlarged and not exhausted either: oo * 2 = oo. |