Galileo's Paradox
in [Math]
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From: Lester Zick on 1 Dec 2006 14:48 On Fri, 01 Dec 2006 11:39:57 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >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? If you do, Tony, then what you've defined as "points" are in point of fact infinitesimals not 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? You could do this but the result wouldn't be finite in numerical terms and anything you might try to do between them and finites would cause the universe to collapse because they just aren't there on the same line and have different properties because they are finitely infinite. This is the price to be paid for absurdities like the real number line and putting infinities on them. "Infinite" means "not finite" and you just can't do finite arithmetic with "not finites". This includes even simple processes like numeric comparison of smaller and larger. You can't have "non finites" and "finites" on the same line in conceptual terms because "finites" have some numeric metric and "non finites" don't. In other words even in the elementary case of arithmetic infinities infintesimals can't be added, subtracted, or compared in size to finites because their metric is completely different and is described in finitely unrelatable terms unless some metric can be established between finites and non finites through a mechanism like calculus and comparison through L'Hospital's rule. In other words just because you say "bigun" doesn't indicate if it is a finite "bigun" or not and just because you use the phrase "number of" doesn't indicate you have any finite number subject to arithmetic in finite terms. ~v~~
From: MoeBlee on 1 Dec 2006 14:51 MoeBlee wrote: > Tony Orlow wrote: > > Bob Kolker wrote: > > > Tony Orlow wrote: > > > > > >> Are x and y ordered? The Cartesian plane is ordered in two dimensions, > > >> not a linear order, but a 2D ordered plane with origin. > > > > > > > > > The plane is not a linearly ordered set of points. > > > > > > Bob Kolker > > > > That's what I just said. > > So what kind of ordering do you contend it is? > > MoeBlee P.S. And WHAT ordering are you referring to? The lexicographic ordering that Kolker mentioned or some other ordering?
From: Six on 1 Dec 2006 15:04 On Thu, 30 Nov 2006 22:11:37 +0000 (UTC), stephen(a)nomail.com wrote: >Six wrote: >> On Thu, 30 Nov 2006 17:44:46 +0000 (UTC), stephen(a)nomail.com wrote: > ><snip> > >>>Six wrote: >>> >>>> I suspect though that there may be no proof of the matter either >>>> way, as has been hinted at in other parts of this thread. It may come down >>>> to this: that someone who wants to take a different, but still very >>>> reasonable (maybe more reasonable) appoach to this paradox, would need to >>>> demonstrate that some interesting and viable mathematics can result from >>>> it. This would certainly involve having precise defintions and so forth; it >>>> is just that they would be more or less different ones. I certainly do not >>>> have the wherewithall to even begin such a task. But it's interesting to >>>> speculate. And it's good to keep an open mind. >>> >>>You are more than welcome to come up with a definition of 'how many' >>>that you like. If other people like it, they may use it. It is not >>>going to change the fact that the naturals and the squares have the >>>same cardinality, nor the fact that the squares are a proper subset >>>of the naturals. >>> >> That is exactly what it is going to change. > >How can that possibly change? There exists a bijection between the >naturals and the squares, therefore the two sets have the same cardinality. >Every square is a natural, and some naturals are not squares, therefore >the squares are a proper subset of the naturals. Those two simple facts >cannot change no matter what definition of 'how many' you come up with. > >> Your insistence that I define my terms in advance is back to front. >> There is a primitive intuition that naturals exceed squares and that they >> are equinumerous. There is no simple confusion here. Though undoubtedly >> there is something to be learned. > >If you are not going to define your terms, then what you say is meaningless. >You say 'how many' but you refuse to say what you mean by that. How is anyone >supposed to understand you? > >> It seems to me that your understanding of the unclarity about how >> many is a phantom product, a reflection of the set theory you accept so >> uncritically. You want to pretend the ambiguity about how many pre-exists, >> whereas in fact it is a creation of the mathematics. Conventional set >> theory has no unique. proprietary rights over this paradox. and it seemed >> to me that, whatever else it does and no doubt does very well, it deals >> with this paradox rather poorly. > >> Thanks, Six Letters > >I am not claiming any unique proprietary rights over this paradox. >I am simply claiming that it is only a paradox because you are relying >on vague notions of 'how many' and 'size'. There is absolutely nothing >paradoxical about the statement > Every element of S is an element of N, but N contains elements not in S, and > There exists a bijection between S and N. >That is all we are saying when we say that > S is a proper subset of N, and > S has the same cardinality as N. >That is all set theory says on the subject. The only problem is that you have >some vague notion of 'size' or 'how many' that you are applying. >But you apparently refuse to even look at your notion of 'size' or 'how many', >and instead would rather complain about others being close minded. > >Stephen Unfortunately we don't seem to be getting anywhere. If I can think of a more persuasive way of getting my point across, I will reply. Anyway, I appreciate your interest. Six Letters.
From: MoeBlee on 1 Dec 2006 15:36 Six wrote: > On Thu, 30 Nov 2006 22:11:37 +0000 (UTC), stephen(a)nomail.com wrote: > > >Six wrote: > >> On Thu, 30 Nov 2006 17:44:46 +0000 (UTC), stephen(a)nomail.com wrote: > > > ><snip> > > > >>>Six wrote: > >>> > >>>> I suspect though that there may be no proof of the matter either > >>>> way, as has been hinted at in other parts of this thread. It may come down > >>>> to this: that someone who wants to take a different, but still very > >>>> reasonable (maybe more reasonable) appoach to this paradox, would need to > >>>> demonstrate that some interesting and viable mathematics can result from > >>>> it. This would certainly involve having precise defintions and so forth; it > >>>> is just that they would be more or less different ones. I certainly do not > >>>> have the wherewithall to even begin such a task. But it's interesting to > >>>> speculate. And it's good to keep an open mind. > >>> > >>>You are more than welcome to come up with a definition of 'how many' > >>>that you like. If other people like it, they may use it. It is not > >>>going to change the fact that the naturals and the squares have the > >>>same cardinality, nor the fact that the squares are a proper subset > >>>of the naturals. > >>> > >> That is exactly what it is going to change. > > > >How can that possibly change? There exists a bijection between the > >naturals and the squares, therefore the two sets have the same cardinality. > >Every square is a natural, and some naturals are not squares, therefore > >the squares are a proper subset of the naturals. Those two simple facts > >cannot change no matter what definition of 'how many' you come up with. > > > >> Your insistence that I define my terms in advance is back to front. > >> There is a primitive intuition that naturals exceed squares and that they > >> are equinumerous. There is no simple confusion here. Though undoubtedly > >> there is something to be learned. > > > >If you are not going to define your terms, then what you say is meaningless. > >You say 'how many' but you refuse to say what you mean by that. How is anyone > >supposed to understand you? > > > >> It seems to me that your understanding of the unclarity about how > >> many is a phantom product, a reflection of the set theory you accept so > >> uncritically. You want to pretend the ambiguity about how many pre-exists, > >> whereas in fact it is a creation of the mathematics. Conventional set > >> theory has no unique. proprietary rights over this paradox. and it seemed > >> to me that, whatever else it does and no doubt does very well, it deals > >> with this paradox rather poorly. > > > >> Thanks, Six Letters > > > >I am not claiming any unique proprietary rights over this paradox. > >I am simply claiming that it is only a paradox because you are relying > >on vague notions of 'how many' and 'size'. There is absolutely nothing > >paradoxical about the statement > > Every element of S is an element of N, but N contains elements not in S, and > > There exists a bijection between S and N. > >That is all we are saying when we say that > > S is a proper subset of N, and > > S has the same cardinality as N. > >That is all set theory says on the subject. The only problem is that you have > >some vague notion of 'size' or 'how many' that you are applying. > >But you apparently refuse to even look at your notion of 'size' or 'how many', > >and instead would rather complain about others being close minded. > > > >Stephen > > Unfortunately we don't seem to be getting anywhere. If I can think > of a more persuasive way of getting my point across, I will reply. Stephen's comments are right on the mark. His latest rejoinder nails it, even though it and his other followups should not have been needed. There's no sensible concern that YOU persuade HIM. MoeBlee
From: Virgil on 1 Dec 2006 15:46
In article <456FEEEF.5070409(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 1:32 PM, Bob Kolker wrote: > > Eckard Blumschein wrote: > > > >> > >> > >> I consider Dedekind wrong, and he admitted to have no evidence in order > >> to justify his basic idea. > > > > What sort of evidence? Surely not empirical evidence. Mathematics done > > abstractly has no empirical content whatsoever. > > > > Bob Kolker > > Serious mathematicans have to know the pertaining confession. Dedekind > wrote: "bin ausserstande irgendeinen Beweis f�r seine Richtigkeit > beizubringen". In other words, he admitted being unable to furnish any > mathematical proof which could substantiate his basic assumption. That conclusion assumes something not in evidence, that no one else has been able to do what Dedekind said he had not done. > Consequently, any further conclusion does not have a sound basis. > Dedekind's cuts are based on guesswork. So are everyone else's equally based on "guesswork", as without ASSUMING something, one cannot deduce anything. |