Galileo's Paradox
in [Math]
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From: Virgil on 11 Dec 2006 15:44 In article <457D59C5.90007(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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. If we do away entirely with what EB calls DA2, We still have the same set of Dedekind-cut reals as before, with each such real being a finite, therefore countable, set, but the set of all of them being, by Cantor's first proof, uncountable.
From: MoeBlee on 11 Dec 2006 15:48 Six wrote: > >Only if '>' is left vague and undefined. > > > This is what gets me nettled straightaway. Surely this is something > to be established, rather than stated as a matter of fact at the outset. If it's a question of whether a certain formal definition is appropriate to capture an informal, intuitive notion, then yes, debate can go on for a long time. But at a certain point, we'll want to draw up a definition. If someone has another definition, then we can always add that and then anyone is free to choose which he thinks better captures the informal, intuitive notion. > >Once we make clear definitions, we see that there is a difference > >between one set being injectable (or injectable but not equinumerous) > >into another and one set being a subset (or being a proper subset) of > >another. > > I understand the difference, but I'm not convinced of its relevance. I don't know why it wouldn't be relevant. It seems to be at the crux of what you're talking about. > Do you think it was ever a paradox, before these definitions were > dreamed up? The word 'paradox' is itself informal. I do recognize that the fact that the primes are 1-1 with the naturals struck (and still strikes) people as paradoxical. > Galileo thought it a paradox, presumably. If it was ever a > paradox, do you not think there is the tiniest little scope for some very > clever mathematicians (not me) to debate how well the definitions measure > up to the original intuitions. Could be they were just all stupid back > then. I think it's fine to see how things compare with intutions. I suspect that just about anyone who has seriously studied set theory has thought about the possibility of another definition along the lines you'd like to see invented. Even I thought about that when I first studied some set theory (and I'm only a beginning hobbyist-student of the subject). But I don't know of such a viable proposal. It does seem to come down to seeing that bijectability and proper subsethood are just two different things. > >> Saying > >> that N = E says no more than that they are both infinite. > > > >WRONG. Saying that N is equinumerous with E (not =) is saying there > >exists a bijection between them. There are sets that are both infinite > >but without a bijection between them. > > You are jumping in here prematurely. The conceit here is that Cantor has > not yet arrived, if your imagination is up to it. Okay, I'll play. But is this an historical question of what mathematicians actually thought before Cantor, or a conceptual question of how we would view this matter if we had not yet conceptualized the basics of set theory of infinite sets? > If equinumerous is defined in the usual way, yes. It's a pity > though that word was appropriated. In a non-technical sense, meaning equal > in number, I would say it is not clear that N and E are equinumerous. Other terms in use are 'equipollent', 'equipotent' and 'have the same power'. Also, "bijectable" is used in these threads. I use 'equinumerous' usually, but any time I do use it, you may regard 'equipollent' to be used instead. 'equipollent' would be just as good for me, except it's a bit old-fashioned. > >They are all bijectable with one another. In Z set theory there is no > >object that is the order of infinity for any cardinality in the sense > >of every set of a certain cardinality being a member of that order. > > I'm not trying to introduce any new concepts here. I'm just making the > distinction between the countable and uncountable infinities, and > suggesting that discoveries about the latter need have no direct bearing > on how the former are to be understood in themselves. Okay, but I don't know what you have in mind specifically. > Imaginatorium has discussed below subsets and mappings, and there > are things I'm still trying to get to grips with here which may well make a > difference. But as I see things at the moments there is a primitive > intuition about size which is the same whichever we talk about. > > Consider these two mappings from N to O (odd numbers): > > the (non-injective) mapping: > 1,2 -> 1 > 3,4 -> 3 > 5,6 -> 5 > etc > and: > > the (bijective) mapping: > 1 -> 1 > 2 -> 3 > 3 -> 5 > etc. > > > According to the first, we are inclined to think there are twice as > many members in N than O. According to the second mapping, we are inclined > to think there are exactly as many members in O as in N. > What gives the bijective mapping more weight? Because it is indeed 'bi'; it goes both ways. (Ha! I didn't even intend that to be a pun on sexuality.) > I would draw the conclusion that it makes no sense to think of N or > any denumerable set as having a fixed size. N is a piece of elastic, or > more accurately a piece of elastic with no resting state. We know it's > being stretched out to the rationals, and shrunk again when it goes down to > the even numbers, or squares. But there is no way of measuring these > stretchings. N is the ruler. Okay. You might find a way to formalize that in set theory or some other theory of your invention. > There are some radical consequences, which I think just have to be > accepted. We cannot say that N is base-invariant. On the other hand we > cannot say either that it has a different size for each base. Following a > nice argument from Hughes in zuhair's infinity thread (though I expect > Hughes to show me I've mangled it), we cannot even say that N is the same > size as itself, I'm not familiar with the argument, but, yes, I don't imagine he argued that we can't show that N does not have the same size as itself. > but fortunately we cannot say either that it has a > different size from itself (else we would be in trouble). Indeed. MoeBlee
From: Virgil on 11 Dec 2006 15:56 In article <457D6707.4010208(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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. Actually is is anti-Cantorian nonsense. TO is just of a different faction of anti-Cantorianism nonsense purveyors than EB. Cantor was able to show > himself that all natural and rational numbers do not have a different > "size". Infinity is not a number. No mathmatician says "infinity" in that context. They will call some sets "infinite", but that is not the same thing. > It has been understood like something > which cannot be enlarged and not exhausted either: Infinite sets can be enlarged in the sense that for each of them there are proper supersets, at least in ZFC and NBG. They can be exhausted by taking away all their members.
From: MoeBlee on 11 Dec 2006 15:58 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> 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: > >>>>> > >>>>>> 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, > > > > "Assumption"? Why do you say "assumption"? > > > > What in math is not an assumption, or built upon assumption? What are > axioms but assumptions? He has postulated that he can form an extended > system by extending statements about N to *N, and works out the details > and conclusions of that assumption. Why do you ask? > > >> 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 need for omega. It's illegitimate schlock, like I said. > > > > Do you really think Robinson is talking about ordinals? > > > > Did you even read what I said? Of course he's not talking about omega > and the ordinals, he's talking about a sensible approach to the infinite > and infinitesimal for a change. Sheesh! Then your response about Robinson was a COMPLETE NON SEQUITUR. Sheesh! MoeBlee
From: Virgil on 11 Dec 2006 15:59
In article <457D6A64.7010008(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 8:52 PM, Virgil wrote: > > In article <4576EDC6.6090307(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/5/2006 11:48 PM, Virgil wrote: > >> > In article <457596BC.3040307(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > > >> >> On 12/4/2006 10:47 PM, David Marcus wrote: > >> > > >> >> >> Standard mathematics may lack solid fundamentals. At least it is > >> >> >> understandable to me. > >> >> > > >> >> > If it is understandable to you, then convince us you understand it: > >> >> > Please tell us the standard definitions of "countable" and > >> >> > "uncountable". > >> >> > >> >> I do not like such unnecessary examination. > >> > > >> > Then stop trying to examine others. > >> > >> I examined others here? > > > > You try, but as you are proceeding from a false assumption: > > that your own understanding of mathematics is superior to that of > > thousands of others who have spent much more time and effort and talent > > in gaining their understanding than you have. > > Concerning time you may be correct. However, do not underestimate the > talents by Galilei, Spinoza, Gauss, Kronecker, Poincar�, Wittgenstein > and many others. Just as Hilbert could improve on Euclid's geometry, those who follow can improve on what those who have gone before have wrought, but not the reverse. |