Galileo's Paradox
in [Math]
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From: Virgil on 12 Dec 2006 16:22 In article <457EC9DE.5020000(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 9:56 PM, Virgil wrote: > > > 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. > > Yes. Infinity is a fiction. It cannot be achieved by conting. Infinite > is almost the opposite. > > >> 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. > > So infinite sets are potentially infinite. Quite reasonble. Only they are all simultaneously actually infinite as well. So that the distinction between "prtential" and "actual"is no distinction as any set which is either is both. > > > They can be exhausted by taking away all their members. > > One more indication for my insight that ZFC does not really cover pi. > > Unfortunately, I will be hindered for a while to continue our discussion. EB's misforune is our good fortune. But his place will no doubt quickly be filled by some other, equally kooky and trollish. There never seems to be a shortage of them.
From: stephen on 12 Dec 2006 16:32 MoeBlee <jazzmobe(a)hotmail.com> wrote: > Eckard Blumschein wrote: >> Uncountable is the opposite of countable. Therefore it has also to >> include non-sets. > Oy vey. Blumschein POSSIBLY qualifies as even more hopeless than Orlow. > MoeBlee I wonder if even is the opposite of odd, or is odd the opposite of even? Which one of them includes all non-integers? Is 3/2 odd or even? How about pi, or the Statue of Liberty? Can Blumschein tell us? :) Stephen
From: MoeBlee on 12 Dec 2006 16:44 stephen(a)nomail.com wrote: > MoeBlee <jazzmobe(a)hotmail.com> wrote: > > Eckard Blumschein wrote: > >> Uncountable is the opposite of countable. Therefore it has also to > >> include non-sets. > > > Oy vey. Blumschein POSSIBLY qualifies as even more hopeless than Orlow. > > > MoeBlee > > I wonder if even is the opposite of odd, or is odd the opposite of > even? Which one of them includes all non-integers? Is 3/2 odd > or even? How about pi, or the Statue of Liberty? Can Blumschein tell us? :) Blumschein can't even state the axiom of extensionality. But he's a fount of important criticisms of set theory nonetheless, he thinks. MoeBlee
From: Six on 14 Dec 2006 09:21 On 11 Dec 2006 12:48:49 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >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. OK >> >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. It may well be. I think I know what you are saying. If they are two different things, then the two 'sides' of the paradox don't actually pull in opposite directions. This is fine unless both of the following are true: 1) Cardinality (based on bijection) is a kind of count; it is understood to be a measure (of how many). I take it that is the conventional point of view. 2) Subsethood has also an implication for measure, for how many. And I'm still not convinced this isn't true. (Remember I am trying to give equal weight to both, so it is not a question of throwing out cardianlity but of reaching a correct interpretation of it.) Imaginatorium talked of subsethood not being a good measure of size because of partial order and so forth, but even after looking up set ordering I still don't understand this point. It's unfair to ask you explain what another poster is saying, but I'm wondering if this is your point of view too. I'm not even convinced that subsethood is essential to stating the paradox, which is why I tried to present it as a comparison of mappings. >> 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. OK >> 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. OK >> >> 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. That's very sporting of you. > 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? The latter, certainly. Or a useful rhetorical device along those lines. >> 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. Yes, I have come across those. I just wanted a word for 'equal in number' which didn't already presuppose that bijectability gave the answer. >> >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. Just to acknowledge Cantor's discoveries, and that bigger pieces of elastic are needed. >> 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.) Is it really that simple? Isn't that just the bias towards bijection of which zuhair complained (though ultimately he realized his solutions were futile), and which others perceive or think they perceive? Consider the binary relation P (to be thought of as 'pairs with') which is to operate on sets of natural numbers with one or two elements. We will say A P B iff |A| not = |B|, and: {a, b} P {c} iff c is odd, a = c, and b = a + 1 {d} P {e, f} iff d is odd, d = e and f = d + 1 This, I hope, expresses the mapping between: 1.2 -> 1 3,4 -> 3 5,6 -> 5 etc. It is a bijective mapping (if I've done it right). We can construct an analogous mapping, P*, to express the correspondence between: 1 -> 1 2 -> 3 3 -> 5 4 -> 7 etc. but straightaway we know it is not going to be a bijective mapping as we want {3} P* {5}, but we also want {3} P* {2}. I would like to go on here and say that the absence of a bijection is a good measure of artifice in bijections between 'unequal' (by one half of the paradox) sets. However, if I have learnt anything in this forum, it is not to follow one's flights of fantasy until one has waited for the assumptions on which they are base to be torn to shreds. >> 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. If you are not kidding me, I would love to know how to even begin to do that. Even IF (ie big if) I was right, as I say I don't think it makes any difference to anything. Except possibly to arrive at a better picture of infinity. In very elementary treatments one often sees the sentiment that it is wrong to think of infinity as a number, usually for all the wrong reasons. The thrust of my argument is that is wrong to think of denumerable infinity as any kind of count, EXCEPT, as it turns out, relative to non-denuerable infinities. It is only the existence of those (bigger pieces of elastic) which gives meaning to aleph_0 being a count. Does that make sense? snip Very grateful for your post. At last I feel I'm coming close to nailing this thing. Six Letters
From: Tez on 14 Dec 2006 12:24
Eckard Blumschein wrote: [snip] > >> Yes. The power set algorithm does not change what mathematicians still > >> used to call cardinality. 2^oo=oo. > > > > "The" power set "algorithm"? Could you write out "the" algorithm here > > for us so that you and I can flesh out your claims in detail? > > If you have N elements, you will get 2^oo combinations > (cf. Pasqual's triangle) That doesn't look like an algorithm to me. Usually, given an algorithm, one can take an input, and compute the algorithm's output. Let's call the "powerset algorithm" pow(s) (for some input set s). How does your description of the powerset algorithm help me determine pow( {1, 2, 3, 4, 5} ) ? Step-by-step please. > >> > >> Yes. You cannot apply the algorithm until you have all numbers. [snip] > One can delay operations but not shift them into future. Oh. So one *can* apply the algorithm before you have all numbers, by (what you call) delaying. Glad we resolved that. > > "Address"? Rational "order"? If we're going to play with words, I'm > > going to ask: Why don't entities with numerical addresses with an > > irrational order "exist"? > > In mathematics existence means common properties. What about the reals, > we have to distinguish genuine and putative reals. > > Genuine reals are those which were assumed for DA2 and are indeed > uncountable since they have an actual infinite number of decimals. This isn't even wrong. No specific reals are assumed in Cantor's diagonal argument. What is assumed is a general mapping from naturals to reals. If you think this is a sufficient definition of "genuine real", can you show me, using your definition, how you can demonstrate that pi is a real number? Hell, just show me, using your definition, that sqrt(2) is a real number. Or just pick a "genuine real", and show me how you use your definition to determine your chosen real's genuineness. > Putative reals are defined by Dedekind's cut, nested intervals, Cauchy > limits if the tiny word fictitious is missing and the like. So can a real be both genuine and putative? Perhaps you've overlooked the fact that the set of genuine real numbers is the same as the set of putative real numbers. Strike that. Your characterisation of genuine reals means that the set of genuine reals is the empty set. -Tez |