Cantor Confusion
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From: imaginatorium on 5 Feb 2007 04:08 David Marcus wrote: > Virgil wrote: > > In article <MPG.20300e437f666b9e989c3d(a)news.rcn.com>, > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > > > Virgil wrote: > > > > In article <1170582481.992415.182840(a)s48g2000cws.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > Every potentially infinite set is finite. > > > > > > > > False by any standard definition of finiteness. What definition of > > > > finiteness does WM claim? > > > > > > How do you know it is false? What definition of "potentially infinite" > > > are you using? > > > > "Not finite" where "finite" of a set is > > Either > > I. A set is finite if it has a natural number as cardinality, presuming > > the set of natural numbers has been defined as a minimal inductive set. > > Or > > II. A set is finite if every injection to itself is a surjection. > > So, you would define "potentially infinite" = "infinite". That makes the > most sense to me. But, it seems that WM has > > potentially infinite => finite. > > I wonder if he also has > > finite => potentially infinite. This is the statement: finite : implies : potentially infinite where I have used colons to separate the verb from its subject and object. I'm sure we've all noticed that cranks don't like (clear) definitions, but I have also noticed that they _do_ seem to like adverbs. Mathematics really doesn't use adverbs - well, of course there _are_ adverbs, like "aysymptotically", but not to qualify basic verbs like "is", or "implies", or "is a subset of". Either X is a subset of Y, or it isn't; the only way that the vastly extended chains of reason that maths is built on can work is if all of the basic verbs are unconditional. Here, though, I'd guess WM's line is more like: finite : potentially implies : infinite A finite set _might_ be infinite, when the conversational ambience makes it a more amenable state of affairs to consider. Or something. * * * I suppose I'm not alone in wondering about this "potential infinity" stuff. The fact that mathematics doesn't use the term does not _necesssarily_ imply that it is nonsense - I wonder if the following is a fair summary: Consider the natural numbers as an example. "Actual infinity" would be the place at the end of the natural numbers. Well, the natural numbers don't have an end (yeah, yeah, just one end, at the beginning, but not in the opposite direction). So therefore "Actual infinity" doesn't exist. Meanwhile, the natural numbers don't end, therefore when we consider how many there are we get stuck, because "potentially" we could count as many as we like, and never end. Umble mumble, this [what exactly? ed] is "potential infinity". I'm enjoying Dik's mini-review - I hope he will tell us the error in chap. 7, and the two in chap. 8; perhaps at the end. Brian Chandler http://imaginatorium.org conventionalIt's just what is
From: Andy Smith on 5 Feb 2007 04:28 imaginatorium(a)despammed.com writes >David Marcus wrote: >> Virgil wrote: >> > In article <MPG.20300e437f666b9e989c3d(a)news.rcn.com>, >> > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> > >> > > Virgil wrote: >> > > > In article <1170582481.992415.182840(a)s48g2000cws.googlegroups.com>, >> > > > mueckenh(a)rz.fh-augsburg.de wrote: >> > > > >> > > > > Every potentially infinite set is finite. >> > > > >> > > > False by any standard definition of finiteness. What definition of >> > > > finiteness does WM claim? >> > > >> > > How do you know it is false? What definition of "potentially infinite" >> > > are you using? >> > >> > "Not finite" where "finite" of a set is >> > Either >> > I. A set is finite if it has a natural number as cardinality, presuming >> > the set of natural numbers has been defined as a minimal inductive set. >> > Or >> > II. A set is finite if every injection to itself is a surjection. >> >> So, you would define "potentially infinite" = "infinite". That makes the >> most sense to me. But, it seems that WM has >> >> potentially infinite => finite. >> >> I wonder if he also has >> >> finite => potentially infinite. > >This is the statement: > >finite : implies : potentially infinite > >where I have used colons to separate the verb from its subject and >object. I'm sure we've all noticed that cranks don't like (clear) >definitions, but I have also noticed that they _do_ seem to like >adverbs. Mathematics really doesn't use adverbs - well, of course >there _are_ adverbs, like "aysymptotically", but not to qualify basic >verbs like "is", or "implies", or "is a subset of". Either X is a >subset of Y, or it isn't; the only way that the vastly extended chains >of reason that maths is built on can work is if all of the basic verbs >are unconditional. > >Here, though, I'd guess WM's line is more like: > >finite : potentially implies : infinite > >A finite set _might_ be infinite, when the conversational ambience >makes it a more amenable state of affairs to consider. Or something. > > * * * > >I suppose I'm not alone in wondering about this "potential infinity" >stuff. The fact that mathematics doesn't use the term does not >_necesssarily_ imply that it is nonsense - I wonder if the following >is a fair summary: > >Consider the natural numbers as an example. "Actual infinity" would be >the place at the end of the natural numbers. Well, the natural numbers >don't have an end (yeah, yeah, just one end, at the beginning, but not >in the opposite direction). So therefore "Actual infinity" doesn't >exist. > >Meanwhile, the natural numbers don't end, therefore when we consider >how many there are we get stuck, because "potentially" we could count >as many as we like, and never end. Umble mumble, this [what exactly? >ed] is "potential infinity". > (snip) I've been guilty of this. (I think) the classification of "potentially" and "actually" infinite is due to Aristotle, who accepted the idea of a "potential" infinity - something never-ending, like space, or time, or the natural numbers. But he held there could never be an "actual" infinity - a "potential" infinity as a "finished thing". So, suggested translation: Aristotle => you lot potentially infinite => finite (e.g. all natural numbers) actually infinite => infinite (e.g. set of all natural numbers, a decimal representation of pi). -- Andy Smith
From: imaginatorium on 5 Feb 2007 07:05 Andy Smith wrote: > imaginatorium(a)despammed.com writes > >David Marcus wrote: > >> Virgil wrote: > >> > In article <MPG.20300e437f666b9e989c3d(a)news.rcn.com>, > >> > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> > > >> > > Virgil wrote: > >> > > > In article <1170582481.992415.182840(a)s48g2000cws.googlegroups.com>, > >> > > > mueckenh(a)rz.fh-augsburg.de wrote: > >> > > > > >> > > > > Every potentially infinite set is finite. <snip ... stuff about adverbs...> > >I suppose I'm not alone in wondering about this "potential infinity" > >stuff. The fact that mathematics doesn't use the term does not > >_necesssarily_ imply that it is nonsense - I wonder if the following > >is a fair summary: > > > >Consider the natural numbers as an example. "Actual infinity" would be > >the place at the end of the natural numbers. Well, the natural numbers > >don't have an end (yeah, yeah, just one end, at the beginning, but not > >in the opposite direction). So therefore "Actual infinity" doesn't > >exist. > > > >Meanwhile, the natural numbers don't end, therefore when we consider > >how many there are we get stuck, because "potentially" we could count > >as many as we like, and never end. Umble mumble, this [what exactly? > >ed] is "potential infinity". > > > (snip) > > I've been guilty of this. (I think) the classification of "potentially" > and "actually" infinite is due to Aristotle, who accepted the idea of a > "potential" infinity - something never-ending, like space, or time, or > the natural numbers. But he held there could never be an "actual" > infinity - a "potential" infinity as a "finished thing". So, suggested > translation: > > Aristotle => you lot > potentially infinite => finite (e.g. all natural numbers) > actually infinite => infinite (e.g. set of all natural numbers, a > decimal representation of pi). But this still isn't right, is it? Just take the first "translation" line: > potentially infinite => finite (e.g. all natural numbers) This isn't a one-for-one equivalence, obviously. "Finite" only applies to an individual natural, which surely A would not have claimed was other than finite (56, for example). But A might have said something about the "natural numbers being potentially infinite", and one could understand that to mean that the natural numbers go on and on and on, without ending (as, believe it or not, almost everyone seems to agree). The only way I can make any sense of this stuff is to think of mathematics as a sort of "performance". WM for example seems to think that only numbers that have at some time or other been written on a blackboard "actually" exist - others merely wait in the wings, to be brought into existence at the stroke of a piece of chalk. Thus the set of numbers that have been performed up to some point such as Now is (obviously) finite, but it grows by the second. Then obviously the complete set of naturals will never be performed, and perhaps this is the "actual infinity" whose existence is denied. Trouble is, this is a desperately impoverished scheme. Apart from anything else, you can't talk about the "complete set of naturals", since it doesn't "exist", yet you have to admit of its existence, or you can't effectively deny that the set of performed numbers is not in fact the whole lot. Again, if you say that the decimal expansion of pi doesn't "exist", yet there is no single digit of it that does not exist and moreover there is an algorithm for computing that digit. I can't see how this could ever be a productive approach to anything. Brian Chandler http://imaginatorium.org
From: Andy Smith on 5 Feb 2007 09:05 imaginatorium(a)despammed.com writes (snip) >> >> I've been guilty of this. (I think) the classification of "potentially" >> and "actually" infinite is due to Aristotle, who accepted the idea of a >> "potential" infinity - something never-ending, like space, or time, or >> the natural numbers. But he held there could never be an "actual" >> infinity - a "potential" infinity as a "finished thing". So, suggested >> translation: >> >> Aristotle => you lot >> potentially infinite => finite (e.g. all natural numbers) >> actually infinite => infinite (e.g. set of all natural numbers, a >> decimal representation of pi). > >But this still isn't right, is it? Just take the first "translation" >line: > >> potentially infinite => finite (e.g. all natural numbers) > >This isn't a one-for-one equivalence, obviously. "Finite" only applies >to an individual natural, which surely A would not have claimed was >other than finite (56, for example). But A might have said something >about the "natural numbers being potentially infinite", and one could >understand that to mean that the natural numbers go on and on and on, >without ending (as, believe it or not, almost everyone seems to >agree). > Yes, all that I meant was that things that A would have said were "potentially" infinite, you would say are all finite members of an infinite set. >The only way I can make any sense of this stuff is to think of >mathematics as a sort of "performance". WM for example seems to think >that only numbers that have at some time or other been written on a >blackboard "actually" exist - others merely wait in the wings, to be >brought into existence at the stroke of a piece of chalk. Thus the set >of numbers that have been performed up to some point such as Now is >(obviously) finite, but it grows by the second. Then obviously the >complete set of naturals will never be performed, and perhaps this is >the "actual infinity" whose existence is denied. > >Trouble is, this is a desperately impoverished scheme. Apart from >anything else, you can't talk about the "complete set of naturals", >since it doesn't "exist", yet you have to admit of its existence, or >you can't effectively deny that the set of performed numbers is not in >fact the whole lot. > >Again, if you say that the decimal expansion of pi doesn't "exist", >yet there is no single digit of it that does not exist and moreover >there is an algorithm for computing that digit. I can't see how this >could ever be a productive approach to anything. > I agree with you. I was just commenting that the adjectives "potentially" and "actually" have their history (I think) in an Aristotelian philosophical perspective of infinity, which is what Cantor tipped over. I think the lay view (mine until recently) is still Aristotelian - infinity exemplified by a never ending type of process. -- Andy Smith
From: G. Frege on 5 Feb 2007 10:43
On Fri, 2 Feb 2007 01:19:51 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >> >> 1) 1 in M. >> 2) If n in M then n + 1 in M. >> 3) IN ist Durchschnitt aller Mengen M, die (1) und (2) erf�llen. >> > That is an alternative definition, again not circular. > Actually, this is just M�ckenheim-Nonsense, as usual. > > It is better than Peano [WM] > Doesn't such a claim ALERT you? F. -- E-mail: info<at>simple-line<dot>de |