An uncountable countable set
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From: Tony Orlow on 29 Sep 2006 13:51 Virgil wrote: > In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >> Dik T. Winter schrieb: >> >>> > The successor function *is* counting (+1). >>> >>> Wrong. >> After a while you will have run out of the predefined successor, >> unavoidably. > > If that were ever to happen, one would have discovered a largest > possible number. But it does not ever happen, because for every set x > there is a set UNION(x,{x}) which is its successor. > I believe Wolfgang is saying that, once you run out of the starting known successive symbols of your language, your alphabet, you then have to employ an actual number system, using those elements recursively. Since alphabets are generally finite, you can never represent "infinite" quantities, in terms of string length. As you know, I like to represent specific infinite quantities using finite strings, but of course it's only a countable set of infinite numbers, since the infinite sequences are defined using repeating patterns, making them rational fractions of declared infinities. They're T-riffic! :) > > >> Then you have no other choice but to add 1 each time you >> proceed. That is counting. > > That is nonsense. Oh, well, to hell with higher number systems. It's unary. But WM has a point. Without increment as part of the definition of successor, you are not producing the set of naturals with all its properties of numerical measure. Upon increment build addition, then multiplication, exponentiation, etc. >>> > The successors are defined >>> > without counting only over a very restricted domain. > > The domain (but not the set) of all ordinals, which is a very large > domain. I think he means, essentially, a starting set of successors, like an alphabet. Is that close, WM? Tony
From: Tony Orlow on 29 Sep 2006 13:55 Randy Poe wrote: > Tony Orlow wrote: >> Han de Bruijn wrote: >>> Virgil wrote: >>> >>>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>>> >>>>> Randy Poe wrote, about the Balls in a Vase problem: >>>>> >>>>>> It definitely empties, since every ball you put in is >>>>>> later taken out. >>>>> And _that_ individual calls himself a physicist? >>>> Does Han claim that there is any ball put in that is not taken out? >>> Nonsense question. Noon doesn't exist in this problem. >>> >>> Han de Bruijn >>> >> That's the question I am trying to pin down. If noon exists, that's when >> the vase supposedly empties, > > Why does the existence of noon imply there is a time > which is the last time before noon? > > It doesn't. > > - Randy > I never said it did. When did I say that? I will offer this simple logical argument. If the vase ever became empty, it would be because one ball was removed, as per the gedanken, but 10 balls would have been inserted immediately beforehand. The vase would therefore have had to contain -9 balls, which I'm afraid is simply impossible. Don't you? It's a ridiculous set-theoretical result. Tony
From: Tony Orlow on 29 Sep 2006 14:01 Virgil wrote: > In article <451baaa0$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <451b3296(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Randy Poe wrote: >>>>> Tony Orlow wrote: >>>> You must have been a strange 10 year old, like that kid >>>> down the block that used to pull the legs off of roaches. >>> Only those that looked like TO. >>> >>>>>>> So the reason I don't say it's full "an infinitesimal time >>>>>>> before noon" or "some other time before noon" is that >>>>>>> I don't say it's full. >>>>>> But, you do say it's full or empty, right? >>> One can easily say that it is empty at any time at which every ball >>> that was put in has been taken out again. >>> >>> Does TO suggest that at any time after noon there is any ball that was >>> put in that was not also taken out? >> Yes, at any given time 9/10 of the balls inserted remain. > > TO's "yes" is a claim that some ball does not get removed. > Name one ball, by number, which does not get removed, TO. > >>>> If you say it empties, then you would agree that it either fills or it >>>> empties. When does it empty? You say, not before noon. You also say >>>> this does not occur at noon, but after noon there are no balls left. So >>>> when does this occur? >>> When every ball that was put in has also been taken out again. >> At noon or before noon? You're skirting the issue. > > To is the one skirting the issue by claiming that some balls remain but > being unable to name any of them. Your labels are a bunch of hooey. Not being able to name which molecules of water remain in an ocean being filled at 10 liter/min and emptied at 1 liter/min has nothing to do with whether it ever empties. You're playing voodoo.
From: Tony Orlow on 29 Sep 2006 14:03 Virgil wrote: > In article <451babf8(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> It's possible because no distinction is currently made between countable >> infinities, even to the point where a set dense in the reals like the >> rationals is considered equal to a set sparse in the reals like the >> naturals. > > TO would require that changing the order relation on a set changes its > size as if reordering the naturals into a dense set, like the rationals, > would somehow make more of them The order of everything on the real line is quantity. Virgil would require that the real line have a different length for every different infinite set he embedded on the line, so that they could all be the same size sets. It's like wearing uniforms in school.
From: Tony Orlow on 29 Sep 2006 14:04
Virgil wrote: > In article <451bac34(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>>> If the vase is empty at noon, but not before, how can that not be the >>>> moment that it becomes empty? >>> Saying that it is empty is quite different from saying anything about a >>> "last ball". andy does not deny that the vase becomes empty, he just >>> does not say anything about any "last ball out". >> Does that answer the question of **when** this occurs? Of course not. > > It does answer the question of "whether" it occurs. "When" is of lesser > importance. So, you have no answer. And so it's not important. I see. |