Can N be constructed?
in [Logic]
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From: Virgil on 11 Jun 2010 03:18 In article <cdb7f0ca-279a-4c7f-8dc7-911a84c8f2a6(a)t10g2000yqg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Jun., 01:14, "K_h" <KHol...(a)SX729.com> wrote: > > "WM" <mueck...(a)rz.fh-augsburg.de> wrote in message > > > That is a construction. And that > > > construction can be done as > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > > Show two 1's existing in the list and not existing in one line > together. > You cannot do that. There is no need to do that.
From: Virgil on 11 Jun 2010 03:38 In article <b8c1aa63-4a20-4b25-be12-257efe86135b(a)d37g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Jun., 02:32, David R Tribble <da...(a)tribble.com> wrote: > > WM wrote: > > >> Proof: Construct the above list, but remove always line number n after > > >> having constructed the next line number n + 1. > > > > William Hughes wrote: > > >> After any finite number of steps you get a line from the list. > > > > David R Tribble wrote: > > >> Yes, that much is certain. > > > > William Hughes wrote: > > >> After an infinite number of steps you get a line from the list. > > >> The line you get after an infinite number of steps is not > > >> a line from the list. > > > > David R Tribble wrote: > > >> No, I don't think so. After an infinite number of steps, where > > >> at each step a (finite) line is removed from the list, you end > > >> up with no lines at all. > > > > William Hughes wrote: > > > Well this depends on defining what you "end up with" > > > > > If your definition is (the very reasonable) "you end up > > > with any lines that have been written down but not erased", > > > then you end up with no lines as every line you write down > > > gets erased. > > > > > However, I think in this context saying that "you end up > > > with the limit line 111..." �is better. �However, this > > > definition has its problems. �The main one is that you > > > "end up with" a line that you never write down. > > > > > Note, however, that in neither case do you end up with > > > a line from the list. > > > > Yes, exactly. You can't end up with a line that was erased > > at some point, nor with a line that was never written at any > > point. > > > > It's obvious from the beginning that the line 111... does not > > exist in WM's list, so it cannot possibly be the line that you > > end up with. Saying so makes as much sense as saying that > > you end up with the line 101010..., or > > or saying that infinity can be finished. That not only can be done, it has been done. But not, apparently, by WM.
From: WM on 11 Jun 2010 07:49 On 11 Jun., 08:03, Virgil <Vir...(a)home.esc> wrote: > > My proof does not show which line is left. > > It also does not show that any line is left. Sorry, but you seem unwilling or unable to understand my proof. > > > But it shows that finished > > infinity is a non-mathematical notion. > > On the contrary, what can be defined cannot be barred from mathematical > consideration, and "finished" infinity can be and has been defined. To be defined means to be called by a shorter name. To have a name does not prove existence of the named. Think of several Gods which have ceased to have believers. > > > Of course it is always the last > > line that is left, and it is impossible to get rid of a last line > > Except that the process requires passing by every line so that there > cannot be a last one. > > So if tWh finds a last line, WM is talking of a different process. No I am looking at it from another position. If all lines are there, all can be considered and removed. That is the one side. If no line is removed befor the next is written, then not all can be removed. In case of finished infinity both statements must be true. Hence, finished infinity and ZFC and related theories have been contradicted. Regards, WM
From: WM on 11 Jun 2010 07:53 On 11 Jun., 08:15, Virgil <Vir...(a)home.esc> wrote: > In article > <9d4e6d3a-f3c8-47a1-b209-db6519187...(a)w12g2000yqj.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 10 Jun., 22:03, Virgil <Vir...(a)home.esc> wrote: > > > > > That is just the case when constructing N or completeing Cantor's > > > > list. Without completing the list, the diagonal argument is invalid.. > > > > Cantor merely says that no completed list of binary sequences can > > > contain all binary sequences, which is obviously true. > > > He says first of all that an infinite list cna be completed. That is > > matheology. > > Actually that is NOT what Cantor says in his "diagonal" argument. What > Cantor does say is that for every list of infinite binary sequences > presented he can construct a binary sequence not in that list. What would be the result of the construction as long as it is unfinished? > > > > > When WM equates the reality of an axiom system with physical reality, he > > > becomes foolish. > > > Mathematics is physics (V. A. Arnold). > > Physics is Mathematics (Virgil) > > There is much more to mathematics than mere physics, at least in the > eyes of all but physicists and their slaves. Look here for instance, if you want to have the opinion of an expert: http://groups.google.de/group/sci.math.research/browse_frm/thread/4a899e54314704ad# Regards, WM
From: WM on 11 Jun 2010 07:56
On 11 Jun., 09:16, Virgil <Vir...(a)home.esc> wrote: > In article > <55744d1f-57e0-48ee-9913-ebfdfc6d8...(a)u7g2000yqm.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 10 Jun., 22:06, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <bc98ca3a-1b0a-439e-b4c6-8a4a0ec1b...(a)w12g2000yqj.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 10 Jun., 16:13, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > After any finite number of steps the set of remaining lines > > > > > cannot be empty. > > > > > No. After any possible step the set of remaining lines cannot be > > > > empty. > > > > > > Look! Over There! A Pink Elephant! > > > > > > After an infinite number of steps the set of remaining lines > > > > > cannot be empty. > > > > > How would you get to an infinite number of steps when each step has > > > > another finite number? > > > > There is nothing in the relevant axiom system which requires accessing > > > infinite cases only by such step-by-step operations on finite cases. > > > > In fact one such infinite case is built into those axioms > > > The infinite list > > 1 > > 11 > > 111 > > ... > > > need not be built step by step. Nevertheless the proof stands that > > there are not two (or more) lines which are necessary to contain all > > 1's of the list. > > More than two lines will work, if it is enough more!!! > > EVERY infinite set of lines does the trick, but no finite set of lines > does. > > So there is no "smallest" set of lines which works, but there are still > infinitely many sets of lines which will work Try to return to logical thinking. There are enough matheologicians in the world who argue on their beloved belief. If more than one are claimed to be required, then at least two required must be shown. Regards, WM |