constructible = unconstructible
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From: William Hughes on 9 Jun 2010 15:43 On Jun 9, 4:10 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 9 Jun., 20:52, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Jun 9, 3:13 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 9 Jun., 18:37, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Jun 9, 1:31 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > <snip evasion> > > > > > Does the set of nodes > > > > Z= > > > > { > > > > 1 > > > > 11 > > > > 111 > > > > ... > > > > > } > > > > > contain every node in 111...? > > > > > Please start your answer yes or no, > > > > No. > > > So in Wolkenmuekenheim there is a node in the path > > 111... that is not in the set of nodes > > > > Z= > > { > > 1 > > 11 > > 111 > > ... > > > > } > > > > > > Strange place Wolkenmuekenheim > > > Even stranger is your assertion: There are nodes in the list that are > not in a single line. An exceedingly strange assertion, however, I have never made this assertion. - William Hughes
From: WM on 9 Jun 2010 16:32 On 9 Jun., 21:43, William Hughes <wpihug...(a)hotmail.com> wrote: > > Even stranger is your assertion: There are nodes in the list that are > > not in a single line together with all others is meant obviously. That is what you denied. And your denial is nonsense, unless it is nonsense to talk about "all nodes". > > An exceedingly strange assertion, however, > I have never made this assertion. Then all nodes that are in the list are in a single line of the list? Regards, WM
From: William Hughes on 9 Jun 2010 16:58 To be clear, the assertion I made is There is a set of nodes contained in the list that is not contained in any single line of the list. or equivalently. No single line of the list contains all nodes that are contained in the list. This assertion is not strange, it is trivially obvious. - William Hughes
From: WM on 10 Jun 2010 06:19 On 9 Jun., 22:58, William Hughes <wpihug...(a)hotmail.com> wrote: > To be clear, the assertion I made is > > There is a set of nodes contained in the list > that is not contained in any single line of the list. > > or equivalently. > > No single line of the list contains all nodes that > are contained in the list. > > This assertion is not strange, it is trivially obvious. And it is wrong. You see it, when looking for two nodes of the list which are not in one single line. You cannot find such a pair. But if you claim the existnece of infinitely many, then you must show at least two of them. Or even simpler: Every node of the list is in one single line with the first node 1 of each line. Therefore your assertion is wrong. And in order to show it much simpler, I have asked you to produce the list by always deleting the line before that line just constructed - after that line has been constructed. Regards, WM
From: WM on 10 Jun 2010 06:34
On 10 Jun., 11:58, Ulrich D i e z <eu_angel...(a)web.de> wrote: > What does "exist" mean in which context? That is a very difficult question. For the sake of practical maths I would propose: If I define a_n = 10^-n for n in N, then all numbers 0.1 0.01 0.001 .... do exist. > > Let's have a closer look at the phrase > > | 111... had to exist as the last line of the list only in case the > | list had a last line. > > The meaning of the phrase "exist as the last line of the list" > differs from the meaning of the phrase "exist at all". > Correct. But if we divide 1 by 9, then 0.111... does exist according to the above definition. If we write it in a single line, nobody will disagree. If we write it such that every next 1 is placed in a new line, like 0.1 0.11 0.111 .... then everybody will disagree that all 1 are in one line but all 1 "must be there somehow". This leads to the easily disprovable claim that there are at least two lines required to contain all 1's. The solution of this riddle is of course, that there is no finished infinity without and end, i.e., without a last line of the infinite list. Hence, there is no finished infinity at all. > The former phrase means that if the list "had" a last line, then > a property of 111... would be that it was the last line of the > list. > > The pre-condition of the list "having" a last line is stated to > be not true ("[...]but the list cannot have a last line"). > Probably you can conclude from that statement that > " 'being' the last line of the list" is not a property of 111... . > > How do you deduce that the "actually infinite set does not > exist"? Se above. If all actually infinitely many 1 are there, but are not in one line, then it must be possible to find a set of lines, which is containing all 1's. But it is esay to show that all 1's of two lines are in a single line. By complete induction we see that it is impossible to need more than one line. > > How do you deduce from something (111...) not having a > certain property that something (the actually infinite set) > does not exist at all? That is not so new. We can, for instance, conclude from certain results of polarizition experiments (Bell-type experiments) that there are no hidden variables. We can conclude from sqrt(2) being not expressible as a reduced fraction that it is not expressible as any fraction. Regards, WM |