Can N be constructed?
in [Logic]
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From: Virgil on 10 Jun 2010 15:44 In article <d22a93f3-9f13-4480-9aca-da9a58a4a586(a)i28g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 9 Jun., 23:29, David R Tribble <da...(a)tribble.com> wrote: > > WM wrote: > > >> If N can be constructed, then the elements n can be constructed in the > > >> unary system as an infinite sequence of finite sequences of 1's (i.e. > > >> as a list of finite sequences) > > >> �1 > > >> �11 > > >> �111 > > >> �... > > > > >> This list contains all 1's that are contained in 111... > > >> The claim is that no line contains all these 1's. This claim can be > > >> disproved. > > > > >> 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. > > > > Yes, that much is certain. > > > > > Look! Over There! A Pink Elephant! > > > 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. > > > > 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. > > > > This is because every line is removed at some finite step > > in the sequence of infinite steps. There is no point in the > > sequence where a line (finite or otherwise) is not removed > > from the list. After all the steps, an infinite number of lines > > have been removed from the list. There are none left > > That would be correct, unless a removal is not executed before the > next line has been established. > > Therefore the set of remaining lines cannot be empty. Then which lines are left?
From: Virgil on 10 Jun 2010 16:06 In article <bc98ca3a-1b0a-439e-b4c6-8a4a0ec1b49d(a)w12g2000yqj.googlegroups.com>, WM <mueckenh(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.
From: Virgil on 10 Jun 2010 16:11 In article <593c8b7b-c3bb-46bb-8e85-0bab8e3ac2c8(a)i31g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Jun., 16:10, William Hughes <wpihug...(a)hotmail.com> 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. > > > > 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. > > That is wrong. Only every line *before the last one constructed* is > erased. That presumes there will always be a last one, but in many axiom systems there are non-empty ordered sets which do not have a "last one". > Only if "all lines" can be constructed, then all lines are > erased and are not erased. If "all lines have been constructed", which have not been erased? > > This sheds some doubt on the assertion that all lines can be > constructed. To me it only sheds doubt on your false image.
From: K_h on 10 Jun 2010 19:14 "WM" <mueckenh(a)rz.fh-augsburg.de> wrote in message news:172aadb8-7837-4ae0-86fb-f4cecef342c0(a)w12g2000yqj.googlegroups.com... On 9 Jun., 21:54, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jun 9, 4:16 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > In fact I write nothing, but only construct. > For insance I construct by division > 1/9 = 0.111... > > That is a construction. And that > construction can be done as > 0.1 > 0.11 > 0.111 > ... Each number in the list is different from 1/9 by a certain amount, (1/9)/(10^K) for K = 1,2,3,..., and so the Kth number in the list, f(K), is given by : f ( K ) = 1/9 - (1/9)/(10^K) Since each natural number K is finite, the term (1/9)/(10^K) is never zero. Thus, 1/9 appears nowhere in the list. In the limiting case as K-->oo we get f(K)-->1/9. Since ALEPH_0 is infinitely larger than any natural number, after you have constructed the entire list you are no longer in the list. That is, because of its hugeness, ALEPH_0 constructions take you to a place, called 1/9, that is beyond the list. _
From: David R Tribble on 10 Jun 2010 20:32
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 any other line that was never part of the list at any point. |