An uncountable countable set
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From: Virgil on 7 Sep 2006 19:15 In article <45005aae(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >>> In fact, mapping the naturals in [1,Big'un] to the reals in [Lil'un,1] > >>> using the mapping function f(x)=x/Big'un yields Ross' Finlayson numbers, > >>> and is perfectly consistent with IFR. Not only do we obviously have > >>> Big'un^2 reals on the line because we have the sum of Big'un unit > >>> intervals each containing Big'un reals, ... > > > > David R Tribble wrote: > >>> That implies that there are BigUn naturals in the real number line, > >>> so that |N| = |R|. You state this as though it's a fact, but what is > >>> your proof? > > > > Tony Orlow wrote: > >> What? No. I mapped the reals in [Lil'un,1] (Lil'un is successor to 0, in > >> the Finlayson system. He calls it iota, which is finite) to the naturals > >> in [1,Big'un] (the real line). The reals in the unit interval are the > >> image of the naturals on the entire line. > > > > Which implies that there are the same number of naturals in the > > real number line (or in N) as there are reals in (0,1]. That's > > provably false. > > Not for the hypernaturals. For TO's hypernaturals everything is provably false, and provably true. > > > > > Either that or you're omitting an awful lot of reals in (0,1] to get > > your mapping to the naturals to work. Or you're using some alternate > > version of the "real number line" that is not dense in the reals. > > (Based on your previous alleged well-ordering of the reals, and > > your acceptance of Ross's iota ordering, that could very well be > > the case.) > > > > Yes, I am including infinite values on the number line, since it's > "infinitely long". "Infinitely long" merely requires endlessness, which can be, and is, achieved in the real line of standard mathematics without ever having any two points on that line infinitely distant from each other. That TO imagines otherwise reflects his profound misunderstanding.
From: Virgil on 7 Sep 2006 19:18 In article <45005c7c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > The universe is always expanding at the speed of light. How does TO claim to know this? By what yardstick does he measure the daily diameter of the universe to work out how fast it is expanding?
From: Dik T. Winter on 7 Sep 2006 20:39 In article <45005670$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: .... > > But you stated "that you can specify which finite number of iterations" > > etc. > > That is something different. A number is computable when there is a Turing > > machine such that, when given an arbitrary number n, it will calculate the > > 1-st through n-th digit. There is nothing about a specification of a > > finite number of iterations. > > Well, if you set up a Turing machine such that it will take an infinite > number of operations to get to any finite digit, then you have probably > not designed it well. You do not seem to understand. If you set up a Turing machine such that given a number n as input calculates the first n digits of some number can be very well designed. But can you proof that it will stop? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 7 Sep 2006 20:35 In article <45004b9b$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: .... > > > So, while ...111 may not be considered a standard natural, I see no > > > reason why it should not be considered, say, an extended natural. > > > > Why not use the proper name such numbers already have? 2-adics. > > No reason, except that 2-adics could possibly have infinite bit > positions. They have not. I have yet to come across basic objects in mathematics that have infinite posititions in their representation. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: imaginatorium on 7 Sep 2006 22:47
Virgil wrote: > In article <45001e94(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > Virgil wrote: > > > In article <44fe1d38(a)news2.lightlink.com>, > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > So that if TO wants justifications, they are available, but they are > > > often too technical for someone of TO's level of understanding to > > > comprehend, and even if not, tend to be buried in papers that are > > > otherwise highly technical in precisely the ways TO object to all the > > > time. > > > > > > > In other words, you justify the viability of the axioms by the fact that > > they have gone through a long evolutionary process and survived in their > > current form, rather than having died out. The same can be said for the > > platypus. Quack! ;) > > The platypus, despite having what looks much like a duck's bill, does > not quack. What noise does it make, then? Brian Chandler http://imaginatorium.org |