infinity ...
in [Math]
|
Prev: math
Next: The proof of mass vector.
From: Virgil on 25 Oct 2005 22:55 In article <MPG.1dc855247151905798a563(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble said: > > Virgil said: > > >> If each subset of *N is to be represented by an infinite binary sequence > > >> of digits with 1 in some position representing the presence of a member > > >> *N and 0 representing its absence, then one element sets must be > > >> represented by strings with one 1 in them. > > > > > > > Tony Orlow wrote: > > > This makes sense, but doesn't jibe with what you said before, as far as I > > > can > > > tell. Yes, each singleton set in P(*N) will map to a natural whose binary > > > representation has a single bit. This only includes N out of 2^N subsets. > > > > > > > Your scheme listed these mappings: > > f(0) = {} > > f(1) = {0} > > f(2) = {1} > > f(3) = {0,1} > > f(4) = {2} > > f(5) = {0,2} > > f(6) = {1,2} > > f(7) = {0,1,2} > > f(8) = {3} > > ... > > > > This list defines only the mappings of finite naturals in *N to > > the finite subsets in P(*N). Your list does not show any infinite > > naturals, nor does it show any infinite subsets, nor does it show any > > subsets containing infinite naturals. > I was asked about that and gave examples for the sets of odds and evens, > which > map to 2N/3 and N/3. There are other examples. It should suffice to say the > same patter continues without end. As there is no "pattern" sufficient to go to the aleged end, it does not suffice. If it did suffice, TO would be able to give the argument that is mapped into the set of all elements not mapped into their images. > > > > Your list shows an infinite number of finite naturals mapping to an > > equally infinite set of finite subsets. That's easy. > > There is no infinite set of finite naturals, as Albrecht has tried > valiantly to illustrate. The set of finite naturals is Dedekind infinite, which is quite infinite enough outside TOmatics. > > > But it's not > > a bijection between *N and P(*N). It's not a bijection between > > *N and P(N), or even between N and P(N). It is a bijection between > > N and some of the members of P(N). Obviously that's not good enough. > That's not what it is. I have declared it to be between *N and P(*N). To declare something does not make it so, at least outside of TOmatics. Outside of TOmatics, declarations require proofs before they need be accepted. > It includes such numbers in *N as 0:01.....0101 for the evens and > 0:1010...1010 for the odds. You have no reason to think this set is > limited to finite values. And we have o reason to think it a bijection until TO can find some member of *N which maps to the set of all members of *N not in the set they map to. > > Even if we were to grant you that the list somehow includes infinite > > naturals in *N mapping to subsets, you would still be missing a lot > > of subsets in P(*N). > Like which, for instance? The set of all members of *N not in the set they map to. > > > > So fill in the blanks, and show us those other mappings. Otherwise > > your "bijection" is incomplete and is not, in fact, a bijection > > between *N and P(*N). > WHich other mappings do you want? What would satisfy you? A mapping which maps some member of *N to the set of all members of *N not in the set they map to.
From: Virgil on 25 Oct 2005 22:57 In article <MPG.1dc8559aeb65202f98a564(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble said: > > Tony Orlow wrote: > > >> So, which element of the power set does not have a natural mapped to it? > > > > > > > Virgil said: > > >> {x in S:x not in f(x)} > > > > > > > Tony Orlow wrote: > > > Oh yeah, the entire set, last element and all. What element was that > > > again? > > > > Your mapping does not include any natural that maps to the entire set > > *N, which it must do in order to be called a bijection, since *N is one > > of the members of P(*N). Show us that one single mapping, please. > It would obviously be the natural with an infinite unending strings of 1's, > right? Ah, but you want to know, how MANY 1's? We want to see a mapping from *N to P(*N) which maps some member of *N to the set of all members of *N not in the set they map to. > > > > But none of us can figure out where you keep pulling this "last > > element" gibberish from. Just let it go, Tony. > The entire set includes the last element, and the mapping includes a bit for > it. What position should I put this bit in? Isn't there another bit after it? We want to see a mapping from *N to P(*N) which maps some member of *N to the set of all members of *N not in the set they map to. Anything else is irrelevant.
From: Virgil on 25 Oct 2005 23:01 In article <MPG.1dc855c87373a60498a565(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > William Hughes said: > > > > Tony Orlow wrote: > > > Virgil said: > > > > <snip> > > > > > > > So, which element of the power set does not have a natural mapped to > > > > > it? > > > > > > > > {x in S:x not in f(x)} > > > Oh yeah, the entire set, last element and all. What element was that > > > again? > > > > > > No, just the entire set. The set does not have a last element. > > > > - William Hughes > > > > > Then there is no last bit to the natural that maps to it, so it cannot be > truly > identified, can it? As there is no "it' to identify, the lack of an identity is not a problem. What we want is to see a mapping from *N to P(*N) which maps some member of *N to the set of all members of *N not in the set they map to. Anything less is irrelevant to the validity of TO's alleged bijection from *N to P(*N)..
From: Virgil on 25 Oct 2005 23:25 In article <MPG.1dc85919c8f54c3c98a568(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David Kastrup said: > > This does not make aleph_0 a member of the set of naturals. > > A decree does not make it correct either. The first option is better, > to admit that the set size is equal to the largest member, but that > there is no identifiable largest member and therefore no > idnentifiable set size, which makes sense given the lack of endpoint > for measure. Only in TOmatics where self-contradiction is acceptable,is such self-contradiction acceptable.
From: Virgil on 25 Oct 2005 23:29
In article <MPG.1dc85a4d4881712598a56a(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble said: > > So I'm giving you set S, which obviously does not contain any > > infinite numbers. So by your rule, the set is finite, right? > > > > > If it doesn't contain any infinite members, it's not infinite. That only holds in TOmatics where TO's definition of finite allows unbounded sets to be finite. Everywhere else, where unbounded sets must be infintie, and Dedekind infinite sets are all infinite, it does not hold. > Those terms > differ by more than a constant finite amount, but rather a rapidly growing > amount greater than 1. There is no way you have an infinite number of them > without achieving infinite values within the set. You can everywhere except in TOmatics. And what goes on in TOmatics is entirely irrelevant to what goes on in standard mathematics. |