An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 25 Aug 2006 15:42 In article <44ef2769(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <44ed99b7(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > TO has claimed it, but not proved it. > > TO claims a lot but never proves any of it. > > Since the set of digit positions in any positional notation for the > > members of N is indexed by N, the cardinality of the set of bit > > positions equals the cardinality of its index set. > > Since the bit strings are the power set of the set of bit positions False. At least unless one means the SET of INFINITE bit strings has the same cardinality as the power set of the set of bit positions. But since no natural requires an infinite bit string, that is irrelevant. > the set of naturals is power set to the set of bit positions. False! TO deliberately conflates the set of infinite bit strings with the set of finite bit strings. So anything based on this false equivalence, is irrelevant. > >> Yes, given the current understanding, any unbounded set is infinite. > > > > Find any dictionary anywhere that says otherwise, that allows TO's > > self-contradictory unbounded but finite. > > Dictionaries reflect the widest and most accepted usage of words. They also, if they are any good, include most, if not all, special meanings. So if it cannot be found in any of them, even the OED, it is because it ain't so. > > >>> A set is infinite if it is not finite. A set is Dedekind infinite if > >>> it can be mapped to a proper subset of itself, and it is Dedekind > >>> finite if it is not Dedekind infinite. When we assume the axiom of > >>> choice, the two notions are identical. Without that axiom there > >>> can be infinite sets that are Dedekind finite. Do you want to know > >>> more about set theory? > >>> > >>> Now, using, this terminology (pretty standard), what is a "finite but > >>> unbounded" set? > >> Using that system, the phrase is senseless, but that system is not the > >> real universe. It's a concoction. > > > > TO's concoctions are even less sensible and less part of any "real" > > universe. > > > > In every standard dictionary, finite means being bounded or having ends, > > and endless means infinite. > > > > TO thinks he can get away with saying endless means having ends, but TO > > is, as usual, wrong. > > When you have the set of reals in [0,1] there are distinct endpoints to > the set. The question when dealing with an infinite set is whether > enumeration of the elements therein can ever terminate as a process. > Where there is a distinct range and the enumeration never completes, the > set is infinite. Unfortunately, because the enumeration of the naturals > never ends, that set is considered actually infinite, when it's really > only potentially so. The endlessness is due, not to actual infinitude, > but to the non-boundary between finite and infinite. In other words, TO is claiming that everyone in the world except TO is wrong about non-finite sets being non-finite. NOT bloody likely!!!
From: Virgil on 25 Aug 2006 15:50 In article <44ef2f76(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <44ed9fcb(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Dik T. Winter wrote: > >>> In article <ecig8i$ta$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > >>> > Dik T. Winter wrote: > >>> > Yes, the second is really a proof about power set and/or symbolic > >>> > representations of quantites. > >>> > >>> Not at all. There is no question about "representation of quantities". > >> In the second proof of uncountability of the reals, there are no digital > >> strings? What list is he deriving the antidiagonal from, a shopping list? > > > > Lists whose entries are taken from any set of two distinct objects. I > > think Cantor used two distinct letters, but he could have as easily used > > the set of binary digits, 2 = {0,1}. > > What things seem like to TO and what things are in actuality bear little > > resemblance. > > To those who view reality form over here, there are only slight > differences. ;) That must be another of TO's scale arguments, that when TO gets sufficiently far from things everything looks the same. > > > > >>> Quite a lot. When was the last time you did read a book about set > >>> theory? > >> Quite a while back. I wonder how many books on set theory Cantor read? > > > > In any case, it appears that Cantor wrote more than TO has read. > > And a lot has happened in between. I wonder how much Euler Cantor read. > At some point, I'd really like to get into Euler's stuff. And later, > Tesla's, in collaboration, of course. Maybe with Archimedes. ;) Heh! TO should learn some logic before trying serious mathematics. With only his present ineptitude in logic , its a wonder he could make it through grade school arithmetic. Assuming, of course, that he actually did.
From: Tony Orlow on 25 Aug 2006 20:53 MoeBlee wrote: > Tony Orlow wrote: >>> > > A set is infinite if it is not finite. A set is Dedekind infinite if >>> > > it can be mapped to a proper subset of itself, and it is Dedekind >>> > > finite if it is not Dedekind infinite. When we assume the axiom of >>> > > choice, the two notions are identical. Without that axiom there >>> > > can be infinite sets that are Dedekind finite. Do you want to know >>> > > more about set theory? >>> > > >>> > > Now, using, this terminology (pretty standard), what is a "finite but >>> > > unbounded" set? >>> > >>> > Using that system, the phrase is senseless, but that system is not the >>> > real universe. It's a concoction. >>> >>> Well, that is just a statement that (in my opinion) is senseless. >> Only in the context of the Dedekind definition of infinity, > > How rude. The man just explained to you the difference between Dedekind > infinite and infinite, and you just came back with an ignorant remark > that is ignorant for the very fact that it ignores the explanation just > given you. I beg your pardon? What is the difference between Dedekind infinite and infinite? I have been saying there is another sense in which the Dedekind definition is not always appropriate, and have been told that it's the only standing definition, and you're telling me he just explained the difference between the only mathematical definition of infinite, and infinite? You must have meant "finite", in which case, thank you, I am well aware of the Dedekind distinction. I am ignoring nothing. I am trying to explain to YOU the difference between Dedekind infinite and infinite. And you accuse ME of being rude? Well, I never.... > > MoeBlee > > > which one is >> not obligated to consume wholesale. Until one can prove that transfinite >> set theory is "correct", no one is obligated to accept the theory at all. >> >> TOny >
From: Tony Orlow on 25 Aug 2006 20:55 MoeBlee wrote: > Tony Orlow wrote: >> No, it all rests on the notions of identity and equality. As Leibniz >> pointed out, when the properties of two objects are all exactly the >> same, then they are the same object. So, when we say two numbers are >> equal, that means all properties of the two are equal. > > Ha! The fallacy of reversing implication right there! An example of > just about the most basic fallacy. When you say "a=b", you say "A a A b P(a)=P(b)". The two are equivlent statements, and therefore imply each other. > > No, the indiscernibility of identicals does NOT imply the identity of > indiscernibles. You need both implications; you can't derive one from > the other. And, in first order logic, one direction can be posited only > in the semantics not in the axioms. You prove two quantities equal by showing there is no difference, do you not? > > MoeBlee >
From: Tony Orlow on 25 Aug 2006 20:57
MoeBlee wrote: > Tony Orlow wrote: >> Yes, and the universe is consistent by definition, so math should be >> consistently overall as well. > > Unless the universe is a set of sentences, the notion of consistency > does not even apply. The universe is governed by the properties of the elements within it, which properties are statements true about those elements. > >> Yes, personally I want concepts of infinity to be compatible with the >> rest of math, as a logical extension. There is something amiss in set >> theory in this respect. > > "Logical extension". You have no idea what you're talking about. But, I do. > > MoeBlee > Tony |