SCI.MATH POLL - uncountable infinity
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From: Transfer Principle on 6 Jun 2010 02:31 On Jun 5, 6:14 am, herbzet <herb...(a)gmail.com> wrote: > |-|ercules wrote: > > Because the most widely used proof of uncountable infinity is the > > contradiction of a bijection from N to P(N), which is analagous to > > the missing box question. > I'm not sure that this proof is really a "proof of uncountable infinity" > anyway. A finitist, for example, would reject the notion that the naturals > constitute an infinite set in the first place, but I see no reason > why she would reject the proof that for any set S, |S| < |P(S)|. MoeBlee pointed this out too: "Sure, but without the power set axiom, we can still prove that for any S, if S has a power set, then there is no surjection from S onto its power set, which is the "essence" of Cantor's theorem." Of course, whenever posters mention this, I immediately point to the theory NFU. NFU proves the existence of non-Cantorian sets, and a non-Cantorian set is precisely a set S such that card(S) < card(P(S)). The simplest example of such a set is the set V of all possible sets -- a set whose existence is provable in NFU (but not ZFC, of course). It's easy to find a surjection from V to P(V) -- since P(V) = V, the identity _bijection_ suffices. Therefore, any poster who doesn't like Cantor's Theorem ought to consider NFU instead of ZFC.
From: Sylvia Else on 6 Jun 2010 06:45 On 5/06/2010 3:57 PM, Barb Knox wrote: > In article<86od4nFja0U1(a)mid.individual.net>, > "|-|ercules"<radgray123(a)yahoo.com> wrote: > > [SNIP] > > Ah, the Townsville looney returns. > > Mate, you just gotta stop licking those cane toads. > > I haven't heard of anyone licking them, but they boil up a treat. Sylvia.
From: master1729 on 6 Jun 2010 06:43 lwalke3 wrote : > On Jun 5, 6:14 am, herbzet <herb...(a)gmail.com> wrote: > > |-|ercules wrote: > > > Because the most widely used proof of uncountable > infinity is the > > > contradiction of a bijection from N to P(N), > which is analagous to > > > the missing box question. > > I'm not sure that this proof is really a "proof of > uncountable infinity" > > anyway. A finitist, for example, would reject the > notion that the naturals > > constitute an infinite set in the first place, but > I see no reason > > why she would reject the proof that for any set S, > |S| < |P(S)|. > > MoeBlee pointed this out too: > > "Sure, but without the power set axiom, we can still > prove that for > any > S, if S has a power set, then there is no surjection > from S onto its > power set, which is the "essence" of Cantor's > theorem." > > Of course, whenever posters mention this, I > immediately > point to the theory NFU. NFU proves the existence of > non-Cantorian sets, and a non-Cantorian set is > precisely > a set S such that card(S) < card(P(S)). > > The simplest example of such a set is the set V of > all > possible sets -- a set whose existence is provable in > NFU (but not ZFC, of course). It's easy to find a > surjection from V to P(V) -- since P(V) = V, the > identity _bijection_ suffices. > > Therefore, any poster who doesn't like Cantor's > Theorem > ought to consider NFU instead of ZFC. not really. we can have a set of all sets , a so-called 'universe' together with the aleph's. to avoid finitism we define countable infinity as : aleph_0 + 1 = aleph_0 then cantors diagonal argument / theorem / powerset gives us - with ^ the combinatorial interpretation of power - 2^aleph_x = aleph_(x+1) giving : 2^aleph_0 = aleph_1 and 2^aleph_n = aleph_(n+1) AND FROM THE ABOVE FOLLOWS !!! 2^aleph_aleph_0 = aleph_(aleph_0+1) = aleph_aleph_0 thus aleph_aleph_0 is the universe in a consistant non-finitist set theory. since the universe has the unique cardinality of the set of all sets , no larger cardinalities can exist , thus aleph_aleph_0 is the unique largest possible. unless you believe in nonsense like 2^aleph_w = aleph_(w-1) perhaps. NFU is not neccessary , tommy's TST might be used , and even ZF(C) might still be used. the trouble is that many ZFC fans dont realise they are using the large cardinal axiom or support that axiom without thinking* -- *as i did above --. ( so-called ZFC experts often dont realize they assume or use the large cardinal axiom , that is the so-called experts on sci.math ( not university ) ) regards tommy1729 " We have to prove with nonsense that nonsense is nonsense " Han de Bruin and tommy1729
From: |-|ercules on 6 Jun 2010 23:17 "herbzet" <herbzet(a)gmail.com> wrote > In any case, this thread appears to be closed, aside from tying > up any loose ends that may remain. Since it has been established > that proofs of the existence of higher orders of infinity do not > in general rely on the proof that |S| < |P(S)| for any set S, there > really is no way anyone is going to answer the poll question in the > affirmative. Not even the OP, apparently, is interested in further > discussion here. This is a joke. I thought I'd heard 10,000 reasons skeptics don't need to investigate claims, but sci.math wants the King Con Crown. You refuse to acknowledge the powerset proof of higher infinity is silly (at least looks silly) because there's other proofs of higher infinity! Fck me! ONE AT A TIME! Even if I meticulously worded my argument in your NULL HYPOTHESIS anti anything lingo and traced down all your good for nothing ZFC religion a - x - i - o - m - s you'd still not understand the claim and wiggle and worm out of that too. Here's an idea. Try answering the question! Given a set of labeled boxes containing numbers inside them, can you possibly find a box containing all the label numbers of boxes that don't contain their own label number? Herc
From: herbzet on 6 Jun 2010 23:34
|-|ercules wrote: > "herbzet" <herbzet(a)gmail.com> wrote > > > In any case, this thread appears to be closed, aside from tying > > up any loose ends that may remain. Since it has been established > > that proofs of the existence of higher orders of infinity do not > > in general rely on the proof that |S| < |P(S)| for any set S, there > > really is no way anyone is going to answer the poll question in the > > affirmative. Not even the OP, apparently, is interested in further > > discussion here. > > This is a joke. I thought I'd heard 10,000 reasons skeptics don't need > to investigate claims, but sci.math wants the King Con Crown. Actually, I don't subscribe to sci.math. You may avoid my replies by not cross-posting to sci.logic. > You refuse to acknowledge the powerset proof of higher infinity is silly > (at least looks silly) because there's other proofs of higher infinity! Sorry, that's not what you asked. I'm not a mind reader, you know. > Even if I meticulously worded my argument Say, that's a thought -- why don't you give it a try? > in your NULL HYPOTHESIS > anti anything lingo and traced down all your good for nothing ZFC religion > a - x - i - o - m - s you'd still not understand the claim and wiggle and worm > out of that too. You're foaming at the mouth a bit. -- hz |