SCI.MATH POLL - uncountable infinity
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From: dannas on 4 Jun 2010 13:19 "|-|ercules" <radgray123(a)yahoo.com> wrote in message news:86sbllF4jpU1(a)mid.individual.net... > "jbriggs444" <jbriggs444(a)gmail.com> wrote... >> On Jun 4, 2:09 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: >>> "Transfer Principle" <lwal...(a)lausd.net> wrote >>> >>> >>> >>> >>> >>> > On Jun 2, 6:00 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >>> >> No explanations or you will spoil the poll, just TRUE or FALSE. >>> >> Is the 'missing box (set)' central to the powerset proof of >>> >> uncountable infinity? >>> >>> > Ah, a poll. I remember making a big deal about a poll to >>> > determine what most people believe about 0.999.... >>> >>> > Here's how I think this poll should be asked: >>> >>> > "Do you believe that there are more reals than naturals?" >>> >>> > Note that this is _not_ the same as, "do you believe that >>> > _ZFC_ proves that there are more reals than naturals?" For >>> > this isn't open to a vote at all -- there is no debate >>> > that the uncountability of the reals is a theorem of ZFC. >>> >>> > Similarly, "Do you believe that CH is true?" is also a >>> > question that can be asked in a poll. It's often said that >>> > most set theorists believe that CH is false, while many >>> > mathematicians who aren't set theorists believe that CH is >>> > in fact true. I wouldn't mind seeing a poll to confirm >>> > this common opinion. Of course, "Do you believe that ZFC >>> > proves CH?" isn't open to debate, since it's known that >>> > ZFC proves neither CH nor its negation. >>> >>> > But the difference between CH and the uncountability of >>> > the reals is that the former is undecidable in ZFC, while >>> > the latter is provable in ZFC. Those for whom ZFC is the >>> > preferred theory are likely to question the legitimacy of >>> > any poll in which a majority believe in any statement >>> > refuted by ZFC. I suspect that they'd consider CH to be a >>> > legitimate poll question, but not the uncountability of >>> > the reals (even though the former asks whether card(R) is >>> > greater than aleph_1, while the latter asks whether it is >>> > greater than aleph_0). >>> >>> > So far, it doesn't appear that Herc's poll is working. In >>> > the same way, I'd like to see a poll that asks whether >>> > one believes that 0.999... is equal to 1 (which isn't the >>> > same as asking whether _ZFC_ proves it), but I doubt that >>> > those who work in ZFC will accept any such poll (unless >>> > it establishes that 0.999... is indeed 1). Any poll that >>> > doesn't establish what ZFC proves is automatically flawed. >> >> ZFC doesn't prove much about whether 0.999... is equal to 1. There's >> a question of notation to be ironed out before the question of >> equality >> can be taken up. >> >> Once you've ironed out the notation, there may not be much left >> to prove. >> >>> Just answer the question. this isn't sci.math.zfc >>> >>> For a (infinite) list of uniquely numbered boxes containing (possibly >>> infinite amount of) fridge magnet numbers >> >> Pretty ambiguous wording... >> >> >>> 1/ Is there a box that contains the numbers of all the boxes that don't >>> contain their own number? >> >> Suppose there is a labelled "1" containing a fridge magnet in the >> shape of the number 1. >> Suppose that all other boxes in the room (if any) also contain a >> fridge magnet in the >> shape of their number. >> >> Does box number 1 box satisfy the condition intended in the question? >> >> I know it satisfies the condition _stated_ in the question. >> >> >> What about box number 6 if we swap fridge magnets with box number 9? >> >> >> What about the fact that a room with an infinite number of boxes or a >> box with an infinite number of fridge magnets >> are both (as far as we can tell) physical impossibilities? Does this >> mean that the question assumes a >> contradiction so that all possible answers are equally, vacuously >> correct. > > > What are you complaining about? There's some boxes with numbers in them, > that's it! > what about the boxes that dont? Why leave them out? Isn''t there an infinity of them? > It's meant to simplify the problem and allow you to conceptualize it, not > open up turgid interpretations > to avoid the frickin simple question. > > Herc
From: MoeBlee on 4 Jun 2010 13:47 On Jun 4, 11:06 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > Your example about the boxes is an analogy of a proof in certain > formal systems that no set is equinumerous with its power set, and > also an analogy with an argument, aside from any formal system, that > many mathematicians take as convincing toward the conclusion that no > set is equinumerous with its power set, and such proofs, along with > other principles, lead to a proof that there exist sets that are > uncountable. P.S. The other principles I have in mind are (1) that every set has a power set and (2) that there exists an infinite set. So Cantor's theorem (which depends only on the axiom schema of separation and the axiom of extensionality, if I recall) along with the principles that every set has a power set (the power set axiom) and that there exists an infinite set (derivable from the axiom of infinity) provide that there exists an uncountable set. The proof relies only on ordinary logic for mathematics (and even a narrower version that permits only intuitionistic inferences) along with the above mentioned principles (or axioms). Of course, one may decline to accept the very modest logic or decline to accept the principles (axioms) used. But that the statement "there exist uncountable sets" does follow by said logic from said axioms is (upon formalization) machine checkable and, even more basically, checkable by ordinary human inspection. I don't know specifically which rule(s) of logic or set theoretic axiom(s) you decline, if any. MoeBlee
From: MoeBlee on 4 Jun 2010 14:05 On Jun 4, 12:55 am, Transfer Principle <lwal...(a)lausd.net> wrote: > it's known thatZFCproves neither CH nor its negation. It's known to you? You know that ZF(C) is consistent? MoeBlee
From: Transfer Principle on 4 Jun 2010 14:37 On Jun 4, 11:05 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 4, 12:55 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > it's known thatZFCproves neither CH nor its negation. > It's known to you? You know that ZF(C) is consistent? How about this: ZFC proves _neither_ or _both_ of CH and its negation (Goedel and Cohen)? Here, I was trying to distinguish between those statements like CH, which are undecidable in ZFC, and those like "R is uncountable," which are definitely _decidable_ in ZFC. In particular, the results of a poll which asks "Is CH true?" are more likely to be accepted by those who use ZFC than the results of a poll "Is R uncountable," especially if a majority votes "no."
From: Transfer Principle on 4 Jun 2010 14:52
On Jun 3, 11:09 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "Transfer Principle" <lwal...(a)lausd.net> wrote > > So far, it doesn't appear that Herc's poll is working. In > > the same way, I'd like to see a poll that asks whether > > one believes that 0.999... is equal to 1 (which isn't the > > same as asking whether _ZFC_ proves it), but I doubt that > > those who work in ZFC will accept any such poll (unless > > it establishes that 0.999... is indeed 1). Any poll that > > doesn't establish what ZFC proves is automatically flawed. > Just answer the question. this isn't sci.math.zfc > For a (infinite) list of uniquely numbered boxes containing (possibly infinite amount of) fridge magnet numbers > 1/ Is there a box that contains the numbers of all the boxes that don't contain their own number? I would say no, then. > 2/ Can the result of 1/ be used to prove the existence of higher infinities than 1,2,3...oo infinity? It depends on the assumptions. What it would prove is that if there is a box for _every_ possible combination of numbered magnets, then the set of all boxes would be of a higher infinity than the natural numbers. But, if there is some limitation on what combinations of magnets exist (e.g., each box can contain only _finitely_ many numbered magnets), then there need not be any higher infinities. > For any set of indexed subsets of natural numbers > 3/ Is there an indexed subset of naturals that contains all the indexes of the subsets who's > index is not an element of it's own subset? > 4/ Can the result of 3/ be used to prove the existence of higher infinities than 1,2,3...oo infinity? 3/ and 4/ are analogous to 1/ and 2/. I definitely answer no to 3/. As for 4/, to make this more precise, it only proves the existence of higher infinities if we're allowed to have a _powerset_, or set of all sets of naturals. Then the powerset is of a higher infinity, but if there's no powerset, we can't prove higher infinity. In ZFC, there is a Powerset axiom, and so ZFC does prove that higher infinities exists. But, as Herc points out, this isn't sci.math.zfc, so if he doesn't accept the Powerset axiom, then more power to him. (Note that first-order PA has no Powerset axiom.) Without the Powerset axiom, we can't prove that there even exists a powerset of omega, much less that the set is of a higher infinity. |