SCI.MATH POLL - uncountable infinity
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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.
From: MoeBlee on 4 Jun 2010 15:02 On Jun 4, 1:52 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > 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. 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. MoeBlee
From: MoeBlee on 4 Jun 2010 15:04 On Jun 4, 1:37 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > 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)? I see. So do you have any confidence that ZF is consistent? MoeBlee
From: David R Tribble on 4 Jun 2010 15:46
|-|ercules wrote: > The powerset proof is exactly this: > > Assume a large/infinite room full of boxes with fridge magnets in the boxes that are any natural > number, and the boxes have a unique number written on them. > > "Which box contains the numbers of all the boxes that don't contain their own number ?" > > is proven (by Cantor) to be nonexistent. > > Is the following statement TRUE or FALSE? > [...] As has been pointed out in several other posts (that I didn't bother reading), you need to be more specific. A. Do the boxes contain a finite or infinite number of magnets? B. Does any natural label on any of the magnets within any given box occur more than once within the box? |