SCI.MATH POLL - uncountable infinity
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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?
From: dannas on 4 Jun 2010 15:48 "David R Tribble" <david(a)tribble.com> wrote in message news:e569c1b1-cb12-4c11-9f86-0933d55153ef(a)o4g2000vbo.googlegroups.com... > |-|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? what kind of glue holds on the labels? do the magnets stick to each other? How can you read the numbers inside a closed box?
From: |-|ercules on 4 Jun 2010 17:04
"MoeBlee" <jazzmobe(a)hotmail.com> wrote > On Jun 2, 8:00 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > >> 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. > > That's garbled. Cantor didn't prove a QUESTION to be nonexistent. > > Also, you've left out the crucial "and only" clause. Maybe what you > mean is this: > > Suppose there is a room with boxes in it, such that each box in the > room has one or more (or, could be zero or more, too) numbers in it, > and each box in the room has a label number. Is there a box in the > room that has in it all and only the label numbers of boxes that do > not have in them their own label number? > > There is no such box, since if there is such a box, then the label > number of the box is in the box if and only if the label number of the > box is not in the box. > >> Is the following statement TRUE or FALSE? >> >> << The fact that there is no box that contains the numbers of all the boxes >> >> << that don't contain their own number proves that higher infinities exist. >> > > That assumes a fact that you've miststated. > > A correct statement is: There is no box that contains the label > numbers of all AND ONLY those boxes that don't contain their own label > number. > > Also, the word 'prove' is ambiguous. In formal mathematics, we prove > sentences relative to formal systems, while also 'prove' means to > provide convincing basis for belief (or something to that effect). > > 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. > > What's your point in asking the question? > > MoeBlee Now reword this into a true statement. The proof of higher infinities than 1,2,3..oo infinity relies on the fact that there is no box that contains all and only all the label numbers of all the boxes that don't contain their own label number. Herc |