SCI.MATH POLL - uncountable infinity
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From: Transfer Principle on 4 Jun 2010 01:55 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.
From: Dingo on 4 Jun 2010 02:08 On Thu, 3 Jun 2010 22:55:32 -0700 (PDT), Transfer Principle <lwalke3(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. All it proves is that you nerds all need to get a life.
From: |-|ercules on 4 Jun 2010 02:09 "Transfer Principle" <lwalke3(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. 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? 2/ Can the result of 1/ be used to prove the existence of higher infinities than 1,2,3...oo infinity? 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? Herc PS the answers are all True or False.
From: jbriggs444 on 4 Jun 2010 08:40 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.
From: |-|ercules on 4 Jun 2010 08:59
"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! 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 |