SCI.MATH POLL - uncountable infinity
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From: MoeBlee on 4 Jun 2010 18:41 On Jun 4, 4:04 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "MoeBlee" <jazzm...(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. You've not pointed out anything I've said that is untrue. Boiled down, pretty much all I'm saying is that with a certain system of logic and certain axioms there exists a certain sentence that is formally derivable. It's not even controversial to anyone who knows what a formal derivation is. > The proof of higher infinities than 1,2,3..oo infinity I don't know what EXACTLY you mean by the notation "1,2,3..oo infinity". Meanwhile, in set theory, we do prove that w (the set of natural numbers) is strictly dominated by the power set of w. > 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. (1) There is no mention of "boxes" and "labels" in a proof of Cantor's theorem. (2) As an analogy, however, yes, we prove that there is no box as you describe. (3) However, we dont' "rely" on that fact so much as we PROVE it. (43) Nice to see you took my advice to include the "and only" clause, but your second and third 'all' is unneeded. You merely need to say: There is no box that contains all and only the label numbers of the boxex that don't contain their own label number. MoeBlee
From: |-|ercules on 4 Jun 2010 18:49 "MoeBlee" <jazzmobe(a)hotmail.com> wrote ... > On Jun 4, 4:04 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> "MoeBlee" <jazzm...(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. > > You've not pointed out anything I've said that is untrue. > > Boiled down, pretty much all I'm saying is that with a certain system > of logic and certain axioms there exists a certain sentence that is > formally derivable. It's not even controversial to anyone who knows > what a formal derivation is. > >> The proof of higher infinities than 1,2,3..oo infinity > > I don't know what EXACTLY you mean by the notation "1,2,3..oo > infinity". Meanwhile, in set theory, we do prove that w (the set of > natural numbers) is strictly dominated by the power set of w. > >> 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. > > (1) There is no mention of "boxes" and "labels" in a proof of Cantor's > theorem. (2) As an analogy, however, yes, we prove that there is no > box as you describe. (3) However, we dont' "rely" on that fact so much > as we PROVE it. (43) Nice to see you took my advice to include the > "and only" clause, but your second and third 'all' is unneeded. You > merely need to say: There is no box that contains all and only the > label numbers of the boxex that don't contain their own label number. > > MoeBlee You ignored my request to reword the paragraph. Are you saying if there WAS a box with such a property then proof of higher infinities still works? If not, then proof of higher infinities relies on that fact. What don't you understand about what infinity is referred to here: 1,2,3...oo Are you saying this paragraph is FALSE then? 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 the boxes that don't contain their own label number. Herc
From: MoeBlee on 4 Jun 2010 19:10 On Jun 4, 5:49 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "MoeBlee" <jazzm...(a)hotmail.com> wrote ... > > > > > On Jun 4, 4:04 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > >> "MoeBlee" <jazzm...(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. > > > You've not pointed out anything I've said that is untrue. > > > Boiled down, pretty much all I'm saying is that with a certain system > > of logic and certain axioms there exists a certain sentence that is > > formally derivable. It's not even controversial to anyone who knows > > what a formal derivation is. > > >> The proof of higher infinities than 1,2,3..oo infinity > > > I don't know what EXACTLY you mean by the notation "1,2,3..oo > > infinity". Meanwhile, in set theory, we do prove that w (the set of > > natural numbers) is strictly dominated by the power set of w. > > >> 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. > > > (1) There is no mention of "boxes" and "labels" in a proof of Cantor's > > theorem. (2) As an analogy, however, yes, we prove that there is no > > box as you describe. (3) However, we dont' "rely" on that fact so much > > as we PROVE it. (43) Nice to see you took my advice to include the > > "and only" clause, but your second and third 'all' is unneeded. You > > merely need to say: There is no box that contains all and only the > > label numbers of the boxex that don't contain their own label number. > > > MoeBlee > > You ignored my request to reword the paragraph. No I didn't.You asked me to reword something (by the way, WHICH paragraph?) into a "true" statement. So I said that you haven't said what is untrue in what I wrote. I'm not going to go further with you in your game. All you need to do is say what you consider untrue in what I wrote. > Are you saying if there WAS a box with such a property > then proof of higher infinities still works? It is simply illogical to say there is such a box. You can see that for yourself. So, the question of "if there were such a box" is aside the point and is outside of any logical conception. To prove that there is no surjection from a set onto its power set, we show that if there were then there would be such an impossible "box" (to use your analogy), so since a box is impossible, there is no surjection from a set onto its power set. > What don't you understand about what infinity is referred to here: > > 1,2,3...oo For me to understand, all you need to do is state the context of your notation and define it. Without that, I can only guess that you mean either 1) the set of natural numbers, as it extends without end or 2) the set of natural numbers, as it extends without end, plus some object or another you're calling 'oo'. > Are you saying this paragraph is FALSE then? > > 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 the boxes that > don't contain their own label number. Uncountability refers to sets that are neither one-to-one with w (read 'omega'; i.e., the set of natural numbers, i.e., N, i.e. {0 1 2 ... } nor with any member of omega. You don't need the notation: "1, 2, 3, ... oo", which is ambiguous as to whether you mean "{1 2 3 ...} or {1 2 3 ... oo}" where "oo" stands for some object. All you need to refer to is w or to N (for the purpose at hand it doesn't even matter whether we include 0 or not). And again, I'll explain the proof in terms of your boxes: If the power set of w is countable, then (in analogy) there exists such a box as you describe, but we also prove that there exists no such box, so we prove that the power set of w is not countable. MoeBlee
From: |-|ercules on 4 Jun 2010 19:18 "MoeBlee" <jazzmobe(a)hotmail.com> wrote ... > On Jun 4, 5:49 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> "MoeBlee" <jazzm...(a)hotmail.com> wrote ... >> >> >> >> > On Jun 4, 4:04 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> >> "MoeBlee" <jazzm...(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. >> >> > You've not pointed out anything I've said that is untrue. >> >> > Boiled down, pretty much all I'm saying is that with a certain system >> > of logic and certain axioms there exists a certain sentence that is >> > formally derivable. It's not even controversial to anyone who knows >> > what a formal derivation is. >> >> >> The proof of higher infinities than 1,2,3..oo infinity >> >> > I don't know what EXACTLY you mean by the notation "1,2,3..oo >> > infinity". Meanwhile, in set theory, we do prove that w (the set of >> > natural numbers) is strictly dominated by the power set of w. >> >> >> 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. >> >> > (1) There is no mention of "boxes" and "labels" in a proof of Cantor's >> > theorem. (2) As an analogy, however, yes, we prove that there is no >> > box as you describe. (3) However, we dont' "rely" on that fact so much >> > as we PROVE it. (43) Nice to see you took my advice to include the >> > "and only" clause, but your second and third 'all' is unneeded. You >> > merely need to say: There is no box that contains all and only the >> > label numbers of the boxex that don't contain their own label number. >> >> > MoeBlee >> >> You ignored my request to reword the paragraph. > > No I didn't.You asked me to reword something (by the way, WHICH > paragraph?) into a "true" statement. So I said that you haven't said > what is untrue in what I wrote. > > I'm not going to go further with you in your game. All you need to do > is say what you consider untrue in what I wrote. > >> Are you saying if there WAS a box with such a property >> then proof of higher infinities still works? > > It is simply illogical to say there is such a box. You can see that > for yourself. So, the question of "if there were such a box" is aside > the point and is outside of any logical conception. > > To prove that there is no surjection from a set onto its power set, we > show that if there were then there would be such an impossible > "box" (to use your analogy), so since a box is impossible, there is no > surjection from a set onto its power set. > >> What don't you understand about what infinity is referred to here: >> >> 1,2,3...oo > > For me to understand, all you need to do is state the context of your > notation and define it. Without that, I can only guess that you mean > either > > 1) the set of natural numbers, as it extends without end > > or > > 2) the set of natural numbers, as it extends without end, plus some > object or another you're calling 'oo'. > >> Are you saying this paragraph is FALSE then? >> >> 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 the boxes that >> don't contain their own label number. > > Uncountability refers to sets that are neither one-to-one with w (read > 'omega'; i.e., the set of natural numbers, i.e., N, i.e. {0 1 2 ... } > nor with any member of omega. You don't need the notation: > > "1, 2, 3, ... oo", which is ambiguous as to whether you mean "{1 2 > 3 ...} or {1 2 3 ... oo}" where "oo" stands for some object. > > All you need to refer to is w or to N (for the purpose at hand it > doesn't even matter whether we include 0 or not). > > And again, I'll explain the proof in terms of your boxes: > > If the power set of w is countable, then (in analogy) there exists > such a box as you describe, but we also prove that there exists no > such box, so we prove that the power set of w is not countable. > > MoeBlee For someone who posted the only clear post from sci.math regulars 2 posts ago, you really excelled yourself in jumping into obscurity and ducking and weaving away from the requests and questions. I'll wait and see if someone else takes the bait. >> 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 the boxes that >> don't contain their own label number. TRUE OR FALSE Herc
From: herbzet on 4 Jun 2010 23:38
|-|ercules wrote: > I'll wait and see if someone else takes the bait. > > >> 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 the boxes that don't contain their own label number. > > TRUE OR FALSE Um, false, so far as I know. We have 1) |N| < |P(N)| 2) |P(N)| <= |R| -------------- .: |N| < |R| but neither of Cantor's proofs that |N| < |R| involves either of premises (1) or (2), as far as I can recall. Perhaps someone will refresh my memory on who first observed that premise (2) is true. -- hz |