CANTOR'S TWO PROOFS
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From: |-|ercules on 16 Jun 2010 23:39 CANTOR'S POWER PROOF! Superinfinity is based on the circular reasoning "no box contains the box numbers that don't contain their own box number". No I don't like it either but there is a new sequence because we *construct* it like so: CANTORS DIFFERENT PROOF! Defn: digit 1 is different, and digit 2 is different, digit 3 is different, ... Proof: digit 1 is different, and digit 2 is different, digit 3 is different... Therefore it's a different number! Superinfinity must be true because there are TWO proofs of superinfinity! We invented natural numbers, partitioned the space between natural numbers recursively with one of ten options. It's ridiculous to define a different number as "the other nine options ad infinitum". Herc -- "And God posted an angel with a flaming sword at the gates of Cantor's paradise, that the slow-witted and the deliberately obtuse might not glimpse the wonders therein." ~ Barb Knox "There are more things in Cantor's paradise, Horatio, than are dreamt of by your computers." ~ Barb Knox
From: Mike Terry on 17 Jun 2010 16:19 "|-|ercules" <radgray123(a)yahoo.com> wrote in message news:87tjn9Fg79U1(a)mid.individual.net... > CANTOR'S POWER PROOF! > Superinfinity is based on the circular reasoning > "no box contains the box numbers that don't contain their own box number". Why do you say this is circular reasoning? It is not. You need to break the phrase down into conceptual chunks: First: "box numbers that don't contain their own box number" This is not circular. The first reference to "box number" refers to a label on the outside of the box, and the second "box number" refers to the numbered balls (or whatever) inside the boxes. The way to think of this (for those who "don't get it") is to imagine the question first for a specific box. For each box number, it is perfectly well defined whether or not the corresponding box contains it's own box number. Well, let's try it out! Remember, the numbered boxes and their contents are all to be fixed at the outset (no cheating and moving balls around half way through :) So suppose you give me the the boxes, and box 7 happens to contain balls {1, 5, 3838383838}. Here we go: Does box 7 contain ball number 7? Um, NO. What's circular about this? (Answer= nothing) And suppose box 22 contains balls {1, 2, 3, 4,...21, 22, 23...}. Does box 22 contain ball number 22? Um, YES. Again, nothing circular going on, even if there are infinitely many balls in a box. I.e. for each number n we can straight-forwardly answer "yes/no" to the question "does box n contain a ball numbered n?". So there is a definite (non-circularly defined) set of "box numbers that don't contain their own box number". In your terminology we could imagine we have a NEW box with exactly the balls corresponding to this set. (Note, it's a NEW box we're talking about here, not one of the numbered boxes. I think this is probably where you're confused, although you wouldn't acknowledge this even if I'm right.) Now we have our new box, we can ask the sensible, non-circular, question: "Does any box contain precisely the same balls as our new box?" This is not a circular question, and we can simply prove the answer is NO, which is analagous to Cantors power set proof. > > No I don't like it either but there is a new sequence because we *construct* it like so: > > CANTORS DIFFERENT PROOF! > Defn: digit 1 is different, and digit 2 is different, digit 3 is different, ... > Proof: digit 1 is different, and digit 2 is different, digit 3 is different... That's not a proof, or even a coherent statement (surprise surprise :-) For a start, digit 1 of what is different from what, and so on? I suppose you're just bringing to mind the basic idea of Cantor's diagonal proof. OK... > Therefore it's a different number! > > Superinfinity must be true because there are TWO proofs of superinfinity! > > We invented natural numbers, partitioned the space between natural numbers > recursively with one of ten options. > > It's ridiculous to define a different number as "the other nine options ad infinitum". > Another incoherent statement (surprise surprise). Are you expecting people to agree or disagree with this? Regards, Mike.
From: |-|ercules on 17 Jun 2010 17:57 "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote > "|-|ercules" <radgray123(a)yahoo.com> wrote in message > news:87tjn9Fg79U1(a)mid.individual.net... >> CANTOR'S POWER PROOF! >> Superinfinity is based on the circular reasoning >> "no box contains the box numbers that don't contain their own box number". > > Why do you say this is circular reasoning? It is not. You need to break > the phrase down into conceptual chunks: > > First: > > "box numbers that don't contain their own box number" > > This is not circular. The first reference to "box number" refers to a label > on the outside of the box, and the second "box number" refers to the > numbered balls (or whatever) inside the boxes. > > The way to think of this (for those who "don't get it") is to imagine the > question first for a specific box. For each box number, it is perfectly well > defined whether or not the corresponding box contains it's own box number. > > Well, let's try it out! > > Remember, the numbered boxes and their contents are all to be fixed at the > outset (no cheating and moving balls around half way through :) > > So suppose you give me the the boxes, and box 7 happens to contain balls {1, > 5, 3838383838}. > > Here we go: Does box 7 contain ball number 7? Um, NO. > > What's circular about this? (Answer= nothing) > > And suppose box 22 contains balls {1, 2, 3, 4,...21, 22, 23...}. Does box > 22 contain ball number 22? Um, YES. Again, nothing circular going on, even > if there are infinitely many balls in a box. > > I.e. for each number n we can straight-forwardly answer "yes/no" to the > question "does box n contain a ball numbered n?". > > So there is a definite (non-circularly defined) set of "box numbers that > don't contain their own box number". In your terminology we could imagine > we have a NEW box with exactly the balls corresponding to this set. (Note, > it's a NEW box we're talking about here, not one of the numbered boxes. I > think this is probably where you're confused, although you wouldn't > acknowledge this even if I'm right.) > > Now we have our new box, we can ask the sensible, non-circular, question: > > "Does any box contain precisely the same balls as our new box?" > > This is not a circular question, and we can simply prove the answer is NO, > which is analagous to Cantors power set proof. I see "no box contains the box numbers (of boxes) that don't contain their own box number" as a self evident truth due to the (seemingly apparent) self reference. How can a box contain the box numbers (of boxes) that don't contain their own box number? The question is self defeating, it's a self evident impossibility. It's a trivial negative. If a box contained the box numbers, of boxes that don't contain their own box number, then it's number would not belong. There's no need for a 10 line proof, it's a self evident fact. How you derive superinfinty from a simple self reference is disturbing. > >> >> No I don't like it either but there is a new sequence because we > *construct* it like so: >> >> CANTORS DIFFERENT PROOF! >> Defn: digit 1 is different, and digit 2 is different, digit 3 is > different, ... >> Proof: digit 1 is different, and digit 2 is different, digit 3 is > different... > > That's not a proof, or even a coherent statement (surprise surprise :-) > > For a start, digit 1 of what is different from what, and so on? > > I suppose you're just bringing to mind the basic idea of Cantor's diagonal > proof. OK... Ahh you recognize the form of Cantor's proof. DEFINE the first digit as something different, and the second digit as something different... PROOF: the first digit is different, and the second digit is different... All you have done is given a rudimentary *specific* different digit in your DEFINITION (e.g. d(n,n) = 4 iff ....) and then proved it was a generally different digit in the PROOF that it's a different number, but your definition and proof could ideally be identical statements. Consider this LOGIC LOOP. D = 1 WHILE TRUE set digit D different to digit D of the first real in the given list YOU HAVEN'T PRODUCED A NEW DIGIT SEQUENCE YET D++ WEND How does this make a *new digit sequence* that is not present on the list of computable reals? > >> Therefore it's a different number! >> >> Superinfinity must be true because there are TWO proofs of superinfinity! >> >> We invented natural numbers, partitioned the space between natural numbers >> recursively with one of ten options. >> >> It's ridiculous to define a different number as "the other nine options ad > infinitum". >> > > Another incoherent statement (surprise surprise). Are you expecting people > to agree or disagree with this? OK, you tell me a *general* definition for ALL possible anti-diagonals. Ignore the .999.. = 1.000.. technicality. >> It's ridiculous to define a different number as "the other nine options ad infinitum". Herc
From: |-|ercules on 17 Jun 2010 18:04 "|-|ercules" <radgray123(a)yahoo.com> wrote > Consider this LOGIC LOOP. > > D = 1 > WHILE TRUE > set digit D different to digit D of the first real in the given list > YOU HAVEN'T PRODUCED A NEW DIGIT SEQUENCE YET > D++ > WEND > > How does this make a *new digit sequence* that is not present on the > list of computable reals? Correction: of the Dth real. Herc
From: Mike Terry on 17 Jun 2010 19:59
"|-|ercules" <radgray123(a)yahoo.com> wrote in message news:87vk1gFekcU1(a)mid.individual.net... > "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote > > "|-|ercules" <radgray123(a)yahoo.com> wrote in message > > news:87tjn9Fg79U1(a)mid.individual.net... > >> CANTOR'S POWER PROOF! > >> Superinfinity is based on the circular reasoning > >> "no box contains the box numbers that don't contain their own box number". > > > > Why do you say this is circular reasoning? It is not. You need to break > > the phrase down into conceptual chunks: > > > > First: > > > > "box numbers that don't contain their own box number" > > > > This is not circular. The first reference to "box number" refers to a label > > on the outside of the box, and the second "box number" refers to the > > numbered balls (or whatever) inside the boxes. > > > > The way to think of this (for those who "don't get it") is to imagine the > > question first for a specific box. For each box number, it is perfectly well > > defined whether or not the corresponding box contains it's own box number. > > > > Well, let's try it out! > > > > Remember, the numbered boxes and their contents are all to be fixed at the > > outset (no cheating and moving balls around half way through :) > > > > So suppose you give me the the boxes, and box 7 happens to contain balls {1, > > 5, 3838383838}. > > > > Here we go: Does box 7 contain ball number 7? Um, NO. > > > > What's circular about this? (Answer= nothing) > > > > And suppose box 22 contains balls {1, 2, 3, 4,...21, 22, 23...}. Does box > > 22 contain ball number 22? Um, YES. Again, nothing circular going on, even > > if there are infinitely many balls in a box. > > > > I.e. for each number n we can straight-forwardly answer "yes/no" to the > > question "does box n contain a ball numbered n?". > > > > So there is a definite (non-circularly defined) set of "box numbers that > > don't contain their own box number". In your terminology we could imagine > > we have a NEW box with exactly the balls corresponding to this set. (Note, > > it's a NEW box we're talking about here, not one of the numbered boxes. I > > think this is probably where you're confused, although you wouldn't > > acknowledge this even if I'm right.) > > > > Now we have our new box, we can ask the sensible, non-circular, question: > > > > "Does any box contain precisely the same balls as our new box?" > > > > This is not a circular question, and we can simply prove the answer is NO, > > which is analagous to Cantors power set proof. > > > I see "no box contains the box numbers (of boxes) that don't contain their own box number" > as a self evident truth due to the (seemingly apparent) self reference. Good. > > How can a box contain the box numbers (of boxes) that don't contain their own box number? It obviously can't. You've got it! (Maybe) :) > > The question is self defeating, it's a self evident impossibility. > > It's a trivial negative. > > If a box contained the box numbers, of boxes that don't contain their own box number, > then it's number would not belong. > > There's no need for a 10 line proof, it's a self evident fact. > Well you're right in that Cantor's proof is NOT complicated, just like you're claiming it shouldn't be - it's almost self evident by your own words. The proof is just a few lines... > How you derive superinfinty from a simple self reference is disturbing. It seems to me you're agreeing that Cantor's proof is correct. However, you don't like the implications? Do you agree that there cannot be a bijection between a set A and it's power set P(a)? That is what Cantor's result says... > > > > > > > >> > >> No I don't like it either but there is a new sequence because we > > *construct* it like so: > >> > >> CANTORS DIFFERENT PROOF! > >> Defn: digit 1 is different, and digit 2 is different, digit 3 is > > different, ... > >> Proof: digit 1 is different, and digit 2 is different, digit 3 is > > different... > > > > That's not a proof, or even a coherent statement (surprise surprise :-) > > > > For a start, digit 1 of what is different from what, and so on? > > > > I suppose you're just bringing to mind the basic idea of Cantor's diagonal > > proof. OK... > > Ahh you recognize the form of Cantor's proof. > > DEFINE the first digit as something different, and the second digit as something different... > PROOF: the first digit is different, and the second digit is different... > > All you have done is given a rudimentary *specific* different digit in your DEFINITION > (e.g. d(n,n) = 4 iff ....) > > and then proved it was a generally different digit in the PROOF that it's a different number, > but your definition and proof could ideally be identical statements. I can't understand what you're saying here. I can't match anything you're saying with Cantor's diagonal argument for the reals, which I think is what you must be talking about? > > Consider this LOGIC LOOP. > > D = 1 > WHILE TRUE > set digit D different to digit D of the first real in the given list > YOU HAVEN'T PRODUCED A NEW DIGIT SEQUENCE YET > D++ > WEND Cantor's proof shows the existence of a new real not in the original list. It is NOT a computer program that starts at digit position 1 and then moves on to position 2 , then position 3 etc.... It simply defines a real number by virtue of defining its digit sequence in terms of the original list. It then shows the number is not in the list. All YOU are doing is looking at the corresponding FINITE PREFIXES of the missing real number digit sequence. You are pointing out that each of these is in the list, which is correct assuming an appropriate starting list (like your list of the computable reals, but there are much simpler lists you could have used). So we have defined a new real d (the "antidiagonal", in your terms) which is not in the list, but Prefix(d,n) IS in the list for each n. So what? Prefix(d,n) is not equal to d, so there is NO CONTRADICTION. Yes - YOU are right that Prefix(d,n) is in the list for each n, but also we are right that d is NOT in the original list. > > How does this make a *new digit sequence* that is not present on the > list of computable reals? The FINITE digit sequences Prefic(d,n) are all in the original list, but the INFINITE digit sequence for d is not in the list. How can this be? It just obviously is. (As proved mathematically in Cantor's proof, if you don't like me saying "it just obviously is".) A better question is "How can you think that every Prefix(d,n) being in the list means that d is also in the list?" This is simply a basic logic error. If you think d is in the list, you would have to say where in the list you thought it was, and you know you can't, because it's just obviously not there! Really - tell me, do you truly believe that there is a natural number n, such that the real number at position n in the list matches d? Really? Regards, Mike. > > > > > >> Therefore it's a different number! > >> > >> Superinfinity must be true because there are TWO proofs of superinfinity! > >> > >> We invented natural numbers, partitioned the space between natural numbers > >> recursively with one of ten options. > >> > >> It's ridiculous to define a different number as "the other nine options ad > > infinitum". > >> > > > > Another incoherent statement (surprise surprise). Are you expecting people > > to agree or disagree with this? > > > OK, you tell me a *general* definition for ALL possible anti-diagonals. Ignore > the .999.. = 1.000.. technicality. So you agree your statement was incoherent, as you've not done anything to clarify it. Also, I'll decline your incoherent request. I'm sure you know how the "antidiagonal" is defined, and I'm sure you know that the antidiagonal is only determined once the list of reals is fixed. > > >> It's ridiculous to define a different number as "the other nine options ad infinitum". > > Herc > |