phyti Posted May 23 Share Posted May 23 (edited) Ann and Bill independently work on examples of Cantor's diagonal argument. Ann: 111001... 000111... 101100... 110011... 000110... 010110... transforms diagonal D1 to alternating '01' sequence T1, 010101... which can't appear in any list per the cda. Bill: 000110... 010110... 110001... 000111... 101100... 110011... transforms diagonal D2 to alternating '10' sequence T2, 101010... which can't appear in any list per the cda. If D1 and D2 appear in any list, they must be members of the complete list. T1=D2. I.e. a missing sequence is only relative to an individual list. Edited May 23 by phyti Link to comment Share on other sites More sharing options...

Genady Posted May 23 Share Posted May 23 5 minutes ago, phyti said: which can't appear in any list per the cda There is no problem, because the cda does not say that a constructed sequence does not appear in any list, but in the given list. Quote If s_{1}, s_{2}, ... , s_{n}, ... is any enumeration of elements from T, then an element s of T can be constructed that doesn't correspond to any s_{n} in the enumeration. Cantor's diagonal argument - Wikipedia Link to comment Share on other sites More sharing options...

phyti Posted Wednesday at 06:24 PM Author Share Posted Wednesday at 06:24 PM That's what I said. "I.e. a missing sequence is only relative to an individual list". Here is what Cantor said. "If E_{1}, E_{2}, …, E_{v}, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E_{0} of M, which cannot be connected with any element E_{v}." "From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E_{1}, E_{2}, …, E_{v}, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M." From the example T1 which is missing from Ann's list is included as D2 in Bill' list. Thus it is not missing from M. Missing from a subset does not imply missing from the complete set. Link to comment Share on other sites More sharing options...

Boltzmannbrain Posted Saturday at 06:39 PM Share Posted Saturday at 06:39 PM On 5/24/2023 at 12:24 PM, phyti said: That's what I said. "I.e. a missing sequence is only relative to an individual list". Here is what Cantor said. "If E_{1}, E_{2}, …, E_{v}, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E_{0} of M, which cannot be connected with any element E_{v}." "From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E_{1}, E_{2}, …, E_{v}, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M." From the example T1 which is missing from Ann's list is included as D2 in Bill' list. Thus it is not missing from M. Missing from a subset does not imply missing from the complete set. CDA is about cardinality (size) of the two sets (countable vs uncountable). I am not sure exactly what the problem you have with CDA. I am thinking about an analogy to your issue to see what happens if we apply your issue on a much smaller scale. Assume Bill and Ann want to do the same thing with a set L vs a smaller set S. They want to know which set is bigger, but they don't want to count the elements. I will keep this in binary-type numbers vs natural numbers Let's say L = {000, 111, 110, 011, 100, 001, 010}, and S = {2, 4, 5}. (Keep in mind that the correlated elements don't have to be same things. K = {5, 7, 1} has three elements and P = {@, U, 9} also has three elements; their cardinalities are the same). Ann might get T1 = 101 and Bill might get T2 = 110. The set L can still be larger than the set S even though T1 doesn't equal T2. And clearly, we can always find an element of L that can not match to an element of S. I see no issue here. Or am I missing your point? Link to comment Share on other sites More sharing options...

phyti Posted 13 hours ago Author Share Posted 13 hours ago ; "If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0 of M, which cannot be connected with any element Ev." "From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M." Cantor believes there is a hierarchy of infinities, a communication from above which he is obligated to explain to the world. Cantor is selling his idea of transfinite numbers to anyone who will listen. In the above quotes he states there are always elements of M which differ from those in any list, thus there cannot be a one to one correspondence with the set of natural numbers N, even though N is inexhaustible. The Ann and Bob example contradicts his idea, since one will have an infinite list containing the sequence E0 that the other infinite list does not contain. Link to comment Share on other sites More sharing options...

Genady Posted 12 hours ago Share Posted 12 hours ago 43 minutes ago, phyti said: In the above quotes he states there are always elements of M which differ from those in any list, thus there cannot be a one to one correspondence with the set of natural numbers N, even though N is inexhaustible. The Ann and Bob example contradicts his idea, since one will have an infinite list containing the sequence E0 that the other infinite list does not contain. No, there is no contradiction. As per Cantor, for any list there is an element in M which differs from those in that list. Link to comment Share on other sites More sharing options...

Genady Posted 11 hours ago Share Posted 11 hours ago IOW, Cantor does not say, ∃ element∈M, ∀ list L, element∉L He says, ∀ list L, ∃ element∈M, element∉L Link to comment Share on other sites More sharing options...

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