(phi( <[1] 2 3 4...> ) & An (phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> ))) -> phi( <[1 2 3 4...]> )
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From: |-|ercules on 30 Jun 2010 20:59 Behold the Cantor killer! phi( <[1] 2 3 4...> ) & An (phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> )) -> phi( <[1 2 3 4...]> ) Prefix Induction Schema If a property holds for the 1st element in a sequence, and if it holds for the first n elements in a sequence then it holds for the first n+1 elements in a sequence, then the property holds for all elements in the sequence. It can be used to prove all possible sequences of digits are listable, namely this limit exists (the sequences are infinite). As the length of the list of computable reals->oo, the length of all possible digit sequences on the list->oo. Herc
From: Dingo on 30 Jun 2010 21:17 On Thu, 1 Jul 2010 10:59:28 +1000, "|-|ercules" <radgray123(a)yahoo.com> wrote: This has no relevance whatsoever to aus.tv - stop posting to it, and others please delete aus.tv when replying to it. >Behold the Cantor killer! > >phi( <[1] 2 3 4...> ) & An (phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> )) >-> >phi( <[1 2 3 4...]> ) > > >Prefix Induction Schema > >If a property holds for the 1st element in a sequence, and if it holds for the first n elements in a sequence >then it holds for the first n+1 elements in a sequence, then the property holds for all elements in the sequence. > >It can be used to prove all possible sequences of digits are listable, namely this limit exists (the sequences are infinite). > >As the length of the list of computable reals->oo, the length of all possible digit sequences on the list->oo. > >Herc
From: George Greene on 1 Jul 2010 01:30 On Jun 30, 8:59 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > Behold the Cantor killer! > > phi( <[1] 2 3 4...> ) & An (phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> )) > -> > phi( <[1 2 3 4...]> ) > > Prefix Induction Schema That IS NOT AN induction schema, DUMBASS. THAT is UNSOUND. There is an EASY list that KILLS THAT SCHEMA. It's the list of all the finite prefixes of Pi followed by ALL ZEROS. Phi, in this case, would be, "when evaluated as a decimal, comes up LESS THAN Pi.". For all these sequences, THEY ARE less than Pi, because they end in all 0s. But the conclusion, Phi(<[1 2 3 4 ...]>), IS FALSE, because the nth digit of the sequence is the nth digit of Pi, so <[1 2 3 4 ...]> EQUALS Pi, and IS NOT less than Pi. So your schema IS UNSOUND, DUMBASS. ANY sequence that approaches a limit over INFINITY, without actually reaching it, you know like, 1/2 3/4 7/8 15/16 31/32 ... ad naus.... KILLS YOUR *SCHEMA*. > > If a property holds for the 1st element in a sequence, and if it holds for the first n elements in a sequence > then it holds for the first n+1 elements in a sequence, then the property holds for all elements in the sequence. That's right. ALL INDIVIDUAL elements of the sequence (if the sequence is inductive). It does NOT hold for the COLLECTIVE INFINITY of all the elements in the sequence. It holds for EVERY element of the sequence, EACH element of the sequence, and ANY element of the sequence. IT DOES NOT hold for THE INFINITE SEQUENCE, DUMBASS. It INSTEAD holds for infiniteLY MANY *indivdidual*different* SINGLE elements -- NOT for an INFINITE COLLECTION TAKEN AS A WHOLE, DUMBASS.
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