ArisKatsaris comments on Logical Pinpointing - Less Wrong

62 Post author: Eliezer_Yudkowsky 02 November 2012 03:33PM

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Comment author: ArisKatsaris 14 November 2012 09:07:57AM 2 points [-]

Did you know that the set of finite strings of a single symbol alphabet also satisfies the Peano Axioms?

Surely the set of finite strings in an alphabet of no-matter-how-many-symbols satisfies the Peano axioms? e.g. using the English alphabet (with A=0, B=S(A), C=S(B)....AA=S(Z), AB=S(AA), etc would make a base-26 system).

Comment author: [deleted] 15 November 2012 01:06:20PM 0 points [-]

Single symbol alphabet is more interesting, (empty string = 0, sucessor function = append another symbol) the system you describe is more succinctly described using a concatenation operator:

  • 0 = 0, 1 = S0, 2 = S1 ... 9 = S8.
  • For All b in {0,1,2,3,4,5,6,7,8,9}, a in N: ab = a x S9 + b

From these definitions we get, example-wise:

  • 10 = 1 x S9 + 0 = SSSSSSSSSS0
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