[plt-scheme] Natural numbers

From: Jos Koot (jos.koot at telefonica.net)
Date: Wed Mar 11 21:31:25 EDT 2009

The following formulation of the axioms of Peano is (almost exactly) copied from Hofstadter's book Goedel, Escher, Bach:

1) Genie is a djinn.
2) Every djinn has a meta, which is also a djinn.
3) Genie is not the meta of any djinn.
4) Different djinns have different metas.
5) If genie has property X and each djinn with property X relays the property to its meta, then all djinns have property X.

Read "djinn" as "natural number" and "meta of djinn" as "djinn + 1".
You can read "Genie" as any arbitrary number, for example pi, which give us pi, pi+1, pi+2, etc.

Sometimes an additional restriction is formulated: The set of djinns is the smallest set that satisfies the axioms.

When using natural numbers for counting, I agree with Matthias that the most natural starting point is 0. You get odd arithmetics when not starting with 0. For example in music a quart (4) *plus* a quint (5) is an octave (8). You dont want 4+5=8, I think (it should be 3+4=7). Apparently the nomenclature of intervals in music were formulated before the notion of number zero was discovered/invented. 
(For who it concerns: I have put 'plus' between asterixes, because when dealing with frequencies, you must multiply ratios of frequencies, not add them :)

  ----- Original Message ----- 
  From: aditya shukla 
  To: PLT Scheme ML 
  Sent: Thursday, March 12, 2009 12:19 AM
  Subject: [plt-scheme] Natural numbers

  Can someone please explain why natural numbers in chapter 11 of htdp begin with 0 ? 



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