Monday, November 19, 2007

6.02 And so we come to numbers. I define

x = Ω0, x Def. and

Ω’ Ων, x = Ων+1, x Def.

According to these symbolic rules [or: rules about signs] we therefore write the series x, Ω’ x, Ω’ Ω’ x, Ω’ Ω’ Ω’ x, …..

thus: Ω0, x, Ω0+1, x, Ω0+1+1, x, Ω0+1+1+1, x, …..

Therefore I write instead of “[x, ξ, Ω’ ξ]”:

“[Ω0, x, Ων, x, Ων+1, x]”.

And I define:

0 + 1 = 1 Def.

0 + 1 + 1 = 2 Def.

0 + 1 + 1 + 1 = 3 Def.

(and so on)

Frascolla (p. 187): “The inductive definition at the outset of 6.02 is given the task of putting the abstract notion of the application of an operation at the bottom of the construction of arithmetic. In sharp opposition to the logicist programme, number is not construed as the number of elements of a class (of the extension of a concept proper), but as the number of applications of a symbolic procedure, whose iteration gives rise to a potentially endless formal series of propositions.”

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