Page:Wittgenstein - Tractatus Logico-Philosophicus, 1922.djvu/159

Rh  And thus we come to numbers: I define
 * x=Ω0'x Def. and
 * Ω'Ωνx = Ων+1x Def

According, then, to these symbolic rules we write the series x, Ωx, Ω'Ωx, Ω'Ω'Ω'x

as: Ω0x, Ω0+1x, Ω0+1+1x, Ω0+1+1+1x,

Therefore I write in place of "[x,ξ,Ω'ξ]",
 * "[Ω0x, Ωνx, Ων+1'x]".

And I define:
 * 0 + 1 = 1 Def.
 * 0 + 1 + 1 = 2 Def.
 * 0 + 1 + 1 + 1 = 3 Def.
 * and so on.

A number is the exponent of an operation.

The concept number is nothing else than that which is common to all numbers, the general form of number.

The concept number is the variable number.

And the concept of equality of numbers is the general form of all special equalities of numbers.

The general form of the cardinal number is: [0, ξ, ξ+1].

The theory of classes is altogether superfluous in mathematics.

This is connected with the fact that the generality which we need in mathematics is not the accidental one.

The propositions of logic are tautologies.

The propositions of logic therefore say nothing. (They are the analytical propositions.)

Theories which make a proposition of logic appear substantial are always false. One could e.g. believe that the words "true" and "false" signify two properties among other properties, and then it would appear as a remarkable fact Rh