There is therefore really a sense in which the philosophy we can talk of a non-psychological I.
The I occurs in philosophy through the fact that the “world is my world.”
The philosophical I is not the man, not the human body or the human soul of which psychology treats, but the metaphysical subject, the limit—not a part of the world.
The general form of truth-function is: [p‾,ξ‾,N(ξ‾)].
This is the general form of proposition.
This says nothing else than that every proposition is the result of successive applications of the operation N′(ξ‾) to the elementary propositions.
If we are given the general form of the way in which a proposition is constructed, then thereby we are also given the general form of the way in which by an operation out of one proposition another can be created.
The general form of the operation Ω′(η‾) is therefore: [ξ‾,N(ξ‾)]′(η‾)(=[η‾,ξ‾,N(ξ‾)]).
This is the most general form of transition from one proposition to another.
And thus we come to numbers: I define
x=Ω0′x Def. and
Ω ′ Ω ν ′ x = Ω ν + 1 ′ x Def.
According, then, to these symbolic rules we write the series x,Ω′x,Ω′Ω′x,Ω′Ω′Ω′x,… as: Ω0′x,Ω0+1′x,Ω0+1+1′x,Ω0+1+1+1′x,…
Therefore I write in place of “[x,ξ,Ω′ξ]”,
“[Ω0′x,Ων′x,Ων+0′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 a 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].