Page 154 - Textos de Matemática Vol. 40

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Appendix
A.4
dabei sind 0, 1 2 bestimmte, von einander verschiedene Dinge. Nun kann ja x alle Dinge durchlaufen; setzen wir also z.B. f = x, so haben wir:
ff=0 , wennff̸=0 ff=1 , wennff=0;
das sind 2 Widerspru¨che und unsere Definition von f ist also formal widerspruchsvoll und daher unzula¨ssig.
Now, it will be useful to clarify what those contradictions are formally based on. Let us think of a thing x which ranges over all things. Now, we define a certain function, which is again a thing f, and the “value of this function for the argument x” is the combina- tion of the two things f x. Depending on whether the combination x x of x with itself is 0 or not, let f x be 1 or 0:
f x = 0, if x x ̸= 0 f x = 1, if x x = 0;
where 0, 1 are two specific, distinct things. Now, x can range over all things; so, if for example, we set f = x, we have:
f f = 0, if f f ̸= 0 f f = 1, if f f = 0;
which are two contradictions and our definition of f is therefore formally contradictory and hence inadmissible.
A first analysis
Unfortunately, Hilbert does not give any further indication why those contra- dictions, i.e., Richard’s, his, and Russell/Zermelo’s paradox, are formally based on such an inadmissible function definition.
Obviously, there is a certain analogy to Russell/Zermelo’s paradox: Read- ing f as characteristic function of a set R, i.e., f x = 0 corresponds to x ∈ R and f x ̸= 0 (or = 1) to x ̸∈ R, f characterizes the “Russell set” R. In gen- eral, self-application (for functions) corresponds to “self-membership” for sets. Thus, self-application leads—together with assumption that all other operations involved in the definition of f are admissible, namely case-distinction and the (in)equality tests—to a contradictory function definition in the same way as (negated) “self-membership” leads to a contradiction.35
35It is also worth mentioning that the “self-membership” of sets can be integrated into