Nam Nguyen wrote: > On 23/08/2013 8:55 AM, Shmuel (Seymour J.) Metz wrote: >> In <email@example.com>, on 08/23/2013 >> at 04:40 AM, quasi <firstname.lastname@example.org> said: >> >>> With Nam's new definition of positive/negative, I think >>> it's immediately provable (subject to some clarification as to what >>> a formula is) that odd(x) is a negative formula. >> >>> Let even(x) <-> Ey(x=2*y). >> >> Let odd(x) <-> Ey(S(x=2*y)). >> >>> Assuming Nam's definition of "formula" supports the claim that >>> even(x) is a positive formula, then odd(x) must be a negative >>> formula since odd(x) is equivalent to ~even(x). >> >> However, even(x) is equivalent to ~odd(x), so even(x) must be a >> negative formula. >> >> You can exclude S, but then you're no longer talking about the natural >> numbers. > > You seem to be confused on the issue, which is that _in the natural_ > _numbers_ while even(x) can be positively expressed _with only_ '*' > can the same be said of odd(x)? > > The correct answer is "NO". > > What is _your answer_ to this question?
What do you take the natural numbers to be? Are you defining them or are you using the usual informal definition?
-- Sorrow in all lands, and grievous omens. Great anger in the dragon of the hills, And silent now the earth's green oracles That will not speak again of innocence. David Sutton -- Geomancies