On 23/08/2013 1:42 PM, Peter Percival wrote: > Nam Nguyen wrote: >> On 23/08/2013 8:55 AM, Shmuel (Seymour J.) Metz wrote: >>> In <firstname.lastname@example.org>, on 08/23/2013 >>> at 04:40 AM, quasi <email@example.com> 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?
What is your answer to this question of mine above?
Anything else is _irrelevant_ .
If you or Ben couldn't give me _a straightforward Yes or No answer_ on this question, let's forget about anything else.
-- ----------------------------------------------------- There is no remainder in the mathematics of infinity.