
Re: A finite set of all naturals
Posted:
Aug 23, 2013 10:12 AM


On 23/08/2013 7:51 AM, Peter Percival wrote: > quasi wrote: >> Peter Percival wrote: >>> Nam Nguyen wrote: >>>> >>>> I certainly meant "odd(x) can _NOT_ be defined as a >>>> positive formula ...". >>> >>> Prove it. >> >> 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). >> >> 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). >> >> So, conceding that, where does he go with it? >> >> quasi > > But who's to say that there isn't a positive formula phi with one free > variable x such that phi(x) <> odd(x)?
> phi may be taken as a definition of odd.
Sure. If in the language of arithmetic  or L(*), you defined:
odd(x) <> Ey(x=2*y)
then _this_ odd(x) would be positive as expressed in _both_ languages.
That still doesn't "invalidate" the _definitions_ .
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI

