
Re: A finite set of all naturals
Posted:
Aug 13, 2013 8:33 PM


On 13/08/2013 10:19 AM, Ben Bacarisse wrote: > Nam Nguyen <namducnguyen@shaw.ca> writes: > <snip> >>>>> Ben Bacarisse wrote: (quoting Nam) > <snip> >>>>>> Def03a: even1(x) <> Ey[x=y+y] >>>>>> Def03b: even2(x) <> Ey[x=2*y] >>>>>> Def03c: even(x) <> (even1(x) \/ even2(x)) > <snip> >> ... _an odd number can not be defined without addition_ while >> an even number can (as per Def03b above). > > odd(x) <> ~Ey[x=2*y]
Yes. There's something though from my original post (link) that has missed being mentioned here but is _quite relevant_ . In that post I had:
'Def01: A formula is "positively assertive", or just "positive", iff the formula contains no negation sign '~' ...'
So my answer to your question below is: > > In what sense does your definition of even2 avoid addition, where this > one of odd does not?
in the sense that the formula defining even2 is a _positive_ formula while the formula for your odd(x) above is _not_ . > > Personally, I'd say that both this and Def03b use addition since > multiplication in PA is usually defined using addition, but my > view of what "without addition" means is not the issue here.
So then with the caveat above about the requirement of positive formula here, can you define odd(x), without addition *and* using _only_ positive formula?
If you can't, then my comments about "overly complex" and Induction nonInduction duality definitions of even(x) but _not_ of odd(x), and the implication thereof in my proof still stands.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI

