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Re: A finite set of all naturals
Posted:
Aug 23, 2013 11:16 AM


On 23/08/2013 8:59 AM, Ben Bacarisse wrote: > Nam Nguyen <namducnguyen@shaw.ca> writes: > >> On 23/08/2013 7:49 AM, Ben Bacarisse wrote: >>> quasi <quasi@null.set> writes: >>> >>>> 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). >>> >>> That does not match my reading of the new definition. It states that a >>> formula is positive if it can be written in a particular form. That >>> odd(x) can be written in at least one form that does not match the >>> requirements for positivity does not mean that it can't be. >>> >>> My first counter example, odd(x) <> Ey[Sx=2*y] seems to me to be to as >>> positive as Nam's version of even, >> >> Right. But remember that odd(x) is a _non logical_ expression, hence >> it does matter (on it being positive or negative) whether or not, say, >> 'S' is part of a language. >> >> In the language L1(S,*), both even(x) <> Ey(x=2*y) and odd(x) <> >> Ey[Sx=2*y] are positive, while in L2(*), only even(x) would be. > > How is 2 defined in L2(*)? What are the axioms for *? Don't both use S?
_Here_ I was merely clarifying the relativity (the language dependence) of "positivity"/"negativity" of nonlogical formulas in general, and not attempting to define something specific such as 2.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI



