```Date: Aug 23, 2013 10:59 AM
Author: Ben Bacarisse
Subject: Re: A finite set of all naturals

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?<snip>-- Ben.
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