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

Nam Nguyen <> 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?