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Topic: A finite set of all naturals
Replies: 7   Last Post: Aug 24, 2013 2:44 AM

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Posts: 2,777
Registered: 12/13/04
Re: A finite set of all naturals
Posted: Aug 23, 2013 11:16 AM
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On 23/08/2013 8:59 AM, Ben Bacarisse wrote:
> 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?

_Here_ I was merely clarifying the relativity (the language dependence)
of "positivity"/"negativity" of non-logical formulas in general, and not
attempting to define something specific such as 2.

There is no remainder in the mathematics of infinity.


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