Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
namducnguyen

Posts: 2,688
Registered: 12/13/04
Re: A finite set of all naturals
Posted: Aug 23, 2013 11:16 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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 non-logical formulas in general, and not
attempting to define something specific such as 2.

--
-----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.