|
|
Re: A finite set of all naturals
Posted:
Aug 12, 2013 11:26 PM
|
|
On 12/08/2013 7:22 PM, Nam Nguyen wrote:
> On 12/08/2013 8:23 AM, Ben Bacarisse wrote:
>> Peter Percival <peterxpercival@hotmail.com> writes:
>>
>>> Ben Bacarisse wrote:
>>>> fom <fomJUNK@nyms.net> writes:
>>>> <snip>
>>>>> Here is the proof he provides when pressed to produce:
>>>>>
>>>>> https://groups.google.com/d/msg/comp.ai.philosophy/AVnJgA5ZIk0/fk1sQQNSYVgJ
>>>>>
>>>>
>>>> I could not resist taking just a little peek... That post claims to be
>>>> "written in the language of arithmetic L(PA)" but it contains this
>>>> definition early on:
>>>>
>>>> Def-03a: even1(x) <-> Ey[x=y+y]
>>>> Def-03b: even2(x) <-> Ey[x=2*y]
>>>> Def-03c: even(x) <-> (even1(x) \/ even2(x))
>>>>
>>>> Does this mean that Ax[even1(x)=even2(x)] is not a theorem of the
>>>> system
>>>> being used,
>
> It actually doesn't mean that; "even1", "even1", "even" are what
> Shoenfield termed as "defined symbols": they are eliminated-able
> by the actual language symbols [of L(PA)]. So there's _nothing wrong_
> with those 3 definitions.
>
>
>>>> or does it mean that the author is inclined to use overly
>>>> complex definitions?
>
> Not "inclined" at all. In fact it was _deliberately_ on my part
> to define the even function (or the like) in this overly complex
> manner, _for a good reason_ : it highlighted the expression-duality
> of even numbers, in that they can be expressed _either_ inductively
> (Def-03a) or non-inductively (Def-03b).
>
> This inductive and non-inductive expression duality was used in other
> definitions (Def-07a and Def-07a, for "There are infinitely many
> examples of property P"), and plays a key role in my proof about
> certain structure theoretical truth-impossibility. The other posters
> (fom, e.g.) didn't recognize that but, to answer your question, it
> was _NOT_ an accidence the these kind of definitions were done
> that complex way.
More than once, I was asked what the difference between the Goldbach
Conjecture and its weaker form that an odd number greater than 7 is
the sum of three odd primes.
The point is though the essences of the 2 conjectures are drastically
different: _an odd number can not be defined without addition_ while
an even number can (as per Def-03b above).
And of course arithmetically, induction, from a binary operation point
of view, would not be possible without addition, designated as '+'.
It's the other non-inductive, if not anti-inductive, operation known
as multiplication that primes could be possibly defined arithmetically.
Of course.
--
-----------------------------------------------------
There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI
|
|
