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Topic: A finite set of all naturals
Replies: 32   Last Post: Aug 15, 2013 4:07 PM

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 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: A finite set of all naturals
Posted: Aug 12, 2013 9:22 PM

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:
>>>>

>>>
>>> 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
was _NOT_ an accidence the these kind of definitions were done
that complex way.

>>> Either way, I didn't find it an encouraging start.
>>>
>>> Maybe I don't understand the notation, because just a few lines further
>>> on I see:
>>>
>>> Def-05c: aGC(x) <->
>>> (even(x) /\ (SS0<x)) -> Ap1p2[prime(p1) /\ prime(p2) /\
>>> (p1+p2<x \/ x<p1+p2)]
>>>
>>> but isn't Ap1p2[prime(p1) /\ prime(p2) /\ (p1+p2<x \/ x<p1+p2)] simply
>>> false?
>>>
>>> <snip>

>>
>> Yes, since not every natural number is a prime, and a conjunct claims
>> otherwise, the whole is false. Perhaps
>>
>> Ap1p2[[prime(p1) /\ prime(p2)] -> (p1+p2<x \/ x<p1+p2)]
>>
>> was meant.

>
> That was my guess too, but the reader shouldn't have to guess. For one
> thing, other errors, they may not be so obvious.

Yes it was an overlook on my part then (the same with my definition
of primes), but that would take only a simple fix (which I indicated
before somewhere it will be done in the next formal presentation.

No consequence to my overall proof though.
>
> Since the post is more than a year old, these sort of issues may have
> been corrected,

Agree.

> but it's a shame there is no TeX-formatted version (as
> you suggested) for easy reading. The journal will definitely want one.
>

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

NYOGEN SENZAKI

Date Subject Author
8/12/13 Ben Bacarisse
8/12/13 Peter Percival
8/12/13 fom
8/12/13 Ben Bacarisse
8/12/13 namducnguyen
8/12/13 namducnguyen
8/12/13 antani
8/12/13 namducnguyen
8/13/13 Marshall
8/13/13 quasi
8/13/13 namducnguyen
8/13/13 quasi
8/13/13 namducnguyen
8/13/13 namducnguyen
8/13/13 quasi
8/13/13 namducnguyen
8/14/13 quasi
8/14/13 namducnguyen
8/14/13 quasi
8/14/13 namducnguyen
8/15/13 namducnguyen
8/15/13 Virgil
8/15/13 namducnguyen
8/15/13 antani
8/13/13 antani
8/13/13 Peter Percival
8/13/13 Ben Bacarisse
8/13/13 namducnguyen
8/14/13 Peter Percival
8/14/13 Shmuel (Seymour J.) Metz