
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: >>>> >>>> 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: >>> >>> Def03a: even1(x) <> Ey[x=y+y] >>> Def03b: even2(x) <> Ey[x=2*y] >>> Def03c: 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 eliminatedable 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 expressionduality of even numbers, in that they can be expressed _either_ inductively (Def03a) or noninductively (Def03b).
This inductive and noninductive expression duality was used in other definitions (Def07a and Def07a, for "There are infinitely many examples of property P"), and plays a key role in my proof about certain structure theoretical truthimpossibility. 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.
>>> 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: >>> >>> Def05c: 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 TeXformatted version (as > you suggested) for easy reading. The journal will definitely want one. >
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI

