On 12/08/2013 7:22 PM, Nam Nguyen wrote: > On 12/08/2013 8:23 AM, Ben Bacarisse wrote: >> Peter Percival <firstname.lastname@example.org> 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.