[FOM] Mathematical explanation
Richard Heck
rgheck at brown.edu
Sat Oct 29 12:53:43 EDT 2005
Neil Tennant wrote:
>FOMers might be interested to learn that Frege himself, in the (two volumes of the) Grundgesetze, never actually gave a satisfactory definition of addition on the natural numbers that would have met his own stringent criteria for adequacy of definitions. His comments in Vol.II, section 33, which Avron has echoed in his (2) above, do not amount to such a definition.
>
>
(2) was: If S and T are two disjoint sets, the cardinality of S is 3,
and the cardinality of T is 4, then the cardinality of their union is 7.
The theorem Frege proves (469, but translated into a standard
formulation of Frege arithmetic) is:
~(Ex)(Fx & Gx) & ~(Ex)(Hx & Jx) & Nx:Fx = Nx:Hx & Nx:Gx = Nx:Jx -->
Nx:(Fx v Hx) = Nx:(Gx v Ix)
Frege translates the proposition as: "The sum of two propositions is
determined by them".
Anyone caring to see Frege's proof and associated discussion can access
a (poor) English translation at
http://bobjweil.com/heck/philosophy/pdf/grundgesetze/gg_ii_b_xi.pdf.
(For the rest of Grundgesetze, see
http://bobjweil.com/heck/philosophy/grundgesetze.php.)
The difficulty to which Neil alludes results from the fact that Frege
never proves the existence of sums: What (469) proves is only their
uniqueness. The difficulty, as Frege pretty much notes, arises from the
fact that (469) concerns infinite cardinals as well as finite ones. To
prove existence, we would need to prove:
(F)(G)(EH)(EI)[F~H & G~I & ~(Ex)(Hx & Ix)],
where "~" means 1-1 correspondence. This both implies and would follow
from:
(EF)(EG)(F~[x:x=x] & G~[x:x=x] & ~(Ex)(Fx & Gx)).
I take it that this will not be provable without some form of choice. I
also take it that someone other than me will be able to say exactly how
much choice is needed. Anyone?
It's arguable on independent grounds that Frege was aware that there
were results in this area that he needed and could not prove using the
resources available to him. See my discussion of his discussion of
Dedekind's proof that every infinite set is Dedekind infinite in
http://bobjweil.com/heck/pdf/published/FiniteAndInfinite.pdf.
Richard
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Richard G Heck, Jr
Professor of Philosophy
Brown University
http://bobjweil.com/heck/
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