[FOM] Frege on Addition

[Richard Heck]

There was some discussion of this topic on the list a while ago---I
recall Neil Tennant had something to say about it---but I can't now find
that discussion on the archives. In any event, I don't remember the
question being answered definitively, and so I'm raising it again. I'm
trying to finish a book on /Grundgesetze/ and would like to be able to
say something definite on the topic.

Frege proposes to define addition of cardinal numbers in terms of
disjoint union. He proves that sums are unique but does not prove that
they exist. It is both necessary and sufficient for this to prove that
the domain can be partitioned, that is, that:
    (F)(G)(EU)(EV)[Nx:Fx = Nx:Ux & Nx:Gx = Nx:Vx & ~(Ex)(Ux & Vx)]
I take it that this will not be provable in Frege arithmetic:
second-order logic plus HP. It is clear that it would follow from global
well-ordering, but the partition theorem seems not to imply anything
nearly that strong. So the question is: What can be said about what
partition requires? Does it entail some form of choice, for example?

Richard Heck



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Richard G Heck, Jr
Professor of Philosophy
Brown University
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