[FOM] Two questions about Peano arithmetic

[David Isles]

I would appreciate  any help on the following two questions:

   1. Within a first-order system of Peano arithmetic, such as that given
   in Kleene’s book, which includes function constants only for ‘ (successor),
   + (addition), and x (multiplication), is there a proof different from the
   obvious one (his Theorem 157) which shows the existence of a common
   multiple of the integers in the sequence 1,2,...,n for any n?
   2. Is it possible in a system of Peano arithmetic as described in
   Question 1, to formalize the theory of the beta-function (Kleene page 44,
   Remark 1) without having proved the existence of a common multiple of the
   integers in the sequence 1,2,...,n for any n?

  Thanks, David Isles
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