[FOM] arithmetic without product

[José Manuel Rodriguez Caballero]

Consider the following theorem:

For any positive integer k, there is a positive integer n having at least k
> representations as sum of consecutive positive integers.


For example, for k = 3 we have n = 9 = 4+5 = 2+3+4.

For the statement of this theorem we need the addition. If we also
have the multiplication,
then our theorem can be proved using a well-known argument due to J. J.
Sylvester. On the other hand, if we have not multiplication and we restrict
ourselves to first order logic, the existence of a proof of this result is
not so clear. Is there an "arithmetic with just addition" where this
theorem can be stated but it cannot be proved?

Kind Regards,
Jose M.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20180721/e55af5ab/attachment.html>