What I gathered, form a cursory reading of the paper is that the old
man described the algebraic properties of the rational numbers, by
actually describing what a ring is, as the corresponding wikipedia
article is explicitly referenced in the text (though the word ring is
not given). Hence, he must have specified that the addition has a
special element called zero wich is an additive identity.
Hence any model that fits the description given by the old man,
i.e. that is actually a ring, must have an element called zero with
the required properties. This is certainly true of the rationals, and
it must be true of what you (as the character of the story) call the
integers since you have no disagreement with the old man.
And, as stated in the article, it will actually be true of any model
meeting the description of the old man. So, for any model you care to
call "numbers", the statement "(5) No number is equal to zero" is clearly
However, I am bothered by statements (1), (3), and (4) that refer to
the number 2, as the axioms for a ring do not define what 2 might be.
It is not supposed to have been defined in the discussion, and it is
therefore difficult to assign any meaning to a statement using it.
They should have at least agreed that 2 is a notation for the result