[FOM] Real Numbers

[John Pais]

John Pais wrote:

> On p. 19:
>
> " 7.15 Definition. omega is the set of natural numbers.
> .....
>
> It is a philosophical quibble whether the elements of omega are the
> *real* natural numbers (whatever that means). The important thing is that
> they satisfy the Peano Postulates, namely...
>
> Given the natural numbers with the Peano Postuales, one may temporarily
> forget about ordinals and proceed to develop elementary mathematics."
>
> On p. 35:
>
> "Section 11. The real numbers
>
> 11.1 Definition. Z is the ring of integers. Q is the field of rational
> number, R is the field of real numbers, C is the field of complex
> numbers.
>
> Any reasonable way of defining these from the natural numbers [omega]
> will do, but for definiteness we take Z = (omega x omega) / ~ , where <
> n, m > is intended to represent n-m, the equivalence relation is defined
> appropriately, Z is the set of equivalence classes, and operations + and
> * are defined appropriately. Q = (Z x (Z - {0})) / ~ =  where < x, y > is
> intended to represent x / y. R = {X in P(Q): X != 0 and X != Q and for
> all x in X for all y in Q ( y < x implies y in X)}.
>
> So R is the set of left side Dedekind cuts. C = R x R, with field
> operations defined in the usual way."
>
> As Kunen indicates, this program of foundations is aware of, but not
> concerned with or interested in, philosophical quibbles regarding
> identity. For mathematicians, mathematical identity *is* isomorphism.

Sean C Stidd wrote:

> > For mathematicians, mathematical identity *is* isomorphism.
>
> Taken at face value, this implies that the evens are the same as the odds
> are the same as the naturals,

Note the context of my original statement, which is the specification of
standard algebraic structures such as the natural numbers satisfying the
Peano Postulates (Kunen's Definition 7.15), and the ring of integers Z (this
letter is a standard abbreviation for the algebraic structure < Z, + , * >
satisfying the axioms for a ring as in Kunen's Definition 11.1), etc.
(Of course, applied too loosely it is an overstatement.)

An isomorphism of two algebraic structures of the same type must first be a
morphism, i.e. a mapping of the underlying sets that preserves the algebraic
operation(s), and second a bijection of the underlying sets. So, no two of
the sets you describe above are isomorphic as Peano Systems--the first two
aren't even closed under the (inherited) successor operation. Similar
mathematical distinctions can be made regarding the corresponding subsets of
the ring of integers.

> Do we really need isomorphism to be central for the semantics or
> metaphysics of mathematics on account of its undisputedly central
> importance for contemporary mathematical practice? If so, why?

It depends on whether or not what you call 'the semantics or metaphysics of
mathematics' is descriptive of and faithful to mathematical practice and its
purposes, or about something altogether different, e.g. folk arithmetic,
numerical philosophy, ...

Best wishes,
John Pais