[FOM] Real Numbers

[John Pais]

Hartley Slater wrote:

You claim:

> .................. using von Neumann ordinals, or any other
> series of sets to represent the natural numbers is a category
> mistake. But while Benacerraf pointed out that, because of the
> multiplicity of such possible series, no one series could be the
> natural numbers, it is not, by contrast, that the natural numbers
> *can* be this series, or that series.

You claim:

> They cannot be *any* series of
> sets:

You claim:

> the theory of number must preceed the theory of sets, because
> one needs the concept of number to decide which specific predicates
> determine sets - the count predicates.

You claim:

>  So the natural numbers are
> not sets period.

In contrast to this 'numerical philosophy', with its strange sounding and
spurious claims and characterizations such as the above, all most all
mathematicians and logicians that do foundation of mathematics, start
with set theory ab initio. For example, in the Introduction to Ken
Kunen's Set Theory on p. xi, it asserted that:

"Set theory is the foundation of mathematics. All mathematical concepts
are defined in terms of primitive notions of set and membership... From
such axioms all known mathematics may be derived... "

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.

So, though mathematicians do hear the claims and protestations of
numerical philosophers, such as those starting at the beginning of this
email, they just don't take these claims seriously since they're
essentially devoid of mathematical content, and hence orthogonal to the
practice of mathematics.

> .................... For there is no plural in the second
> expression, and that is very significant, since it means that the 'X'
> there is not a count, but a mass term.  I spelt out some while ago
> how that makes Set Theory inapplicable to the case, and what must
> replace it, in a series of postings 'natural language and the F of
> M'.

> ....but also, on mathematics which puts Set Theory aside, Harry
> Bunt's 'Mass Terms and Model Theoretic Semantics', C.U.P. Cambrtidge
> 1985 passim.

I beg your pardon, but, though Harvey asked you nicely twice to do so,
you declined to give us enough of a sketch of Bunt's work so that we
could fairly evaluate whether or not it could both address the concerns
of numerical philosophy and serve as a replacement foundation for
mathematics such as that laid out in Kunen's book.

I did some Google searching and found Bunt's website and a few
references, but I couldn't find any papers. I don't plan on buying the
book (I need more convincing), and I think at this point in the
discussion everyone would like to see details supporting for example your
claim regarding: "mathematics which puts Set Theory aside".  It just
seems that if Bunt's work (1985) offered a serious replacement foundation
for mathematics, then it would be more well-known by now. So, though I
remain skeptical, "my brain is open".

Best wishes,
John Pais