[FOM] Real Numbers

[Harvey Friedman]

Reply to Heck 10:10AM  5/3/03.

>Debates over Benacerraf's problem always seem to lead to someone's 
>something like the following, which Wiman says here:
>
>The real numbers are not equivalence classes of Cauchy sequences. 
>They can be, but they can also be Dedekind cuts, binary numbers, 
>real closed fields, one dimensional continua, or a number of other 
>things.
>
>I always feel puzzled when I get to that point, however. What is the 
>modality here? In what sense /can/ the reals be any of these things, 
>though they are not? I understand what it means to say, for example, 
>that, though I am not a mathematician, I might have been, but that 
>is surely not the sort of thing one has in mind here. If I am not a 
>mathematician, then perhaps I /could/ be one, but what would it mean 
>to say that I /can/ be one? Surely if the reals /are not/ Dedekind 
>cuts, then they /cannot/ be Dedekind cuts, any more than I can be 
>George Bush if I am not George Bush.
>
>We can interpret analysis in set theory. That is true. But why 
>should it mean that the reals "can be" sets?
>
>Now granted, there is a point at which one might be trying to get 
>using such language, and it is a point with real mathematical 
>content. But I do not see that it has been clearly expressed, either 
>here or elsewhere.
>

This is an old, but interesting issue. Of course, it is one that 
mathematicians have learned to be totally uninterested in. Also, it 
has fueled attempts to give categorical foundations and other 
structuralist attempts. All of these get quite complicated, and when 
they don't, they look too much like set theory to really serve as an 
alternative foundation for mathematics.

My favorite way of trying to deal with this unpleasant philosophical 
situation, is as follows.

We have variables of only one sort, but with the following 7 
nonlogical symbols (in addition to the logical symbols not, and, or, 
implies, iff, forall, therexists, =).

Sets. (Unary predicate symbol).
Membership. (Binary relation symbol).
Ordered pairing (Binary function symbol).
Real numbers. (Unary predicate symbol).
0,1. (Constant symbols).
<. (Binary relation symbol for ordering of reals).
+. (Ternary relation symbol for addition on reals).

1. Everything is exactly one of: a set, an ordered pair, or a real number,
2. Only sets can have an element.
3. If two sets have the same elements then they are equal.
4. <x,y> = <z,w> iff (x = z and y = x).
5. 0,1 are distinct real numbers.
6. +(x,y,z) implies x,y,z are reals.
7. x < y implies x,y are reals.
8. Usual axioms that reals are an ordered group with 0,1,+,<.
9. Every nonempty set of reals bounded above has a least upper bound.
10. The set of all reals numbers exists.
11. Pairing, union, power set, separation, replacement, foundation, choice.

Rationals, integers, natural numbers, are all defined as certain real 
numbers. Functions are sets of ordered pairs.

One proves that every sentence is provably equivalent to a sentence 
that mentions only epsilon.

This system puts us pretty much where almost all mathematicians are 
when they wish to develop mathematics rigorously at the appropriate 
level of the  undergraduate curriculum (or early graduate).

There are some other interesting ways to deal with this, but let's 
see where the discussion leads us.