[FOM] Tim Gowers work (was: One Real Number)

[Jeremy Clark]

On Saturday, June 28, 2003, at 04:47  PM, Dean Buckner wrote:
>
> Gowers has a Wittgensteinian slant, and has a healthy suspicion of set
> theory.  For example, "If R is an equivalence relation on a set A and x
> belongs to A, then the equivalence class of x is the set E(x)={y in A: 
> xRy},
> but as the proof proceeded, every time I wrote down a statement such 
> as "z
> is an element of E(x)" I immediately translated it into the equivalent 
> and
> much simpler non-set-theoretic statement xRz."
>
> His most interesting stuff is on the idea of definable sets.  He has 
> the
> Wittgensteinian insight that perhaps we can do all analysis using just
> "definable" reals.  And since we obviously can't say what non-definable
> reals are, why bother with them at all?

If we can't say what non-definable reals are, then I would argue that 
it makes
no sense to restrict our attention to definable reals, because we can't 
say
what they are either. We should just talk about reals. *How* we talk 
about them
is, of course, still very much open to question. This is the line of 
constructivive
mathematics: to talk about reals constructively rather than about 
constructive
reals. (Paraphrasing, I think, Fred Richman.)

By the way, the idea of doing away with equivalence classes and just 
referring
to the relationship is also used extensively in constructive 
mathematics.

Jeremy Clark