# [FOM] Re: Tim Gowers work

W.Taylor@math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Wed Jul 2 00:49:40 EDT 2003

```Dean Buckner wrote:

> [Gowers] 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?

This is very, VERY close to the view I have come to adopt on set theory.
I'm sorry to see it called Wittgensteinian though, as what I know of him
I dislike intensely!

> For example, the intermediate value theorem would be true for
> definable continuous functions, and I'm not too worried about any others.

Exactly so!  (Though I would go further and say "there ARE no others!")

> end up speaking exactly the same
> language as you, but mean something slightly different by the phrase
> "for all x", which for me would mean "for all definable x",

Exactly my thoughts.  While noting that this must always be only what
we *mean*, we can never say it directly, because whenever you specify
an (attmptedly complete) bunch of definable objects, there is inevitably
a diagonalization out of it.  There may conceivably be some sort of
limit at omega_1^CK but this is very unclear.  As we finally agreed
in a similar thread in sci.logic a while ago, "definable" objects
are somewhat clear-ish to the intuition, but can never be totally
encompassed therein.    Briefly, "definability" is not definable!

But it remains, IMHO, an excellent intuitve and informal basis on which
to base a "realistic" interpretation of ZF.  It disgrees with AC of course!

>  In fact, I could even say that the reals were uncountable!
> What I'd mean by this in your terms is that there is no
> definable bijection between N and the definable reals, which there isn't
> because then I could apply a diagonal argument and define a real not in

Exactly so, again!

> Anyway, this is my slant on his slant, see for yourselves.  Dean Buckner

Dean, I love his slant, and your slant ON his slant!   Keep it up!

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Bill Taylor                       W.Taylor at math.canterbury.ac.nz
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I do not claim my argument is logical, but simply that I'm right.
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