[FOM] Who was the first to accept undefinable individuals in mathematics?
Vaughan Pratt
pratt at cs.stanford.edu
Wed Mar 18 04:22:21 EDT 2009
Kreinovich, Vladik wrote:
> This description makes the enumeration of algebraic numbers perfectly
> definable.
When I proposed a rational multiple of the Liouville constant as a
transcendental number in [a,b], it was with Bill's mention of Sturm's
theorem in mind. Like Vladik I wondered why Sturm's theorem was needed,
and concluded that Bill wasn't content to *define* a transcendental in
[a,b] but must have wanted to *construct* one. For those puzzled by my
width-2 intervals, my goal was to be impeccably constructive. A
function f(a,b) producing a transcendental in [a,b] cannot be
*constructed* without excluding a set of measure zero (interpret that as
you will). However it can be *defined* everywhere (on the upper left
triangle of R^2 defined by a < b), as Vladik indicated (though again
Liouville's constant gives a simpler definition).
(I'm not actually a constructivist, I'm just trying to play one on FOM.)
Vaughan Pratt
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