From: Misha Gromov
To: "jeffrey E." <jeevacation®gmail.com>
Subject: Re:
Date: Mon, 04 Dec 2017 13:24:12 +0000
Can't say i got it, why "understanding"
On Mon, 4 Dec 2017 07:59:01 -0500, jeffrey E. wrote:
why is transfinite recursion a good model for understanding — the proof that the result is well-defined uses
transfinite induction. Let F denote a (class) function F to be defined on the ordinals. The idea now is that, in
defining F(a) for an unspecified ordinal a, one may assume that F(p) is already defined for all f3< a and thus give
a formula for F(a) in terms of these F(p). It then follows by transfinite induction that there is one and only one
function satisfying the recursion formula up to and including a.
(more will be given later): define function F by letting F(a) be the smallest ordinal not in the set (F(p) I p < a),
that is, the set consisting of all F(p) for p < a. This definition assumes the F(p) known in the very process of
defining F; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact, F(0)
makes sense since there is no ordinal p < 0, and the set (F(3) I p < 0} is empty. So F(0) is equal to 0 (the smallest
ordinal of all). Now that F(0) is known, the definition applied to F(1) makes sense (it is the smallest ordinal not
in the singleton set {F(0)} = {0}), and so on .
it sort of says an approximation to truth. by reduction. alternately we can add other dimensions.
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