<table><tr><th>From:</th><th>Misha Gromov</th></tr><tr><td>Sent:</td><td>Monday, December 4, 2017 1:24 PM</td></tr><tr><td>To:</td><td>Jeffrey E.</td></tr><tr><td>Subject:</td><td>Re:</td></tr></table>

Can't say i got it, why "understanding"

On Mon, 4 Dec 2017 07:59:01-0500, Jeffrey E. wrote:

why is transfinit= recursion <https://en.wikipedia.org/wiki/Transfinite_recursion> a good model for understanding — the proo= that the result is well-defined uses transfinite induction. Let FF to be defined on the ordinals. The i=ea now is that, in defining F(a) for an unspecified ordinal=a, one may assume that F((i) is already defined for all=0 < a and thus give a formula for F(a) in terms of these F((3). It then-follows by transfinite induction that there is one and only one function s=tisfying the recursion formula up to and including a.

(more will be given later): define function F by letting F<em>(α) be the smallest ordinal not in the set {F(β) | β < α}, that is, the set =onsisting of all F(β) for &beta= < α. This definition assumes the F(β) known=in the very process of defining F; this apparent vicious circle i= exactly what definition by transfinite recursion permits. In fact, F<em>(0) makes sense since there is no ordinal =beta; < 0, and the set {F(=beta;)} | β < 0} is empty. So F(0) is equal to 0 (t=e smallest ordinal of all). Now that F(0) is known, the definitio= applied to F(1) makes sense (it is the smallest ordinal not in t=e singleton set {F(0)} = {0}

it sort of says an approximation to truth. by reduction. alt=rnately we can add other dimensions.

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