From: Misha Gromov Sent: Monday, December 4, 2017 1:24 PM To: Jeffrey E. Subject: Re: 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>(a) be the smallest ordinal not in the set {F((i) I P < a}, that is, the set =onsisting of all F((3) for &beta= < a. This definition assumes the F((3) 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;) I a < 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. please note The information contained in this communication is confidential, =ay be attorney-client privileged, may constitute inside information, =nd is intended only for the use of the addressee. It is the property =f JEE Unauthorized use, disclosure or copying of this commu=ication or any part thereof is strictly prohibited and may be unlawfu=. If you have received this communication in error, please notify us =mmediately by return e-mail or by e-mail to [email protected] <mailto:jeevaca=io