From: "jeffrey E." <jeevacation@grnail.com>

From: "jeffrey E." <jeevacation@grnail.com> To: Misha Gromov Subject: Date: Mon, 04 Dec 2017 12:59:01 +0000 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(13) is already defined for all p < 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(13) 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(p) I p C 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. please note The information contained in this communication is confidential, may be attorney-client privileged, may constitute inside information, and is intended only for the use of the addressee. It is the property of JEE Unauthorized use, disclosure or copying of this communication or any part thereof is strictly prohibited and may be unlawful. If you have received this communication in error, please notify us immediately by return e-mail or by e-mail to jeevacation@gmail.com, and destroy this communication and all copies thereof, including all attachments. copyright -all rights reserved EFTA00938893