EFTA02556970

From: jeffrey E. <jeevacation@gmail.com> Sent: Monday, December 4, 2017 1:55 PM To: Misha Gromov Subject: Re: similar to what does it mean to have an ordinal. =AO an ordinal in your language is the"understood object " = you take the sets of all things that are close to it , =ultidimensional sets and undergo recursion in order to c=me close the " understood" the narrowest of definition. =A0 la w On =on, Dec 4, 2017 at 8:24 AM, Misha Gromov rot=": Can't say i got it, why "understanding" On Mon, 4 Dec 2017 07:59:01 -0500, jeffrey E. wrote: why is transfinite recursion chttps://en.wikipedia.org/wiki/Transfinit=_recursion> a good=model for understanding — the proof that the result is well-define= uses transfinite induction. Let F denote a (class) function = to be defined on the ordinals. The idea now is that, in defining (a) for an unspecified ordinal a, one may assume that =03) is already defined for all R < a and thus give a formula for =(a) in terms of these F(f3). It then follows by tra=sfinite induction that there is one and only one function satisfying the r=cursion 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 IF03) I (3 < a}= that is, the set consisting of all F((3) for (3 < a. This definiti=n assumes the F03) known in the very process of defining ; this apparent vicious circle is exactly what definition by transfi=ite recursion permits. In fact, F(0) makes sense since there is n= ordinal (3 < 04=pan>, and the set (F=/em>((3) (3 < 0) is empty. So F(0) is equal t= 0 (the smallest ordinal of all). Now that F(0) is known, the def=nition applied to F(1) makes sense (it is the smallest ordinal no= in the singleton set (<=m>F(0)) = (0)), and so on . it sort of says an approximation to truth. by reduction. alt=rnately we can add other dimensions. please note EFTA_R1_01715812 EFTA02556970 The information contained in this communication is confidential, ma= be attorney-client privileged, may constitute inside information, and =s intended only for the use of the addressee. It is the property of =EE Unauthorized use, disclosure or copying of this communication or =ny part thereof is strictly prohibited and may be unlawful. If you have=received this communication in error, please notify us immediately byreturn e-mail or by e-mail to jeevacation@gmail.com <mailto:jeevacation@gmail.com> , and destroy this communicatio= and all copies thereof, including all attachments. copyright -all righ=s reserved =AO please note The information contained i= this communication is confidential, may be attorney-client privileged,=may constitute inside information, and is intended only for the use =f the addressee. It is the property of J EE Unauthorized use, disclos=re or copying of this communication or any part thereof is strictly pro=ibited and may be unlawful. If you have received this communication =n error, please notify us immediately by return e-mail or by e-mail to =a href="mailtoieevacation@gmail.com" target="_blank">jeevacation@gmai=.com, and destroy this communication and all copies thereof, inc=uding all attachments. copyright -all rights reserved --f403043ad24c5a1583055f840f5d-- conversation-id 22029 date-last-viewed 0 date-received 1512395681 flags 8590195713 gmail-label-ids 7 remote-id 775068 2 EFTA_R1_01715813 EFTA02556971