<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Dear Eric,<div class="">Thank you for the paper by <span style="font-stretch: normal; font-size: 12.5px; line-height: normal;" class="">N</span><span style="font-size: 9px;" class="">EWTON </span><span style="font-stretch: normal; font-size: 12.5px; line-height: normal;" class="">C. A. </span><span style="font-size: 9px;" class="">DA </span><span style="font-stretch: normal; font-size: 12.5px; line-height: normal;" class="">C</span><span style="font-size: 9px;" class="">OSTA. </span></div><div class=""><span style="font-size: 9px;" class="">Some comments here.</span></div><div class=""><span style="font-size: 9px;" class=""><br class=""></span></div><div class=""><span style="font-size: 9px;" class="">1. </span>The question of the completeness of a physical or biological theory is different from the same question for a mathematical theory since one is always ready to take on more empirical evidence in a scientific theory. Also one is probably willing to change the structure of the language as well. With these changes allowed, I do not see how it is possible to apply the Goedelian or Turing arguments. These arguments depend on fixing the language and being able to enumerate all grammatical statements in the language. So I think we are left to understand that a scientific theory is not going to be known to be complete, and more likely it is known to be incomplete in various ways.</div><div class=""><br class=""></div><div class="">2. By the same token mathematics as an ongoing endeavor is not proved to be incomplete. It is only sharply delineated regions of mathematics called formal systems that are either inconsistent or incomplete (if rich enough to handle arithmetic). Mathematics as a whole, goes on just like science, inventing new languages and sometimes inventing or finding new axioms, often bumping into new phenomena. </div><div class=""><br class=""></div><div class="">3. It is worthwhile looking closely at the examplar of incompleteness in the Cantor diagonal argument. If X is a set and P(X) is the set of subsets of X. Suppose that </div><div class="">F:X —> P(X) is a given function. Then for any x in X, either x is in F(x) or x is not in F(x). Cantor forms the set C= { x in X | x is NOT in F(x)}. But then</div><div class="">C can not be equal to F(z) for any z since if z is in F(z) then z is not in C and if z is not in F(z) then z is in C. So C differs from every F(z). This means that there is no way to match all the elements of X with subsets of X. Note that to show that F is incomplete, we first had to be given F. Then we used F via the set C to show that F is incomplete in that not every subset has the form F(z). The same is the case in all the incompleteness results. You have have pin the formal system to the table and look closely at it in order to show that it has true statements that it cannot prove.</div><div class=""><br class=""></div><div class="">4. All incompleteness arguments are variants of 3. This logic is not a tragedy, not very complicated, worth looking at with some examples until it is clear as 2+2=4.</div><div class="">Then one can discuss what it means for a theory to be complete. </div><div class="">Best,</div><div class="">Lou</div><div class=""><br class=""></div><div class=""><div><br class=""><blockquote type="cite" class=""><div class="">On Jun 2, 2025, at 10:28 AM, Stuart Kauffman <<a href="mailto:stukauffman@gmail.com" class="">stukauffman@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div class="">Thank you Erik. Stu<br class=""><br class=""><blockquote type="cite" class="">On Jun 2, 2025, at 12:47 AM, OARF <<a href="mailto:eric.werner@oarf.org" class="">eric.werner@oarf.org</a>> wrote:<br class=""><br class=""></blockquote><span id="cid:B063D350-814E-4A80-90F0-9B2C1235C688"><jonas0,+p153-6.pdf></span><br class="">_______________________________________________<br class="">Fis mailing list<br class=""><a href="mailto:Fis@listas.unizar.es" class="">Fis@listas.unizar.es</a><br class=""><a href="http://listas.unizar.es/cgi-bin/mailman/listinfo/fis">http://listas.unizar.es/cgi-bin/mailman/listinfo/fis</a><br class="">----------<br class="">INFORMACI�N SOBRE PROTECCI�N DE DATOS DE CAR�CTER PERSONAL<br class=""><br class="">Ud. recibe este correo por pertenecer a una lista de correo gestionada por la Universidad de Zaragoza.<br class="">Puede encontrar toda la informaci�n sobre como tratamos sus datos en el siguiente enlace: <a href="https://sicuz.unizar.es/informacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas">https://sicuz.unizar.es/informacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas</a><br class="">Recuerde que si est� suscrito a una lista voluntaria Ud. puede darse de baja desde la propia aplicaci�n en el momento en que lo desee.<br class=""><a href="http://listas.unizar.es">http://listas.unizar.es</a><br class="">----------<br class=""></div></div></blockquote></div><br class=""></div></body></html>