Re: GÖDEL
"Bruno Marchal [email protected] [Fabric-of-Reality]" <[email protected]> Fri, 14 Nov 2014 12:22:50 +0100
| Newsgroups | gmane.science.physics.fabric-of-reality |
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--Apple-Mail-11--429693282 Content-Type: text/plain; charset=WINDOWS-1252; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable On 14 Nov 2014, at 03:45, 'Lee Corbin' [email protected] [Fabric-=20 of-Reality] wrote: > > Those who are interested when G=F6del's theorems should and should=20=20 > *not* be applied ought to read Torkel Franzen=92s wonderful book,=20=20 > =93G=F6del's Theorem: An Incomplete Guide to its Use and Abuse=94. > Torket Franzen book is excellent. Only the title of the book is not=20=20 appropraite as he is concerned only with the misuse of G=F6del, and not = =20 on its use. It does not cite Judson Webb for example, nor my work,=20=20 despite some people pretend there are allusion, and there might be an=20=20 allusion to something I never said, but that some people attribute to=20=20 me (the idea that the computationalist indeterminacy bears on the=20=20 models of PA, which I precisely refute instead). > Study of his book has convinced me that about 95% of the references=20=20 > one comes across to G=F6del's Theorem are fallacious. I agree. > Here is just one fascinating fact: The theorem to formal systems=20=20 > that contain =93a certain amount of arithmetic=94 (it=92s hard to spell = =20 > that out in more detail). One example is the formal system=20=20 > containing this axioms for real numbers. G=F6del's Theorem does not=20=20 > apply, because the real number axioms, surprisingly, cannot =93get=20=20 > their hands=94 on > integers in the sense that you cannot specify =93for all integers=94 the = =20 > way that you can in Peano Arithmetic. > Yes. the first order theory of the real number has shown to be=20=20 decidable, and thus not even Turing complete. So it does not contain=20=20 the theory of natural numbers as a sub-theory. It does not have the=20=20 sufficient amount of arithmetic. You cannot define a natural number from the theory of reals. You can=20=20 intuit this by realizing that Fermat theorem is easy to solve when=20=20 "for all" bears on the reals, and quite difficult on the natural=20=20 numbers, as the proof by Wyles illustrates. Note that the real +=20=20 trigonometric functions or waves gave again the Turing universality.=20=20 You can define a natural number by the solution of sin(2pix) =3D 0, for = =20 example. Nevertheless, there is no problem in "a certain amount of arithmetic",=20=20 which is a standard expression for logician. It means that you can=20=20 define the addition and multiplication laws on the integers or the=20=20 natural numbers, which indeed you cannot do in the first order theory=20=20 of the reals. I made this technically precise with the notion of=20=20 L=F6bian machine in my papers. Bruno > Lee > > > From: mailto:[email protected] > Sent: Thursday, November 13, 2014 7:24 AM > To: [email protected] > Subject: Re: GODEL > > On 04 Nov 2014, at 22:48, [email protected] [Fabric-of-Reality]=20=20 > wrote: > >> >> >> I don't think that Hawkins has come close to proving that the=20=20 >> existence of God is improbable. His case can be summarized when he=20=20 >> says....""when one person suffers from a delusion it is called=20=20 >> insanity. When many people suffer from a delusion it is called=20=20 >> religion." >> > > So the belief in the collapse of the wave is a religion. Why not.=20=20 > Feynman already called it a "collective hallucination", if I=20=20 > remember well. > > > > > >> . >> >> G=F6del=92s mathematical proof is probably the strongest case for the = =20 >> IMPROBABLE existence of God, but this proof simply means that it's=20=20 >> just not possible to identify the processes that gave rise to the=20=20 >> formation "of all that exists" (or "being")....RobtO >> > > The ancient greeks got this too, by the dream argument, and it was=20=20 > the start of science, when we stop to believe that what we see and=20=20 > measure is necessarily the "real thing", which might be just hidden=20=20 > by the appearance. Note that G=F6del provided a proof of the existence = =20 > of God, but it requires you to accept St-Anselmus definition of God,=20=20 > and G=F6del formalization of that definition, and the alethic modal=20=20 > logic S5. Personally, I don't believe in S5, nor do I find the=20=20 > definition of God by St-Anselmus the most enlightening one. > But his incompleteness theorem, together with his non definability=20=20 > of truth (G=F6del-Tarski) makes somehow the arithmetical truth=20=20 > behaving like the God of the Platonist, which is not much more than=20=20 > the truth we search (as opposed to any truth we might believe to=20=20 > have found). > > Bruno > > > >> > > http://iridia.ulb.ac.be/~marchal/ > > > > >=20 http://iridia.ulb.ac.be/~marchal/ --Apple-Mail-11--429693282 Content-Type: text/html; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/htm= l4/strict.dtd"> <html> <head> </head> =20 <body style=3D"background-color: #fff;"> <span style=3D"display:none"> </span> <!--~-|**|PrettyHtmlStartT|**|-~--> <div id=3D"ygrp-mlmsg" style=3D"position:relative;"> <div id=3D"ygrp-msg" style=3D"z-index: 1;"> <!--~-|**|PrettyHtmlEndT|**|-~--> <div id=3D"ygrp-text" > =20=20=20=20=20=20 =20=20=20=20=20=20 <p><br><div><div>On 14 Nov 2014, at 03:45, 'Lee Corbin' <a href=3D"ma= ilto:[email protected]">[email protected]</a> [Fabric-of-Reality]= wrote:</div><br class=3D"Apple-interchange-newline"><blockquote type=3D"ci= te"> <div style=3D"background-color: rgb(255, 255, 255);"> <span> = </span> <div id=3D"ygrp-text"><div> <br class=3D"webkit-block-placehol= der"></div><div dir=3D"ltr"> <div style=3D"FONT-SIZE: 12pt;FONT-FAMILY: 'Bo= ok Antiqua';COLOR: #000000;"> <div>Those who are interested when G=F6del's = theorems should and should *not* be applied ought to read Torkel Franzen=92= s wonderful book, =93G=F6del's Theorem: An Incomplete Guide to its Use and = Abuse=94.</div> <div> </div></div></div></div></div></blockquote></div= ><div><br></div><div>Torket Franzen book is excellent. Only the title of th= e book is not appropraite as he is concerned only with the misuse of G=F6de= l, and not on its use. It does not cite Judson Webb for example, nor my wor= k, despite some people pretend there are allusion, and there might be an al= lusion to something I never said, but that some people attribute to me (the= idea that the computationalist indeterminacy bears on the models of PA, wh= ich I precisely refute instead).</div><div><br></div><div><br></div><div><b= r></div><div><br></div><div><br></div><div><br></div><br><blockquote type= =3D"cite"><div style=3D"background-color: rgb(255, 255, 255);"><div id=3D"y= grp-mlmsg"><div id=3D"ygrp-msg"><div id=3D"ygrp-text"><div dir=3D"ltr"><div= style=3D"FONT-SIZE: 12pt;FONT-FAMILY: 'Book Antiqua';COLOR: #000000;"> <di= v>Study of his book has convinced me that about 95% of the references one c= omes across to G=F6del's Theorem are fallacious. </div></div></div></div></= div></div></div></blockquote><div><br></div><div>I agree. </div><div><= br></div><div><br></div><div><br></div><br><blockquote type=3D"cite"><div s= tyle=3D"background-color: rgb(255, 255, 255);"><div id=3D"ygrp-mlmsg"><div = id=3D"ygrp-msg"><div id=3D"ygrp-text"><div dir=3D"ltr"><div style=3D"FONT-S= IZE: 12pt;FONT-FAMILY: 'Book Antiqua';COLOR: #000000;"><div>Here is just on= e fascinating fact: The theorem to formal systems that contain =93a certain= amount of arithmetic=94 (it=92s hard to spell that out in more detail). On= e example is the formal system containing this axioms for real numbers. G= =F6del's Theorem does not apply, because the real number axioms, surprising= ly, cannot =93get their hands=94 on</div></div></div></div></div></div></di= v></blockquote><blockquote type=3D"cite"><div style=3D"background-color: rg= b(255, 255, 255);"><div id=3D"ygrp-mlmsg"><div id=3D"ygrp-msg"><div id=3D"y= grp-text"><div dir=3D"ltr"><div style=3D"FONT-SIZE: 12pt;FONT-FAMILY: 'Book= Antiqua';COLOR: #000000;"><div>integers in the sense that you cannot speci= fy =93for all integers=94 the way that you can in Peano Arithmetic.</div> <= div> </div></div></div></div></div></div></div></blockquote><div><br><= /div><div><br></div><div>Yes. the first order theory of the real number has= shown to be decidable, and thus not even Turing complete. So it does not c= ontain the theory of natural numbers as a sub-theory. It does not have the = sufficient amount of arithmetic.</div><div><br></div><div>You cannot define= a natural number from the theory of reals. You can intuit this by realizin= g that Fermat theorem is easy to solve when "for all" bears on the reals, a= nd quite difficult on the natural numbers, as the proof by Wyles illustrate= s. Note that the real + trigonometric functions or waves gave again the Tur= ing universality. You can define a natural number by the solution of sin(2p= ix) =3D 0, for example.</div><div><br></div><div>Nevertheless, there is no = problem in "a certain amount of arithmetic", which is a standard expression= for logician. It means that you can define the addition and multiplication= laws on the integers or the natural numbers, which indeed you cannot do in= the first order theory of the reals. I made this technically precise with = the notion of L=F6bian machine in my papers.</div><div><br></div><div>Bruno= </div><div><br></div><div><br></div><div><br></div><br><blockquote type=3D"= cite"><div style=3D"background-color: rgb(255, 255, 255);"><div id=3D"ygrp-= mlmsg"><div id=3D"ygrp-msg"><div id=3D"ygrp-text"><div dir=3D"ltr"><div sty= le=3D"FONT-SIZE: 12pt;FONT-FAMILY: 'Book Antiqua';COLOR: #000000;"> <div>Le= e</div> <div> </div> <div style=3D"FONT-SIZE: small;TEXT-DECORATION: n= one;FONT-FAMILY: "Calibri";FONT-WEIGHT: normal;COLOR: #000000;FON= T-STYLE: normal;"> <div style=3D"FONT: 10pt tahoma;"> <div> </div> <di= v style=3D"BACKGROUND: #f5f5f5;"> <div><b>From:</b> <a title=3D"Fabric-of-R= [email protected]" href=3D"mailto:[email protected]">m= ailto:[email protected]</a> </div> <div><b>Sent:</b> Thursd= ay, November 13, 2014 7:24 AM</div> <div><b>To:</b> <a title=3D"Fabric-of-R= [email protected]" href=3D"mailto:[email protected]">F= [email protected]</a> </div> <div><b>Subject:</b> Re: GODEL<= /div></div></div> <div><font face=3D"Book Antiqua"></font> </div></div= > <div style=3D"FONT-SIZE: small;TEXT-DECORATION: none;FONT-FAMILY: "C= alibri";FONT-WEIGHT: normal;COLOR: #000000;FONT-STYLE: normal;"><span>= </span> <div id=3D"ygrp-mlmsg"> <div id=3D"ygrp-msg"> <div id=3D"ygrp-text"= > <div> <div>On 04 Nov 2014, at 22:48, <a href=3D"mailto:[email protected]= t">[email protected]</a> [Fabric-of-Reality] wrote:</div><br class=3D"App= le-interchange-newline"> <blockquote type=3D"cite"> <div style=3D"BACKGROU= ND-COLOR: rgb(255,255,255);"><span> </span> <div id=3D"ygrp-text"> = <div> </div><p> <strong><font size=3D"2">I don't think that Hawki= ns has come close to proving that the existence of God is improbable. His= case can be summarized when he says....""when one person suffers from a = delusion it is called insanity. When many people suffer from a delusion i= t is called religion."</font></strong></p></div></div></blockquote></div>= <div>So the belief in the collapse of the wave is a religion. Why not. Fey= nman already called it a "collective hallucination", if I remember well.</d= iv> <div> </div> <div> </div> <div> </div> <div> </div>= <br> <blockquote type=3D"cite"> <div style=3D"BACKGROUND-COLOR: rgb(255,25= 5,255);"> <div id=3D"ygrp-mlmsg"> <div id=3D"ygrp-msg"> <div id=3D"ygrp-= text"><p><strong><font size=3D"2">.</font></strong></p><strong><font size= =3D"2"></font></strong><p><strong><font size=3D"2">G=F6del=92s mathematical= proof is probably the strongest case for the IMPROBABLE existence of God= , but this proof simply means that it's just not possible to identify the= processes that gave rise to the formation "of all that exists" (or "bein= g")....RobtO </font></strong></p></div></div></div></div></blockquote> <d= iv> </div> <div>The ancient greeks got this too, by the dream argument= , and it was the start of science, when we stop to believe that what we see= and measure is necessarily the "real thing", which might be just hidden by= the appearance. Note that G=F6del provided a proof of the existence of God= , but it requires you to accept St-Anselmus definition of God, and G=F6del = formalization of that definition, and the alethic modal logic S5. Personall= y, I don't believe in S5, nor do I find the definition of God by St-Anselmu= s the most enlightening one. </div> <div>But his incompleteness theorem, to= gether with his non definability of truth (G=F6del-Tarski) makes somehow th= e arithmetical truth behaving like the God of the Platonist, which is not m= uch more than the truth we search (as opposed to any truth we might believe= to have found).</div> <div> </div> <div>Bruno</div> <div> </div>= <div> </div><br> <blockquote type=3D"cite"> <div style=3D"BACKGROUND= -COLOR: rgb(255,255,255);"> <div id=3D"ygrp-mlmsg"> <div id=3D"ygrp-msg">= <div id=3D"ygrp-text"> <div> </div></div></div><!-- end group email= --></div></div></blockquote><br> <div apple=3D"true"><span class=3D"Apple-= style-span" style=3D"FONT-SIZE: 12px;FONT-FAMILY: helvetica;FONT-VARIANT: n= ormal;BORDER-COLLAPSE: separate;FONT-WEIGHT: normal;COLOR: rgb(0,0,0);FONT-= STYLE: normal;LETTER-SPACING: normal;TEXT-INDENT: 0px;"> <div> <div><a href= =3D"http://iridia.ulb.ac.be/~marchal/">http://iridia.ulb.ac.be/~marchal/</a= ></div> <div> </div></div></span><br class=3D"Apple-interchange-newlin= e"></div><br></div><!-- end group email --></div></div></div> </div></div><= div><br class=3D"webkit-block-placeholder"></div> </div> </div= > <!-- end group email --> </div></div></blockquote><br><div apple=3D"tru= e"> <span class=3D"Apple-style-span" style=3D"border-collapse: separate;col= or: rgb(0, 0, 0);font-family: Helvetica;font-size: 12px;font-style: normal;= font-variant: normal;font-weight: normal;letter-spacing: normal;text-align:= auto;text-indent: 0px;"><div><div><a href=3D"http://iridia.ulb.ac.be/~marc= hal/">http://iridia.ulb.ac.be/~marchal/</a></div><div><br class=3D"webkit-b= lock-placeholder"></div></div></span><br class=3D"Apple-interchange-newline= "> </div><br></p> </div> =20=20=20=20=20 <!--~-|**|PrettyHtmlStart|**|-~--> <div style=3D"color: #fff; height: 0;">__._,_.___</div> =20=20=20=20=20=20=20=20=20=20 =20=20 =20 =20=20=20=20 <div style=3D"clear:both"> </div> <div id=3D"fromDMARC" style=3D"margin-top: 10px;"> <hr style=3D"height:2px ; border-width:0; color:#E3E3E3; background= -color:#E3E3E3;"> Posted by: Bruno Marchal <[email protected]> <hr style= =3D"height:2px ; border-width:0; color:#E3E3E3; background-color:#E3E3E3;"> </div> <div style=3D"clear:both"> </div> <table cellspacing=3D4px style=3D"margin-top: 10px; margin-bottom: 10px= ; color: #2D50FD;"> <tbody> <tr> <td style=3D"font-size: 12px; font-family: arial; font-weight: bo= ld; padding: 7px 5px 5px;" > <a style=3D"text-decoration: none; color: #2D50FD= " href=3D"https://groups.yahoo.com/neo/groups/Fabric-of-Reality/conversatio= ns/messages/25673;_ylc=3DX3oDMTJxbm5tNG51BF9TAzk3MzU5NzE0BGdycElkAzE4ODg2MD= AEZ3Jwc3BJZAMxNzA1MDIwNTg1BG1zZ0lkAzI1NjczBHNlYwNmdHIEc2xrA3JwbHkEc3RpbWUDM= TQxNTk3NTA2MQ--?act=3Dreply&messageNum=3D25673">Reply via web post</a> </td> <td>•</td> <td style=3D"font-size: 12px; 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