Boris Borcic
Apr22-04, 03:26 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>John Baez wrote:\n> Frank Hellmann <C.i.m@gmx.net> wrote:\n>\n>>Are those powerseries divergent in the sense that they run of to\n>>infinity, or non convergent but finite (like summing over an infininte\n>>series of -2 and +2, evidently bounded yet not convergent)?\n>\n>\n> Usually they go like\n>\n> 1 - .5 + .01 - .004 + .00007 - .0007 + .15 - 2 + 23 - 247 + ....\n>\n\nThis "aesthetically" reminds me of a wonder about Goedel\'s results.\n\nFaced with an axiomatic system while knowing any proof of consistency thereof\nis forever out of reach, one is led to contemplate the case where it is\ninconsistent, together with the bound provided by a measure of the minimal\ncomplexity of an (eventual) proof of inconsistency, say by number of\ninferences from axioms.\n\nWe are trained to collapse all verified consequences of a system of axioms to\nequivalent true status, and on another hand from a contradiction we can derive\nanything : with that view an inconsistent system appears totally useless. But\nif we order consequences by minimal numbers of steps to reach them from the\naxioms, and limit ourselves to fewer steps than the number required to reach\nan inconsistency, the system is still applicable to the task of separating\nformulas into disjoint equivalence classes. Can\'t such work remain useful ?\n\nI am wondering whether somebody explored that general alley of thought, or\nrather, I\'d like to know who did since I can\'t believe that nobody did. Also,\nI am curious whether readers see the relationship with divergent QFT power\nseries and if indeed there could be more to it than just aesthetic reminiscence.\n\nRegards, Boris Borcic\n--\n365 ? - 365 = 73*5 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2 !\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>John Baez wrote:
> Frank Hellmann <C.i.m@gmx.net> wrote:
>
>>Are those powerseries divergent in the sense that they run of to
>>infinity, or non convergent but finite (like summing over an infininte
>>series of -2 and +2, evidently bounded yet not convergent)?
>
>
> Usually they go like
>
> 1 - .5 + .01 - .004 + .00007 - .0007 + .15 - 2 + 23 - 247 + ....
>
This "aesthetically" reminds me of a wonder about Goedel's results.
Faced with an axiomatic system while knowing any proof of consistency thereof
is forever out of reach, one is led to contemplate the case where it is
inconsistent, together with the bound provided by a measure of the minimal
complexity of an (eventual) proof of inconsistency, say by number of
inferences from axioms.
We are trained to collapse all verified consequences of a system of axioms to
equivalent true status, and on another hand from a contradiction we can derive
anything : with that view an inconsistent system appears totally useless. But
if we order consequences by minimal numbers of steps to reach them from the
axioms, and limit ourselves to fewer steps than the number required to reach
an inconsistency, the system is still applicable to the task of separating
formulas into disjoint equivalence classes. Can't such work remain useful ?
I am wondering whether somebody explored that general alley of thought, or
rather, I'd like to know who did since I can't believe that nobody did. Also,
I am curious whether readers see the relationship with divergent QFT power
series and if indeed there could be more to it than just aesthetic reminiscence.
Regards, Boris Borcic
--
365 ? - 365 = 73*5 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2 !
> Frank Hellmann <C.i.m@gmx.net> wrote:
>
>>Are those powerseries divergent in the sense that they run of to
>>infinity, or non convergent but finite (like summing over an infininte
>>series of -2 and +2, evidently bounded yet not convergent)?
>
>
> Usually they go like
>
> 1 - .5 + .01 - .004 + .00007 - .0007 + .15 - 2 + 23 - 247 + ....
>
This "aesthetically" reminds me of a wonder about Goedel's results.
Faced with an axiomatic system while knowing any proof of consistency thereof
is forever out of reach, one is led to contemplate the case where it is
inconsistent, together with the bound provided by a measure of the minimal
complexity of an (eventual) proof of inconsistency, say by number of
inferences from axioms.
We are trained to collapse all verified consequences of a system of axioms to
equivalent true status, and on another hand from a contradiction we can derive
anything : with that view an inconsistent system appears totally useless. But
if we order consequences by minimal numbers of steps to reach them from the
axioms, and limit ourselves to fewer steps than the number required to reach
an inconsistency, the system is still applicable to the task of separating
formulas into disjoint equivalence classes. Can't such work remain useful ?
I am wondering whether somebody explored that general alley of thought, or
rather, I'd like to know who did since I can't believe that nobody did. Also,
I am curious whether readers see the relationship with divergent QFT power
series and if indeed there could be more to it than just aesthetic reminiscence.
Regards, Boris Borcic
--
365 ? - 365 = 73*5 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2 !