Marc Nardmann
Apr28-04, 01:30 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>rof@maths.tcd.ie wrote:\n\n> Right; the problem would be solved if our number system had genuine\n> infinitesimals. Then each integer could be assigned a probability\n> 1/aleph_0 and everything would work out ok. Such number systems\n> exist and there is nothing inherent in probability theory which\n> ties it to real numbers except its current formulation.\n\nThere is something very important which ties (countably additive)\nprobability theory to the real numbers; see below.\n\n> I was thinking of the surreals rather than the hyperreals;\n> I believe they have well-defined multiplicative inverses for\n> every non-zero number (including aleph_0) along with the same\n> rules of distributivity as real numbers.\n\nYes, they are equipped with the structure of an ordered field. (The\nfield is not a set but a proper class. But that does not matter for the\ndiscussion here; you can consider suitable subfields of the surreal\nnumbers which are sets.)\n\n> I\'m don\'t know\n> very much about this, but wouldn\'t that be sufficient?\n\nNo. Look at the definition of a real-valued measure: There is a\ncondition about countable additivity. One side of this equation is a\ncountable infinite sum of nonnegative real numbers. A countable sum of\nnonnegative real numbers is well-defined as a supremum in [0,infinity].\nNow try to define the sum of a countable infinite sequence of\nnonnegative surreal numbers and you will see the problem.\n\nIf one can circumvent this problem at all, then the result will probably\nbe neither more practical nor more aesthetical than standard measure\ntheory. Of course, you can prove me wrong in this respect by inventing a\nnice definition of a surreal-valued measure. But that is certainly not\ntrivial.\n\n\n-- Marc Nardmann\n\n(To reply, remove every occurrence of a certain letter from my e-mail\naddress.)\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>rof@maths.tcd.ie wrote:
> Right; the problem would be solved if our number system had genuine
> infinitesimals. Then each integer could be assigned a probability
> 1/\aleph_0 and everything would work out ok. Such number systems
> exist and there is nothing inherent in probability theory which
> ties it to real numbers except its current formulation.
There is something very important which ties (countably additive)
probability theory to the real numbers; see below.
> I was thinking of the surreals rather than the hyperreals;
> I believe they have well-defined multiplicative inverses for
> every non-zero number (including \aleph_0) along with the same
> rules of distributivity as real numbers.
Yes, they are equipped with the structure of an ordered field. (The
field is not a set but a proper class. But that does not matter for the
discussion here; you can consider suitable subfields of the surreal
numbers which are sets.)
> I'm don't know
> very much about this, but wouldn't that be sufficient?
No. Look at the definition of a real-valued measure: There is a
condition about countable additivity. One side of this equation is a
countable infinite sum of nonnegative real numbers. A countable sum of
nonnegative real numbers is well-defined as a supremum in [0,infinity].
Now try to define the sum of a countable infinite sequence of
nonnegative surreal numbers and you will see the problem.
If one can circumvent this problem at all, then the result will probably
be neither more practical nor more aesthetical than standard measure
theory. Of course, you can prove me wrong in this respect by inventing a
nice definition of a surreal-valued measure. But that is certainly not
trivial.
-- Marc Nardmann
(To reply, remove every occurrence of a certain letter from my e-mail
address.)
> Right; the problem would be solved if our number system had genuine
> infinitesimals. Then each integer could be assigned a probability
> 1/\aleph_0 and everything would work out ok. Such number systems
> exist and there is nothing inherent in probability theory which
> ties it to real numbers except its current formulation.
There is something very important which ties (countably additive)
probability theory to the real numbers; see below.
> I was thinking of the surreals rather than the hyperreals;
> I believe they have well-defined multiplicative inverses for
> every non-zero number (including \aleph_0) along with the same
> rules of distributivity as real numbers.
Yes, they are equipped with the structure of an ordered field. (The
field is not a set but a proper class. But that does not matter for the
discussion here; you can consider suitable subfields of the surreal
numbers which are sets.)
> I'm don't know
> very much about this, but wouldn't that be sufficient?
No. Look at the definition of a real-valued measure: There is a
condition about countable additivity. One side of this equation is a
countable infinite sum of nonnegative real numbers. A countable sum of
nonnegative real numbers is well-defined as a supremum in [0,infinity].
Now try to define the sum of a countable infinite sequence of
nonnegative surreal numbers and you will see the problem.
If one can circumvent this problem at all, then the result will probably
be neither more practical nor more aesthetical than standard measure
theory. Of course, you can prove me wrong in this respect by inventing a
nice definition of a surreal-valued measure. But that is certainly not
trivial.
-- Marc Nardmann
(To reply, remove every occurrence of a certain letter from my e-mail
address.)