FOM: re: ``infinite products''

[Matt Insall]

Basically, the answer is that yes, non-standard analysis affords a way of
dealing with ``infinite products'', if I am correctly understanding your
meaning, but it may be not of the type of which you speak.  I assume that by
``omega'' you mean the first infinite ordinal.  That is a significantly
different type of ``infinite number'' than the type available on the
non-standard real line.  However, if by ``the omega power of .5,'' you mean
the limit (in the Cauchy sense) of a sequence of partial products then the
non-standard method of dealing with infinite limits will answer your
questions.  In particular, if nu is any infinite natural number, then .5^nu
is infinitesimal.  The relative sizes of .5^nu and .6^nu are completely
determined by the transfer principle, so .5^nu < .6^nu, and yes, they are
both infinitesimal.  For other products, the question of whether the result
is infinitesimal depends on the properties of the infinite collection of
numbers in consideration.



Matt Insall