🚀 go-pugleaf

Please note that the existence of the set of all natural numbers is
sanctioned by one of the Fathers of the Church.  Note the use of
theological concepts to attack a set-theoretic problem.  But,
seriously, Augustine says some very modern things in this passage if
you think about it.  From St. Augustine, _the City of God_:


19.  The answer to the allegation that even God's knowledge cannot
embrace an infinity of things.

Then there is the assertion that even God's foreknowledge annot
embrace things which are infinite.  If men say this, it only remains
for them to plunge into the depths of blasphemy by daring to allege
that God does not know all numbers.  It is certainly true that numbers
are infinite.  If you think to make an end with any number, then that
number can be increased by the addition of one.  More than that,
however large it is, however great the quantity it expresses, it can
be doubled; in fact, it can be multiplied by any number, according to
the very principle and science of numbers.

Every number is defined by its own unique character, so that no number
is equal to any other.  They are all unequal to one another and
different, and the individual numbers are finite, but as a class they
are infinite.  Does that mean that God does not know all numbers,
because of their infinity?  Does God's knowledge extend as far as a
certain sum, and end there?  No one could be insane enough to say
that.

Now those philosophers who revere the authority of Plato will not dare
to despise numbers and say that they are irrelevant to God's
knowledge.  For Plato emphasizes that God constructed the world by the
use of numbers, while we have the authority of Scripture, where God is
thus addressed, "You have set in order all things by number, measure,
and weight".  And the prophet says of God "He produces the world
according to number"; and the Saviour says in the Gospel, "Your hairs
are all numbered".

Never let us doubt, then, that every number is known to him 'whose
understanding cannot be numbered'.  Although the infinite series of
numbers cannot be numbered, this infinity of numbers is not outside
the comprehension of him 'whose understanding cannot be numbered'.
And so, if what is comprehended in knowledge is bounded within the
embrace of that knowledge, and thus is finite, it must follow that
every infinity is, in a way we cannot express, made finite to God,
because it cannot be beyond the embrace of his knowledge.

				St. Augustine,
				_the City of God_,
				Book XII, ch. 19,
				pp. 496-7 (Pelican paperback)
-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu