Finite or Infinite?
Of course, since all finite Ultrafilters are principal,
it is a mute point that the Ultrafilter used in Anselm's Argument would
have to be infinite. The case for A finite ultrafilter in the principal
case discussed in the previous page would depend upon the language used.
A set may consist of any number of elements. Simply stating that above
the principal least element that There is "one instance of God" we may
add in Gödel's terms "This individual is God-like in every way" as
opposed to merely listing an ever expanding sequence of supersets
listing every individual entailing characteristic.
Of course I can tell you see the flaw, that all
supersets of the principal element must be included to fill between our
least element and the largest superset of all perfections. Yet I stick
behind it because I hid behind the word "language". The statement on the
"God-like in every way" is a property in the singular sense, although it
may entail all of the multitude of perfections we may imagine.
The statement is very nearly a least element for the
ultrafilter in Anselm's argument. The assumption of Anselm that none
greater than God can be conceived to exist is very close to this. It is
not the same however, as there is no way to quantify an idea with no
property other than the name "God" or "Perfect being". Is the statement
that God is a perfect being equivalent to the statement? Without
realising that a "god-like" property is necessarily exemplified if it is
exemplified at all, it is hard to see the difference. Anselm's statement
is an assumption, whilst the latter is a consequence. Both parts of the
same argument, the former is clearly a great part of the latter.
Since the conclusion can't be reached until after the
assumptions are tested, we reduce the ultrafilter to a simple statement
if we employ the conclusion to the method from the start. Simply, it may
not be used simply to confirm itself in the proof. Restated - Anselm's
argument uses a non-principal infinite ultrafilter.
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