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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