Application by Anselm

Just about everywhere ultrafilters are mentioned with respect to Anselm's argument, the argument is said to use the concept of "perfection" as a non-principal ultrafilter. What we may firstly observe is that the ultrafilter must indeed be infinite. We can dream up any number of properties that are "perfections" and heap them upon God. We are not simply restricted to considering His existence only, we may infer any property of a perfect being to God, such as compassion, strength, omnipotence etc. Why therefore, is his existence not a least element? Surely, he needs to exist for the others to make any sense? Since one can treat the characteristics of omnipotence, omnipresence etc as properties of God whether he exists or not, existence would appear not to be a pre-requisite of these. These properties of a god do not depend on his existence in the logical sense therefore.

If however, God's necessary existence is required to justify any perfection, we at least could arrive at the coincidence of all of these in his existence itself. We imagine a series of conceptions of god, a growing collection of perfections leading us to "That greater than which no other can be imagined". There remains uncertainty that the smallest least element is "The existence of God himself." There is some pre-conception upon which we pile our perfection of "conceived as existent" that would indeed be a "smaller" set. Indeed, we already start with "that which is none greater". In truth, the ultrafilter is non-principal. were we to start with "God exists necessarily" we would not be able to find proof for it in the argument itself! Once proven, we may add as many perfections upon it as we might, but not to prove the first as consequent from the method of a principal ultrafilter.

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