Anselm's argument employs that which in mathematical terms is named an ultrafilter. Ultrafilters are part of a branch of maths called 'Model theory' It is part and parcel to the use of an ultrafilter, that if they are correctly applied, then anything that is true for the ultrafilter, is then true in the 'language' of the filtered set. Well, almost! in fact it can only be stated that any property of the ultrafilter is analytically true, "Almost Everywhere". When I first heard of filters, I heard that they were being used to describe the topology of theories (Models). That is, one could imagine that 'the shape' of the filter can be used to determine the local 'shape' of the set to which it applies, no matter where in the set it is applied. Ultrafilters should not be thought of as a present member of a 'language', but rather a method by which we may examine if a system has a property of the filter.

Ultrafilters, Insert Coin
The definition of Filters, Ultrafilters and the various types of the latter.

Application By Anselm
By examination of the ontological arguments for and against God's existence, can we tell if there is a problem in any particular argument from their use of an ultrafilter?

Principal, Or Non-Principal
Though Anselm's argument uses a non-principal ultrafilter, is there an argument with any principal filter?

Finite, Or Infinite?
Is the filter in Anselm's argument finite, or infinite? When we consider merely necessary existence, do we create an ultrafilter with millions of members when we magnify his greatness? Do we try to define any positive property as 'rock solid and existent' in the case of God's existence, that somehow He 'is' His goodness, or that His existence alone must then entail every positive property in Him?.

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