Principal or Non-Principal

In the argument over existential judgements that they are actually statements of an ordinal nature, that whether something exists or not is not a property of the object, but a comment of the number of instances of the object, do we alter the argument without knowing we do so?

Were we to state that we imagine God to be greater if he exists, than he would were he not existent, Can we state that one is better than none? Is Anselm's consequent result that God is necessarily existent reliant solely on considering existence as a property of the object? If we can state that God necessarily exists if he exists, (There is at least one instance of Him) may we assume his existence is a "least element" in a principal ultrafilter? Surely, that for God to have a property of any kind, in the sense of the property being "existent", we at least require the instances of God to be greater than or equal to one. Such "properties of God", love, compassion etc in their perfected forms can no longer be said to "exist" in something that may not, so indeed that there is "one instance of God" like it or not, becomes a least element and base level for any property of the Perfect being, whether it is a clearly existential judgement or not, (such as a personal one, or other abstractions in his nature.) That all such absolutes require an instance flips the Ultrafilter from non-principal to principal.

So does this answer whether God exists? In the least, it shows that the concept of god, a perfect being is well defined and consistent. We require an instance of one or more "Gods", Of course, if there is no God, then he may not have any properties (in instance). Proving That he necessarily has one instance, may we start by considering all properties of God other than existential ones? Referring back to Gödel's ontological argument, we can find many of these "essences", and that an individual has these positive properties exemplified necessarily in him. In fact, all such properties lead to him being "god-like".

We can say these properties are necessarily exemplified, but we hit a stumbling block when we wish to infer that such a being EXISTS and is greater for it, as Anselm did. To the argument of instances, God can necessarily be good without necessarily existing. In reality however, no property of God can be considered as entailed rationally in view of a principal nature of the ultrafilter. So we have two outcomes dependent on instance.

1) There is 1 or more Perfect beings, and the perfection of this Godhead is consistent and well defined.

2) There are no perfect beings, (0 instances) and therefore due to the principal ultrafilter, the concept of a perfect being is inconsistent and entirely unreasonable.

So, we arrive indeed at the same outcome as Charles Hartshorne, that "Either God exists necessarily, or his existence is an impossibility." So, the argument of instances is separable between the zero and positive.

To review, a principal ultrafilter has a "least element", and it is readily deduced that the filter contains every set containing this least element. If the assertion that existence is a matter of number of instances, rather than any property of the object then, at least in terms of necessary existence we may add a slant to the term "necessary" similar to Gödel's method. We may introduce a "least element" of non-zero instance to God, and everything that God would entail upon his existence, as such a principal ultrafilter. We may, in truth consider everything that we can rationally understand about the universe without fault. We take what is true about the ultrafilter, that God 'necessarily exists' as true almost everywhere. Now, the critique of the arguments like the ultimate atheist argument manifest the notion that "existent" is part and parcel with "not conceived as (not existent)" rather than "Not (Not existent)". and their 'negations'. That there must be conceptual freedom to conceive of God's non-existence in error or otherwise is explained by "almost everywhere". Our understanding of the universe is conception. That we may consider existence of an object / being its own property in this manner does not ruin our argument. Descartes ontological proof is the summary of this system.

"Certainly, the idea of God, or of a supremely perfect being, is one which I find within me just as surely as the idea of any shape or number. And my understanding that it belongs to his nature that he always exists is no less clear and distinct than is the case when I prove of any shape or number that some property belongs to its nature. Hence. . . I ought . . . to regard the existence of God as having at least the same level of certainty as I have hitherto attributed to the truths of mathematics.

(Descartes 1984 vol.2:45) - Taken from "The Non-Existence of God." Author; Phillip Everitt (Routledge paperback)

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