Metamath
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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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