What do you mean with special participant? Do you have ability to stop a chain of causality and not participate in it anymore?
A special participant in that its effects are undetermined by any cause through itself. We see this as special because we know the universe more or less operates very deterministically like a clock (if we ignore the quantum level phenomena); that is to say, each effect can be perfectly and immediately determined by a full understanding of a given cause. The causes of a phenomena are therefore the ultimate determiners of the
complete nature of a phenomena. But, the mind is different. The mind doesn’t operate deterministically by cause but rather by its own thoughts, suppositions, biases, and values as well. As such, it has agency over the effect, rather than solely the cause. As such, the mind is a special participant in the causal nexus.
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Different in that his attributes are pure actuality and thus supremely simple and infinite, whilst our are not fully actuality, nor are we supremely simple, and even less are we infinite. Though, I admit with the similarities you described.
Something which exist has substance.
If we are using the same definition of substance, which is the make up of a being (or material, one may say), then I cannot say God would have substance, for he is made up of nothing, but instead soley is himself. But if you simply mean that he has being which can be distinguished to others, then yes.
So what is your objection on Cantor’s proof?
I am not objecting even slightly to Cantor’s proof. In fact, I one hundred percent believe it, implications included. What I am objecting to is the premise which lies at the core of the infinite being refutation on the basis of mathematical paradox, that being that an infinite being can be mathematically represented as an ordinal set. But most theist would deny that entirely, and would say that the conclusion of the Burali-Forti’s paradox is actually very much correct, as God is an infinite not mathematically able to be put into a complete, semi- comprehensive set. They would, instead, say that God is of such a magnitude that one could not possibly contain him in even this format. I would add on to this that his infinity isn’t one of many mathematical infinities, which have distinctions in them because of the nature of cardinal and ordinal numbers, but rather that he just is endless, never ceasing, and uncontainable/ incomprehensible being.
A similar objection to this can be made on the basis of infinite knowledge (to which I showed you in the last post), where the question is how can God be all knowing if you can go infinitely in regress recognizing your own recognition of a given body of truth. The solution simply is that we are thinking of the infinity of God too much as a closed set to which may be rationally represented (like in mathematics) rather than a seemingly endless and, seemingly continual, and seemingly grand ultimate that does not find these issues due to sheer grandeur in being.