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Notes -
I really don't think that's a definition of "omnipotence" that most Christians, past or present, would actually agree with, and if forced to use that definition, I think most would concede that the God they posit is not in fact omnipotent in this sense. I'd be very interested in examples of Christians arguing otherwise, if you've seen any.
I could very easily include the string "in this world, up is down". I couldn't describe the necessary causes and effects of such a change, because logically-invalid linguistic constructions have no necessary causes and effects. Any effects I then attribute to "up is down" in the story are not the cause or result of up being down, they're the result of my direct, arbitrary will.
Notably, the Christian idea that humans have a free will separate from that of God, common to most branches of the faith, logically depends on God not actually having this sort of relationship to our world; ditto for many other parts of Christian theology and philosophy. As I understand it, the Christian conception of God contains a lot of examples of him being shaped by the necessities and interdependencies of what is taken to be baseline, unalterable reality. All of these would flatly contradict the concept of omnipotence as you define the term, which is fair enough, but it is in fact just redefining omnipotence, and my impression is that their definition came first by a number of centuries.
I think there are roughly three options to take:
God is truly omnipotent. When he says All is possible for the Lord, He is being literal. Just because we as finite moral limited beings cannot comprehend how He could do logically paradoxical things that is no bar to Him being able to do so. He is truly omnipotent.
While God says All is possible through the Lord, events in the bible and since and fundamental logic seem to indicate this cannot literally be true. God cannot make paradoxes. He has some limits. Therefore He is not omnipotent.
God is omnipotent but He cannot do certain things, but those don't count towards Him not bring omnipotent because they can't be done at all.
and 2) are I think reasonable approaches. 3) is I think a rationalization of two contradictory positions. Its a cop out.
Indeed, there are Christians who hold position 2) though it doesn't appear too popular. But I think Christians should hold position 1). Faith that what God says is true even if we are unable to wrap our minds around how. And many Christians claim to hold position 1) but when pressed with examples like the ones you bring up they revert to position 3). Because they cannot conceive of how God could do things they find logically impossible.
There are positions in Christianity which hold 1) to be true, but they aren't I think popular in modern times. Their solution to can God create a rock He cannot lift is: Yes, God can do the impossible in creating a rock that He cannot lift. And then he can do the impossible and lift it. For ALL things are possible through the Lord. He is not constrained by logic or anything else. I think 3) is a result of Christians finding it very difficult (and not unreasonably so!) to resolve the logical materialism they see around themselves with the true mystery of God as described in the bible. So this halfway house is created. Its not new, this debate is over a thousand years in the making.
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