OK, here's the first draft of my formal proof, submitted for your scrutiny. I want everyone to be harshly critical, but not to the extent that invalid criticism is levied.
By the way, I posted this at Christianforums.com, which you can find here with lots of dialogue. Credit to the user zippy2006 over there who, despite driving me insane with his bad logic, helped me refine the proof. I didn't post here for you guys to see until I had a formalized proof.
(1) Assume ZFC, which is the Zermelo-Fraenkel collection of axioms with the axiom of choice added.
(2) Observe that the law of non-contradiction, which is assumed in ZFC, is equivalent to the law of excluded middle. This means that no coherent statement can be neither true nor false unless we drop the law of non-contradiction. Explicitly,
(3) Gödel's first incompleteness theorem, which states that any self-consistent, non-trivial logical system S must contain some true statement that is unprovable in S, applies to ZFC.
(4) One such statement is the continuum hypothesis, which states that there exists no set whose cardinality is greater than that of the integers but less than that of the real numbers. Formally, there is no X such that |Z|<|X|<|R|. Observe that X must either exist or not exist according to (2).
(5) A little-g god is, as YouTuber Aron Ra puts it, a magical, anthropomorphic immortal. There can be many little-g gods, and little-g gods cannot be proven to not exist. However, big-G God is the supreme being, and there can only be one or zero of these. (If there were two big-G Gods, and they disagreed on an aspect of creation, whose will would be reflected in reality? The existence of multiple big-G Gods necessitates that they are all of the same will and indistinguishable in any way whatsoever, which is to say that there is only one of them.)
(6) According to (1), big-G God cannot be omnipotent because big-G God cannot create, for example, a circle with a vertex. Omnipotence results in paradoxes, so big-G God is defined as maximally supreme. To be maximally powerful is to be able to perform any task that is not logically impossible. To be maximally knowledgeable is to be aware of any fact that can logically be known. Big-G God's knowledge is immediately available and does not need to pass through channels of memory or thinking.
(7) Therefore, big-G God is immediately aware of every real number and of any possible set of real numbers. This data set is of size aleph-2, placing a lower bound on big-G God's maximal knowledge. Observe that big-G God must be aware of all real numbers because humans can generate any desired real number and big-G God, being maximally knowledgeable, must know everything that is known to humanity.
(8) Now suppose X exists. Then big-G God can canonically embed X into R and then enumerate X. This suffices to prove that X exists, which is a paradox. Observe that X is uncountable, but that this does not mean that no entity, not even big-G God, is capable of considering all elements of X at once; the definition of "uncountable" here simply means that there is no bijection from Z into X. Note that big-G God's ability to perform this task requires no axioms foreign to ZFC.
(9) Therefore X does not exist. However, big-G God could also prove this fact by arranging every subset of R according to cardinality and then demonstrating the gap in the continuum. This is also a paradox. Note again that this task requires no axioms foreign to ZFC, but only the abilities of a supreme being.
(10) To avoid paradox, there must be some set of size aleph-0 or aleph-1 (or perhaps both) of which big-G God is unaware. But according to (7), this cannot be the case if big-G God is a supreme being. Therefore big-G God is not the supreme being, and no such entity can exist. QED.
By the way, I posted this at Christianforums.com, which you can find here with lots of dialogue. Credit to the user zippy2006 over there who, despite driving me insane with his bad logic, helped me refine the proof. I didn't post here for you guys to see until I had a formalized proof.
(1) Assume ZFC, which is the Zermelo-Fraenkel collection of axioms with the axiom of choice added.
(2) Observe that the law of non-contradiction, which is assumed in ZFC, is equivalent to the law of excluded middle. This means that no coherent statement can be neither true nor false unless we drop the law of non-contradiction. Explicitly,
(3) Gödel's first incompleteness theorem, which states that any self-consistent, non-trivial logical system S must contain some true statement that is unprovable in S, applies to ZFC.
(4) One such statement is the continuum hypothesis, which states that there exists no set whose cardinality is greater than that of the integers but less than that of the real numbers. Formally, there is no X such that |Z|<|X|<|R|. Observe that X must either exist or not exist according to (2).
(5) A little-g god is, as YouTuber Aron Ra puts it, a magical, anthropomorphic immortal. There can be many little-g gods, and little-g gods cannot be proven to not exist. However, big-G God is the supreme being, and there can only be one or zero of these. (If there were two big-G Gods, and they disagreed on an aspect of creation, whose will would be reflected in reality? The existence of multiple big-G Gods necessitates that they are all of the same will and indistinguishable in any way whatsoever, which is to say that there is only one of them.)
(6) According to (1), big-G God cannot be omnipotent because big-G God cannot create, for example, a circle with a vertex. Omnipotence results in paradoxes, so big-G God is defined as maximally supreme. To be maximally powerful is to be able to perform any task that is not logically impossible. To be maximally knowledgeable is to be aware of any fact that can logically be known. Big-G God's knowledge is immediately available and does not need to pass through channels of memory or thinking.
(7) Therefore, big-G God is immediately aware of every real number and of any possible set of real numbers. This data set is of size aleph-2, placing a lower bound on big-G God's maximal knowledge. Observe that big-G God must be aware of all real numbers because humans can generate any desired real number and big-G God, being maximally knowledgeable, must know everything that is known to humanity.
(8) Now suppose X exists. Then big-G God can canonically embed X into R and then enumerate X. This suffices to prove that X exists, which is a paradox. Observe that X is uncountable, but that this does not mean that no entity, not even big-G God, is capable of considering all elements of X at once; the definition of "uncountable" here simply means that there is no bijection from Z into X. Note that big-G God's ability to perform this task requires no axioms foreign to ZFC.
(9) Therefore X does not exist. However, big-G God could also prove this fact by arranging every subset of R according to cardinality and then demonstrating the gap in the continuum. This is also a paradox. Note again that this task requires no axioms foreign to ZFC, but only the abilities of a supreme being.
(10) To avoid paradox, there must be some set of size aleph-0 or aleph-1 (or perhaps both) of which big-G God is unaware. But according to (7), this cannot be the case if big-G God is a supreme being. Therefore big-G God is not the supreme being, and no such entity can exist. QED.
Jesus is like Pinocchio. He's the bastard son of a carpenter. And a liar. And he wishes he was real.
