Proof that there is no God

[Nihilist Virus]

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,

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Let

"~" be "not"
"•" be "and"
"v" be "or"
"A" be a coherent statement

The law of non-contradiction, which states that A cannot be both true and false, is expressed as

~(A•~A)

Distribution of the negation operator yields

~A~•~~A

Observe that ~•=v and ~~A=A. Therefore the above reduces to

~AvA

This is the law of excluded middle: A must be true or false.

We now start with

~AvA

Applying a double negation yields

~(~(~AvA))

Distribution gives us

~(~~A~v~A)

This is logically equivalent to

~(A•~A)

Thus

~(A•~A) <==> ~AvA

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(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.

[chimp3]

In terms of explaining reality philosophy has made as much progress as religion in the last 500 years.

[The Valkyrie]

My proof is far simpler.

Every weekday, millions of people pray that there were fewer Monday and more Fridays in the week, or something that will shorten their working week.

And yet it never happens. Still one Monday and one Friday per week.

COme on, "god", you're all powerful, you could make it happen!

[Angrboda]

I was with you, right up until the point you started speaking Swahili.

[Nihilist Virus]

I think the potentially fatal flaw of the proof is the fact that there is not and cannot be a Gödel number for God's proof of X's existence or nonexistence, possibly rendering the ideas here moot.

Gödel numbers are positive integers, and there is no way to exhaustively enumerate X and then ecode that into a Gödel number. Thus God's awareness or exhaustive enumeration of X, presuming in this case that it exists, is inexpressible by Gödel's machinery and thus it can be argued that Gödel's theorem does not apply.

There is also, of course, no way to assign a Gödel number to the idea of arranging all subsets of R according to cardinality, and yet such an arrangement is necessary for God to prove that X does not exist.

An analogy might go like this:

While it is impossible to accelerate up to the speed of light, it is not forbidden to travel at the speed of light. Similarly, while it is forbidden to effectively enumerate or construct X, immediate and inexplicable knowledge of X might not be forbidden.

It still is the case, though, that God cannot be omniscient. It's just that the potential for maximal knowledge might still be there, allowing for the possibility of a supreme being.

[ignoramus]

Can the same logic prove my invisible life giving dragon in my basement isn't real?

That's a very wide net you're throwing there...

[Edwardo Piet]

I followed until point "8)". I feel there are some undefined premises thrown in there. Like what is "R" in that point?

At point "8)" I would have simply said that if omnipotence cannot include doing logically impossible things then G-God is no more powerful than the logical absolutes themselves which makes the logical absolutes as all powerful as G-God which is effectively a second G-God. And since there can't be more than one G-God but there must be if the logical absolutes are equal to G-God then we have a logical contradiction and therefore G-God cannot exist.

A response could be that God is identical to the logical absolutes.... but then God isn't actually a mind or a creator of the universe.

Unless the logical absolutes have a sort of hidden mind that is in fact the mind of G-God.

So I still don't see a full disproof. But who the fuck needs one?

We could say G-God is impossible if and only if the G-God that exists then gave us contra-causal free will... since such a version of free will is logically incoherent... but nowhere in the Bible does it specify that the type of free will he supposedly gave us is contra-causal

[The Valkyrie]

Can the same logic prove my invisible life giving dragon in my basement isn't real?

That's a very wide net you're throwing there...

[Nihilist Virus]

Can the same logic prove my invisible life giving dragon in my basement isn't real?

That's a very wide net you're throwing there...

No it cannot and no it isn't.

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I followed until point "8)". I feel there are some undefined premises thrown in there. Like what is "R" in that point?

At point "8)" I would have simply said that if omnipotence cannot include doing logically impossible things then G-God is no more powerful than the logical absolutes themselves which makes the logical absolutes as all powerful as G-God which is effectively a second G-God. And since there can't be more than one G-God but there must be if the logical absolutes are equal to G-God then we have a logical contradiction and therefore G-God cannot exist.

A response could be that God is identical to the logical absolutes.... but then God isn't actually a mind or a creator of the universe.

Unless the logical absolutes have a sort of hidden mind that is in fact the mind of G-God.

So I still don't see a full disproof. But who the fuck needs one?

We could say G-God is impossible if and only if the G-God that exists then gave us contra-causal free will... since such a version of free will is logically incoherent... but nowhere in the Bible does it specify that the type of free will he supposedly gave us is contra-causal

R refers to the set of all real numbers. Don't bother dissecting the proof - it's flawed as I said.

[Edwardo Piet]

Yeah of course it's flawed lol.