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l33tminion) wrote2005-02-13 11:38 pm
l33tminion) wrote2005-02-13 11:38 pm
Entry tags:
Wondrous Isomorphism
So, here's that philosophical post that's been percolating for a while. I'm tried to reduce it down to be as simple as possible, but it's still pretty long. Still, I think this is rather interesting, so bear with me, please. (Note that I'm not trying to prove anything here, just exploring some ideas I find interesting, and that I hope you will also find interesting.)
Definitions:
1. Formal system- A logical system with a finite number of axioms (things taken to be true without proof) and rules of inference that define a set of theorems (things taken to be true with proof).
2. TNT- Typographical Number Theory, a formal system. The theorems in this system express truths about natural numbers (0, 1, 2, 3, ...)
3. Isomorphism- A correspondence between two things that are similar (functionally the same in some way).
4. Godel's Theorem- Formal systems must either be inconsistent or incomplete.
5. ~ means "not" and theorem definitions are in the form "Name: Definition".
TNT, G, and ~G
TNT theorems can be encoded as natural numbers. Therefore, you can use TNT to say things about TNT. One of these things is Godel's string, the reason that TNT is incomplete:
G: G is not a theorem of TNT.
Another way to view this is in terms of proof:
G: G cannot be proven within TNT.
If G is a theorem, it is false (and, therefore, not a theorem). If G is not a theorem, then G is true, which means that TNT is incomplete.
G, like all TNT strings, has an opposite:
~G; G can be proven within TNT.
So, there are two ways that you can try to complete TNT. You can add G as another axiom or add ~G as another axiom. If you add G, you get a system TNT+G. Unfortunately, this is no more complete because you can still find another theorem that shows this system is still not complete.
Meta-G: Meta-G cannot be proven within TNT+G.
Let's look at the other possibility. If you add ~G to TNT, the system becomes quite interesting. At first you might think the system exhibits direct inconsistency. Within TNT (and like systems) you can prove "If P and ~P then Q" (where P and Q are anything). The practical upshot of this is that a contradiction of the form "P and ~P" (a direct inconsistency) causes such a system to explode! You can prove that everything is true and false at the same time!
~G asserts G is provable, so you'd think that would allow you to prove "G and ~G" causing TNT+~G to disappear in a puff of logic. But that is not so. Despite the assertion that G is provable, you won't find such a proof no matter how long you search. It isn't provable for 0, for 1, for 2, for 3, etc. How does this make sense in terms of natural numbers? The isomorphism the system suggests is that there are some supernatural number or numbers (which cannot be found, which is rather convenient as finding them would explode the system). Supernatural numbers would act exactly like natural numbers, except you could use them to prove G. Because G is provable (although not proven) the system is complete!
However, Godel's theorem asserts this system must be inconsistent, and it is. It asserts that a proof exists that cannot be found. This type of inconsistency is called omega-inconsistency.
Theism and Atheism
Once you have this, there are some really cool isomorphisms that can be found. One is Atheism and Theism:
TNT- Reality (I know that people's views of reality are not consistent, but assume they are for the purpose of this exercise)
G- Atheism: God does not exist (The existence of God cannot be proven)
~G- Theism: God does exist (The existence of God can be proven)
(By God, I mean something supernatural, not necessarily a monotheistic deity.)
Reality+Atheism is just as incomplete a system as Reality.
Reality+Theism asserts that God can be proven, although not by natural means, so it is omega-inconsistent, but complete (and it lacks the system destroying form of inconsistency).
The Nature of Reality
Another thing to consider is what this might say about the nature of reality in terms of formal systems. Let's assume that reality can be represented completely by a complete system. That leaves a few possibilities:
1. Reality cannot be represented by a formal system because it has an infinite number of axioms (reality is chaotic at a fundamental level). This makes reality theoretically impossible to understand fully, even without barriers to knowledge. As a result, I don't like this idea, although it could well be true.
2. Reality cannot be represented by a formal system because it has an infinite number of rules of inference (reality is chaotic at a fundamental level). Like the above.
3. Reality can be represented as a directly inconsistent formal system like TNT+"P and ~P" (reality is supremely chaotic). Like the above, but infinitely more so. Everything is true and false at the same time. (Note that our minds probably contain things of the form "P and ~P" but, fortunately, our minds, unlike TNT, also contain some sort of incompleteness that contains this sort of logical explosion.)
4. Reality can be represented as a system similar to TNT+~G (reality is omega-inconsistent). This suggests the existence of something supernatural.
What does this tell us about the nature of reality? I think it would suggest this: "Either reality cannot be understood, or something supernatural exists." The meaning of that I leave to your interpretation.
Definitions:
1. Formal system- A logical system with a finite number of axioms (things taken to be true without proof) and rules of inference that define a set of theorems (things taken to be true with proof).
2. TNT- Typographical Number Theory, a formal system. The theorems in this system express truths about natural numbers (0, 1, 2, 3, ...)
3. Isomorphism- A correspondence between two things that are similar (functionally the same in some way).
4. Godel's Theorem- Formal systems must either be inconsistent or incomplete.
5. ~ means "not" and theorem definitions are in the form "Name: Definition".
TNT, G, and ~G
TNT theorems can be encoded as natural numbers. Therefore, you can use TNT to say things about TNT. One of these things is Godel's string, the reason that TNT is incomplete:
G: G is not a theorem of TNT.
Another way to view this is in terms of proof:
G: G cannot be proven within TNT.
If G is a theorem, it is false (and, therefore, not a theorem). If G is not a theorem, then G is true, which means that TNT is incomplete.
G, like all TNT strings, has an opposite:
~G; G can be proven within TNT.
So, there are two ways that you can try to complete TNT. You can add G as another axiom or add ~G as another axiom. If you add G, you get a system TNT+G. Unfortunately, this is no more complete because you can still find another theorem that shows this system is still not complete.
Meta-G: Meta-G cannot be proven within TNT+G.
Let's look at the other possibility. If you add ~G to TNT, the system becomes quite interesting. At first you might think the system exhibits direct inconsistency. Within TNT (and like systems) you can prove "If P and ~P then Q" (where P and Q are anything). The practical upshot of this is that a contradiction of the form "P and ~P" (a direct inconsistency) causes such a system to explode! You can prove that everything is true and false at the same time!
~G asserts G is provable, so you'd think that would allow you to prove "G and ~G" causing TNT+~G to disappear in a puff of logic. But that is not so. Despite the assertion that G is provable, you won't find such a proof no matter how long you search. It isn't provable for 0, for 1, for 2, for 3, etc. How does this make sense in terms of natural numbers? The isomorphism the system suggests is that there are some supernatural number or numbers (which cannot be found, which is rather convenient as finding them would explode the system). Supernatural numbers would act exactly like natural numbers, except you could use them to prove G. Because G is provable (although not proven) the system is complete!
However, Godel's theorem asserts this system must be inconsistent, and it is. It asserts that a proof exists that cannot be found. This type of inconsistency is called omega-inconsistency.
Theism and Atheism
Once you have this, there are some really cool isomorphisms that can be found. One is Atheism and Theism:
TNT- Reality (I know that people's views of reality are not consistent, but assume they are for the purpose of this exercise)
G- Atheism: God does not exist (The existence of God cannot be proven)
~G- Theism: God does exist (The existence of God can be proven)
(By God, I mean something supernatural, not necessarily a monotheistic deity.)
Reality+Atheism is just as incomplete a system as Reality.
Reality+Theism asserts that God can be proven, although not by natural means, so it is omega-inconsistent, but complete (and it lacks the system destroying form of inconsistency).
The Nature of Reality
Another thing to consider is what this might say about the nature of reality in terms of formal systems. Let's assume that reality can be represented completely by a complete system. That leaves a few possibilities:
1. Reality cannot be represented by a formal system because it has an infinite number of axioms (reality is chaotic at a fundamental level). This makes reality theoretically impossible to understand fully, even without barriers to knowledge. As a result, I don't like this idea, although it could well be true.
2. Reality cannot be represented by a formal system because it has an infinite number of rules of inference (reality is chaotic at a fundamental level). Like the above.
3. Reality can be represented as a directly inconsistent formal system like TNT+"P and ~P" (reality is supremely chaotic). Like the above, but infinitely more so. Everything is true and false at the same time. (Note that our minds probably contain things of the form "P and ~P" but, fortunately, our minds, unlike TNT, also contain some sort of incompleteness that contains this sort of logical explosion.)
4. Reality can be represented as a system similar to TNT+~G (reality is omega-inconsistent). This suggests the existence of something supernatural.
What does this tell us about the nature of reality? I think it would suggest this: "Either reality cannot be understood, or something supernatural exists." The meaning of that I leave to your interpretation.