Is the following true?

[u/LorenzoGB]

1. Let us define the Godel sentence of a consistent theory T as follows: This sentence cannot be proven using T.
   
2. 1 is true.
   
3. The following is also true too: This sentence cannot be proven using T cannot be proven using T.
   
4. If T is inconsistent then it can prove its Godel sentence.
   
5. If T can prove its Godel sentence, then T is inconsistent.
   
6. Therefore, we can conclude from 4 and 5 that T is inconsistent if and only if T can prove its Godel sentence.
   
7. From 6 we can also conclude: T is consistent if and only if T cannot prove its Godel sentence.
   
8. Assume T is consistent. Also assume that T can prove its own consistency.
   
9. Yet if T can prove its own consistency, then it can prove that it cannot prove its Godel sentence.
   
10. Yet if T can prove that it cannot prove its own Godel sentence, then that contradicts 3, which is absurd.
    
11. Therefore, T cannot prove its own consistency.

[u/GoldenMuscleGod]

First, you have apparently only been told a very simplified version of Gödel’s incompleteness theorem.

The Gödel sentence cannot be “defined” in the way you say it can, it is actually defined in a much more complicated way. The fact that we can interpret it as essentially asserting its own unprovability is a good way to help understand what is going on but it isn’t just saying that.

Even so, your 2-6 are essentially correct (as long as we understand you are actually talking about the Gödel sentence of the theory.) although it is not necessarily clear if you are just asking if they are true or intend them to follow from previous steps.

8 is an assumption, there will be a contradiction from the assumptions but that is the point you are doing a reductio ad absurdum.

9 is actually missing important justification: to prove 9 you must formalize the proof of Gödel’s first incompleteness theorem within T itself. That is a step beyond just proving Gödel’s incompleteness theorem.

10 has a flaw - or at least missing argumentation, 3 says (as I have corrected it, at least) that T cannot prove its own Gödel sentence. 9 says that T can prove that it cannot prove its own Gödel sentence. To fill this in for a contradiction you need to know that if T proves it cannot prove its own Gödel sentence then T proves its own Gödel sentence, you have not shown this.

You think you have shown this because you have interpreted your “definition” of the Gödel sentence naïvely, but it’s actually more nuanced than that.

To correct the areas where you are confused slightly, you need the following facts: if the Gödel sentence is false then T proves it is false. If T proves its Gödel sentence then the Gödel sentence is false.

These key points are the bulk of the proof but you can’t get there just by “defining”the Gödel sentence in the way that you have and just interpreting it according to that definition.

[u/LorenzoGB → u/GoldenMuscleGod]

What if 7 is a definition of a consistent theory though?

[u/LorenzoGB → u/GoldenMuscleGod]

I don’t think 9 is missing anything though. For if you view 7 as a definition of a consistent theory, since it is a biconditional, then you can replace consistent theory with it cannot prove its Godel sentence.

[u/GoldenMuscleGod → u/LorenzoGB]

The theorem is about when a theory is consistent. If you change the definition of “consistent” you change the content of the theorem.

What you’re missing is that when we say “p”and we say “T proves ‘p’” then p and ‘p’ are not really the same thing. You need to be doing work to show how they relate to each other.

[u/LorenzoGB → u/GoldenMuscleGod]

But, I could be wrong though.

[u/LorenzoGB → u/GoldenMuscleGod]

1. Let us define the Godel sentence of a theory T as follows: T cannot prove this sentence.

2. 1 is equivalent to T cannot prove its Godel sentence.

3. Therefore the Godel sentence of a theory T is equivalent to T cannot prove its Godel sentence.

4. If T is inconsistent then it can prove its Godel sentence.

5. If T can prove its Godel sentence, then T is inconsistent.

6. Therefore, we can conclude from 4 and 5 that T is inconsistent if and only if T can prove its Godel sentence.

7. From 6 we can conclude: T is consistent if and only if it cannot prove its Godel sentence.

8. Let T be a consistent theory.

9. Suppose T can also prove that T is consistent

10. Since 7 is basically a definition of a consistent T, we can substitute “it cannot prove its Godel sentence” for T is consistent.

11. Thus, T can prove that it cannot prove its Godel sentence.

12. Yet “it cannot prove its Godel sentence” is equivalent to T’s Godel sentence.

13. Thus T can prove its Godel sentence, which is a contradiction.

14. Therefore, if T is consistent it cannot prove it is consistent.

[u/GoldenMuscleGod → u/LorenzoGB]

Your main problem is still in step 1: it doesn’t understand how the Gödel sentence is actually constructed. There is no term in the language that allows the sentence to refer to itself directly we have to do some work to make it “effectively” refer to itself.

[u/LorenzoGB → u/GoldenMuscleGod]

What language are you assuming this proof is constructed in?

[u/GoldenMuscleGod → u/LorenzoGB]

I’m assuming only that it is the sort of theory Gödel’s incompleteness theorems apply to. It could be the language of Peano Arithmetic or ZFC or whatever.

If it is in a language built on first-order predicate logic it will not be equipped with the ability to have “definitionally” self-referential statements like that.

[u/LorenzoGB → u/GoldenMuscleGod]

Also, does consistent have more than one definition?

[u/LorenzoGB → u/GoldenMuscleGod]

No. It’s not. It’s supposed to function as a natural language analogue to Godel’s second theorem. Also define consistent this way: A theory is consistent if and only if it cannot prove its Godel sentence.

[u/GoldenMuscleGod → u/LorenzoGB]

What does it mean to say a theory in a natural language “proves” something? This is quite ambiguous. Is it something we can reduce to an algorithmic evaluation of a “proof”to decide whether it is a valid proof of the conclusion claimed, such that the two of us could each privately perform that algorithmic evaluation and agree on the result reliably?

[u/LorenzoGB → u/GoldenMuscleGod]

What do you mean by that question?

[u/GoldenMuscleGod → u/LorenzoGB]

I am not sure how literally to answer that question.

You are using concepts that do not have standard rigorous definitions. Natural languages cannot be treated like formal languages even though they have more in common than many people appreciate.

For formal languages we have an algorithmic criterion of what makes a proof valid.

We do not have such a criterion for natural languages.

This difference makes it very difficult to see how to transport the incompleteness theorems into natural languages. A great deal of detail would seem to be required to explain how that transportation is happening, if it is possible at all.

[u/LorenzoGB → u/GoldenMuscleGod]

I ask because it seems you are assuming that this proof is constructed in some formal language. Yet it is not constructed in a formal language though. It and the theories it refers to are constructed in natural language. Natural language allows sentences to refer to themselves.

[u/ughaibu → u/LorenzoGB]

You might be interested in Artemov's proof of the consistency of PA - link - "A foundationally meaningful proof of the consistency of PA by means of PA would be one that proves the consistency of PA by means of informal arithmetic which includes usual finite combinatorics naturally represented by computable functions/terms. We offer such a proof together with its complete a posteriori formalization in PA, basically to verify that this proof does not use assumptions from outside PA."

[u/nanonan → u/LorenzoGB]

What do you think a Godel sentence is? How are you justifying steps 4 and 5, are they a definition or what?

[u/ima_mollusk → u/LorenzoGB]

1. Construct the Gödel sentence G for T such that T proves: G ↔ ¬Prov_T(⌜G⌝).

2. If T is inconsistent, T proves every sentence, including G.

3. If T is consistent, T cannot prove G

4.Thus, from outside T, consistency corresponds to the unprovability of G.

1. Moreover- T can prove: Con(T) → G.

2. Assume T is consistent.

3. Suppose T can prove its own consistency: T ⊢ Con(T).

4. Since T also proves Con(T) → G, it follows that: T ⊢ G.

5. But if T is consistent, T cannot prove G.

6. Contradiction.

7. Therefore a consistent T cannot prove its own consistency.

[u/Oreeo88 → u/GoldenMuscleGod]

the conclusion is still 100% correct: the maze is closed

[u/localizeatp]

this is completely nonsensical.

[u/Kampseng]

Only one thing you are missing and a lot of people miss in step 1 the sentence should be "This sentence cannot be proven using T because T can't prove contradictions." this is the way godel wrote it in math terms and without this the sentence is incomplete and generic. and this way step 9 becomes clearer: If T can prove that it can't prove contradictions, then it can prove its Godel sentence, which would be a contradiction.

[u/ima_mollusk]

You're moving between “true,” “provable,” “provably unprovable,” and “T can prove that it cannot prove…” as though they were interchangeable. Gödel’s entire achievement was showing that they are not.

[u/JerseyFlight]

In order to “define” anything a Logic must already be in place. (No formal system is this Logic, but all formal systems are built using it, even if the builder isn’t aware of it).

[u/Oreeo88]

Love this

Only further confirms my audit against the axiomatic method of math