I think the most exciting part of Gödel's proof is the following part (which I'm citing from "Gödel's Proof" by Ernest Nagel and James Roy Newman, Routledge, 2005, p. 72):

> For, although the formula G is undecidable if the axioms of the system are consistent, it can nevertheless be shown by meta-mathematical reasoning that G is true. That is, it can be shown that G formulates a complex but definite numerical property which necessarily holds of all integers—just as the formula ‘(x) ∼ (x + 3 = 2)’ (which, when it is interpreted in the usual way, says that no cardinal number, when added to 3, yields a sum equal to 2) expresses another, likewise necessary (though much simpler) property of all integers. The reasoning that validates the truth of the undecidable formula G is straightforward. First, on the assumption that arithmetic is consistent, the meta-mathematical statement ‘The formula ‘(x) ∼ Dem (x, sub (n, 13, n))’ is not demonstrable’ has been proven true. Second, this statement is represented within arithmetic by the very formula mentioned in the statement. Third, we recall that metamathematical statements have been mapped onto the arithmetical formalism in such a way that true metamathematical statements correspond to true arithmetical formulas.

The first and the last phrase in this quoted paragraph interest me because I don't understand them. Is there something like a "theorem of metamathematics" I'm missing?

Meta: Crossposted on both r/mathematics and r/logic.

Proof of Gödel's Incompleteness Theorem in seven steps.

Peano Arithmetic (PA) is a formal proof system. Statements and proofs must follow a very strict syntax.

Step 1: Encode sentences in PA as numbers (regardless of if they have a free variable or not).

Step 2: Encode proofs in PA as numbers.

Step 3: If x encodes a sentence F with a free variable and y is a number, let sub(x,y) be the number encoding F(y) (i.e. y substituted into the free variable of F).

Step 4: Let P(x) be the sentence "the sentence encoded by x (has no free variables and) has no proof."

Step 5: Let n encode the sentence (with a free variable) P(sub(x,x)). Then n is simply a number.

Step 6: Note that sub(n,n) encodes P(sub(n,n)).

Step 7: P(sub(n,n)) is true but unprovable.

A proof is defined as a finite sequence of statements, each of which is either an axiom or computably the result of applying a rule of inference to at most three of the previous statements.

On the list of Rules of Inference at Wikipedia, we find no less than 28–30 (or 18 in the table) separate rules of inference that should count.

I'm not looking for an absolute or effective lower bound in some given conception of Peano Arithmetic, just a nice low number (or rather, specific set of rules) that's low enough to be resonable to expound that it's computable (I don't care about primitive recursive, just computable) to check that each given statement follows from the previous (or is an axiom).

To that end, I might redefine a proof as a list of pairs (statement, reason), where the statements are the usual lines of a proof, and the reasons mention either which axiom or exactly which rule of inference and which lines and substitutions were used, exactly in the way in which high-schoolers learn to write up their proofs in Euclidean geometry.

I just don't want to have to go on about umteen separate rules of inference if I don't have to, and if the reader doesn't have to read about it.

I've written about this before, but I thought I'd give it another shot. This is not a complete explanation or a rigorous proof, just a quick overview.

In short:

>Not OK:

>If you plug "If you plug x into itself, the resulting sentence is false" into itself, the resulting sentence is false

>OK (or even unavoidable):

>If you plug "If you plug x into itself, the resulting string of symbols cannot be obtained from the axioms" into itself, the resulting string of symbols cannot be obtained from the axioms

To explain slightly, the first one is equivalent to "This sentence is false", a paradoxical sentence. We don't want paradoxes, and we can set up rules that prohibit us from saying sentences like that.

On the other hand, that second sentence is equivalent to "This sentence is unprovable (i.e. can't be obtained from the axioms)". Gödel's main achievement was to show that any formal language that can express sentences about numbers can express that second sentence. (The key is to encode strings of characters as numbers.)

For example, even seemingly harmless systems like Peano Arithmetic (a formal proof system, in which sentences and proofs must follow a very strict syntax) can express that second sentence. That's why I said "unavoidable": you can disallow the first sentence by limiting what you can say, but you can't disallow the second sentence (unless you're so limited you can barely say anything at all).

Why is this a problem? Well, assume Peano Arithmetic is consistent, meaning that if you can prove something using PA's axioms then it's true (you can't prove a falsehood). If PA can express the sentence "This statement can't be proven from PA's axioms", is that sentence true or false?

Have that as an exam question tomorrow and I have no idea how to answer solve it. Any help is useful.

It seems a lot of what could potentially be unprovable statements(eg. Goldbach's conjecture) could be proven with such a oracle. If there are any, what is the nature of the type of statements that could be unprovable even given such an oracle?

Introduction

Most people here probably have heard about Gödel's incompleteness theorems, and if not then this post may be a nice introduction to these fascinating theorems. First I will give an idea of what these theorems are, and how they are proved. Then in more detail I will explain why we often technically falsely use his second incompleteness theorem.

Hopefully, this will ignite an interesting discussion about the incompleteness theorems. All questions or remarks, be it very basic of advanced, are very much welcomed!

Gödel's first incompleteness theorem

The statement of Gödel's first incompleteness theorem is roughly:

We cannot write down a system of axioms for arithmetic that is complete.

So what does that mean? A system of axioms should be clear I think. Being complete means that every possible statement can be either proved or disproved using these axioms. In other words: for any system of axioms for arithmetic we try to write down, we will be able to find some statement that we cannot prove nor disprove.

Edit: by being able to "write down" a system of axioms I will mean that it is recursive. In other words, it means that there is a computer program that, given a certain statement, is able to check whether or not that statement is part of the axioms. Essentially this means we can actually explicitly write down what the axioms are.

Gödel's first incompleteness theorem, more detailed

Let's be a bit more precise. We will work in Peano Arithmetic, abbreviated as PA. Just Google it if you want a list of its axioms (or try Wikipedia), but the precise list may not be too interesting. The important thing is that we have addition and multiplication and constants 0 and 1. Then the axioms will basically list the basic properties of those operations. For example, we have an axiom saying ∀x(x + 0 = x) and another one saying ∀x¬(x+1 = 0). Furthermore, there is an axiom scheme that says we can do proofs by induction.

Of course, the natural numbers are a very natural model for PA, and we will call this the standard model. It should also be clear that in PA, we can express every natural number. For example, we can express 3 as 1+1+1.

It is not too hard to think of a way to encode logical formulas as numbers. A silly example may be: just write down the formula using a computer, then it is just a string of 1s and 0s, which we can also interpret as a natural number. It does not really matter how

Proofs don't fall out from the sky; there usually is some motivation to thinking that some conjecture is true which then leads to discovery of its proof. So, prior to proving them, what motivated Gödel to think his theorems were true?

Gödel Incompleteness theorem was one of the most important result in mathematics in 20th century.It can be compare to the Heisenberg uncertainty principle in physics more or less.In simple term it states that there are some problems in mathematics which are undecidable or problems that could be true or false but impossible to prove that means unprovable.

Has anyone wondered about a formal reason of why Gödel’s incompleteness theorems have such wide applicability beyond the realms of what they were actually proven for? Do they in anyway connect Kant’s ideas about categories and the way they limit what can be known? I apologize for any incoherencies in my question. I am just a neurobiology student so have very limited knowledge about Mathematical knowledge and Kant’s work as a whole.

Greetings,

I am a math student at the Master's level but never looked at mathematical logic. I'm interested in becoming a little more acquainted with mathematical logic, to the extent that I'd like to have a grasp on the essential points and techniques needed to prove the incompleteness theorem. Can anyone suggest an efficient sequence (could have singleton range) of books/papers to read, to accomplish this goal?

Thanks!

Gödel's incompleteness theorems are the two theorems that are formulated based on various paradoxes in Logic that were postulated in various forms around the World, in the centuries preceding.

Gödel's incompleteness theorems(GIcT here on) puts serious limitations on how much Logic we can squeeze into the Material/Physical World. This could ultimately be applicable to Consciousness which is our window to all the existence. GIcT holds true wherever there are absolute paradoxes.

Now that could pose a particular problem to Mind Uploading and such stuff but in my opinion, that's just looking at one part of implications of the GIcT. GIcT also points towards Panpsychism(Consciousness is all that exists but at different levels) and Immaterialism, which both provide alternatives to Physical Mind Uploading. Besides, GIcT doesn't posit any particular theistic interpretations. It just points to something beyond and the theistic interpretations are still subject to the Logic that we reason even withoit the GIcT.

So in my view, it's just a different approach and doesn't make Mind Uploading impossible. Opinions?

This seems like a chicken and egg situation to me. Gödel's Theorem proves that any axiomatic system is incomplete, you can't prove the axioms, if you can you can prove anything and the system is circular and meaningless, I get that (even if my phrasing is inaccurate, please don't get hung up on this part).

BUT, if Gödel's Theorem was proven using math... shouldn't it mean that is was proven using a certain axiomatic system?, if so, why does it apply to any axiomatic system?, wouldn't it be possible to construct an axiomatic system in which Gödel's theorem does not apply?.

Roses are chocolates, violets are flowers. This statement's not true. I have super powers.