Sure; you can't see how the argument works.

My question is, what do you do next? Do you re-read and study and find alternate presentations, until you see how the argument works; or do you conclude that all the mathematicians and logicians up to you are wrong?

One choice leads to understanding. The other, to psychoceramics. — Banno

It's called meta-mathematical.

Godel claims any given axiomatic mathematical system is incomplete because there's always going to be a mathematical proposition that's both — TheMadFool

No. Not any. But a lot of them. Your questions and observations do credit to an intelligent 15-year-old, but that's not you, is it. The answers are in the material you already have. You just have to work it though, and Godel is twisty-tricky, but beyond that not so difficult, except in some of the more arcane details, which to the substance are not essential.

You can even research his paper in English translation. It's very readable, and with some effort, understandable. But it's the kind of thing you have to do yourself to some degree at least, if you're going to get it. And making uninformed challenges or claims becomes a waste of your time and energy and everyone else's, if you're trying to learn.

I'm asking a simple question and no satisfactory answer has been offered by forum members.

I hope this dogged persistence on my part is only a minor irritation to you all.

First, let's get the facts straight:

Kinds of mathematical truths:

1. Axioms (assumed)

2. Theorems (proven from the axioms)

Godel sentences, if you don't know what they are, are sentences like "this mathematical statement is true but unprovable in a given axiomatic system".

As you can see, Godel sentences are neither axioms nor theorems which leads me to believe they aren't mathematical truths at all unless...there's a third category of mathematical truth. If yes what is this third kind of mathematical truth?

Last but not least, Godel admits that his method of proof has a connection with the Liar Paradox which, to me, should set the alarm bells ringing.

I'm asking a simple question and no satisfactory answer has been offered by forum members. — TheMadFool

The answer was given back here:

You missed it or misunderstood it.

SO, again, what will you conclude?

It is not easy to wrap your head around this. I sort of understand it, but still struggle with it.

I can recommend two sources which might help you.

First, get a hold of Godel, Escher and Bach by Douglas Hofstadter. it's a great read in of itself with all sorts of very cool insights.

Next, you might want to check this out: https://www.youtube.com/watch?v=GbtNQ7yzo9Y&feature=youtu.be

The example here is analogous to Godel - well sort of.

Godel sentences, if you don't know what they are, are sentences like "this mathematical statement is true but unprovable in a given axiomatic system... As you can see, Godel sentences are neither axioms nor theorems which leads me to believe they aren't mathematical truths at all unless..." — TheMadFool

Try "proposition."

The answer was given back here: ↪boethius

You missed it or misunderstood it.

SO, again, what will you conclude? — Banno

@tim wood @EricH

Simply put, there appears to be a contradiction in Godel's proof.

Try using the phrase "true BUT unprovable" in any other topic of discussion, and don't forget philosophical discourse, and see what response you get from your audience.

For instance if I say the proposition "God exists" is "true BUT unprovable" you would think I'm an idiot or insane or both or something even worse for the simple reason that truth of propositions mandates proofs; I can't say a proposition P is true without a proof, no? To understand what I mean you need look no further than Godel himself; after all, he did prove his incompleteness theorems didn't he? How come then that when Godel slips in a claim that contradicts the very principle that he assumes by attempting to prove his theorem viz. making the statement "true but not provable", everyone seems to ignore it?

Indeed, either Godel is wrong, along with all the subsequent mathematicians and logicians who have agreed with his findings and built on them. Or you have made an error.

And again, what will you conclude?

As @boethius said, "true" does not mean "proven".

Indeed, either Godel is wrong, along with all the subsequent mathematicians and logicians who have agreed with his findings and built on them. Or you have made an error.

And again, what will you conclude? — Banno

That's a different issue. Of course, the chances are next to zero that the entire mathematical community is wrong about the truth of Godel's theorems.

Nevertheless, I'd like to know their reactions to the following make-believe conversation between lowly me and the great Kurt Godel:

Me: Godel's incompleteness thoerems are false. There exists a mathematical axiomatic system that's both complete and consistent. Let's call this last statement of mine S.

Godel: Prove it.

Me: Just like Godel sentences that state things like "this mathematical proposition is true but unprovable" are, well, true but unprovable, my statement S is also true but unprovable.

Godel: But I proved the existence of Godel sentences

Me: So, proof is necessary to establish truth?

Godel: Yes

Me: Then explain how, in a Godel sentence that claims "this mathematical proposition is true but unprovable", the mathematical proposition being considered is true.

Godel: <please speak for Godel> because I can't think of how he'd respond.

"true" does not mean "proven" — Banno

Explain this statement.

There can be true statements that have no proof. Incompleteness shows us an example.

There can also be just "true facts" about numbers and arithmetic that are true and there's simply no proof possible.

For instance, the Collatz conjecture we may simply never be able to prove is true, false, or even undecidable, it just stays unknown (beyond what we can check through computer calculation, which wikipedia says we've done to 87 * 2^60 which seems impressible is minuscule compared to "all numbers"). I.e. it can be "true" but also true that no proof nor proving it's undecidable is possible; some things that "resist refutation" can potentially just stay a big question mark indefinitely. The halting problem is a related concept. — boethius

Forget for the moment Godel's case, let me ask you a simple question: Can you say the phrase "true but unprovable" in a conversation without raising eyebrows?

Sure.

I've argued as much elsewhere. Being true is not the same as being proven, justified, or believed

Let me ask you another question... can "This statement is not proven" be false?

Start by assuming that it has been proven...

Then consider what happens when it is assume to be false.

Sure.

I've argued as much elsewhere. Being true is not the same as being proven, justified, or believed — Banno

Proof is both necessary and sufficient for truth unless we're talking about postulates/axioms. Godel's theorems are missing the proof that is both necessary and sufficient to prove Godel sentences.

Let me ask you another question... can "This statement is not proven" be false?

Start by assuming that it has been proven...

Then consider what happens when it is assume to be false. — Banno

Yes, "this statement is not proven" can be false. Ok, I assumed that it has been proven. Now I consider what happens if it is assumed to be false. We'll arrive at a contradiction. So?

Sure. — Banno

So, you don't mind if I say that it's true that you're the robber that carried out the bank heist last Wednesday but that that's not provable?

Godel's theorems are missing the proof that is both necessary and sufficient to prove Godel sentences. — TheMadFool

So, you are right, and mathematics is wrong.

Then your name is most apt.

Yes, "this statement is not proven" can be false. — TheMadFool

...then there is a proof, and it is true.

The only option is to conclude that it is true and hence not proven.

Hence, being proven and being true are not the very same.

Say what you like. It's not true.

Being true is not the same as being proven, justified or believed.

So, you are right, and mathematics is wrong.

Then your name is most apt. — Banno

No, I'm not claiming that I'm right. What I'm bothered by is the contradiction that takes the following form in Godel's theorems:

1. Godel deems proofs necessary to establish truth because he puts forth a "proof" for his theorems

2. Godel sentences are of the form "this statement is true but unprovable" which implies that proofs are not necessary to establish truth because "this statement" is true but unprovable

1 & 2 = Proofs are necessary & Proofs are unnecessary

Godel is contradicting himself.

This is only a side issue though. The main problem is the statement "true but unprovable". This is self-contradictory

Hence, being proven and being true are not the very same. — Banno

This is partially correct but only in the domain of induction, not deduction and math is completely deductive - if the premises are true and the argument is valid, it's impossible for the conclusion to be false. Ergo, being proven in math implies and is implied by truth.

So there it stands, you have the issue, the solution and the choice.

Yes, "this statement is not proven" can be false. Ok, I assumed that it has been proven. Now I consider what happens if it is assumed to be false. We'll arrive at a contradiction. So? — TheMadFool

Please, reconsider. It can only be true.

...then there is a proof, and it is true.

The only option is to conclude that it is true and hence not proven.

Hence, being proven and being true are not the very same. — Banno

In deductive logic, that includes math, true if and only if proven.

OK, let's try something a little bit more complex. informally:

"this statement is not proven" can be provided with and equivalent well-formed formula in the system that Godel constructed.

It cannot be proven within that system

But if you peruse it again, you will see that it cannot be false.

Here's the proof: If it is false, then there is a proof and it must be true, which would be a contradiction. IF it is true, there is no contradiction Hence, it is true.

Now that proof lies outside of the system constructed by Godel.

How's that?

Edit: you have your proof that "this statement is not proven" is true; and a proof that any system of sufficient complexity is either incomplete or inconsistent.

OK, let's try something a little bit more complex.

"this statement is not proven" is a well-formed formula in the system that Godel constructed.

It cannot be proven within that system

But if you peruse it again, you will see that it cannot be false.

Here's the proof, informally: If it is false, then there is a proof and it must be true, which would be a contradiction. IF it is true, there is no contradiction Hence, it is true.

Now that proof lies outside of the system constructed by Godel. — Banno

Things are beginning to clear up for me. Thanks.

Your argument above according to me is:

G (Godel sentence) = this mathematical statement (Godel number xyz) is not provable. For the moment let's forget the truth claim.

Now assume P = there's a proof of G. Then...
1. If P then G
2. if G then ~P [because G claims it can't be proven]
3. If P then ~P......1, 2 HS
4. P.......assume for reductio ad absurdum
5. ~P....3, 4 MP
6. P & ~P.....4, 5 Conj
7.~P.....4 to 6 reductio ad absurdum

~P = there's no proof for G and so G is true and G states that there's a mathematical statement (Godel number xyz) that's not provable: Godel's first incompleteness theorem there. Is this correct?

That'll do. Cheers.

That'll do. Cheers. — Banno

:ok: Thanks

@Banno A better version of your argument would be:

P = there's a proof of G
G (Godel sentence) = the mathematical statement with Godel number xyz is unprovable

1. ~G > P (if G is false then P is true)
2. P > G (if P is true then G is true)
3. G > ~P (if G is true then P is false)
4. ~G....assume for reductio ad absurdum
5. P......1, 4 MP
6. ~G > G.....1, 2 HS
7. ~G > ~P....3, 6 HS
8. ~P........4, 7 MP
9. P & ~P....5, 8 Conj
10. G.....4 to 9 reductio ad absurdum

No need to acknowledge this post unless you see errors.