What I don't like is people spouting about Godel's theorem without knowing what it is. If you're not up to finding out what Godel's theorem is then at least be up to saying so — TonesInDeepFreeze

Godel's theorems prove that, in the form in which we think now, mathematics is either inconsistent or has infinite propositions that can't be proven. It's undecidable which of these are true for Godel.

I don't see how someone can find this to be a satisfactory idea to rest in. Ideas of spiritually are not fairy dreams. They are some of the deepest thought you can have

Your way of thinking is contingent perhaps though, although it seems logical to you. Spiritual pursuits search for higher necessary knowledge and is still philosophy, actually is more philosophy than analytical philosophy.

Not everyone who comes to this forum is into analytical philosophy. You've called Hegel rubbish but some like him, as I do. He certainly thought every truth could be "sublated" until everything is known. If there will still be truths in mathematics that can't be proven, they will be seen as to why this is the case and the whole of truth can find a consistent point of rest. Even if we can't know every higher truth, there can be truth as a whole found in life. I don't see Godel's ideas as consistent with finding THE truth

I don't see Godel's ideas as consistent with finding THE truth — Gregory

And because the unprovable mathematical ideas would have to axioms known by intuition. If they are a connection of ideas and there is no way to get from one to the other, this blocks knowledge as a whole from find the truth of everything

Nothing to be done.

...with a supreme effort @Banno succeeds in pulling off his boot. He peers inside it, feels about inside it, turns it upside down, shakes it, looks on the ground to see if anything has fallen out, finds nothing, feels inside it again, staring sightlessly before him.

Well?

Imagine some unprovable proposition. Can it be *understood* intuitively like axioms are and be taken as axioms?

I'm not sure what you think an unprovable proposition is. Can you give an example?

Gödel offered a proof that math is either inconsistent or incomplete and that the dilemma is undecidable. So unless math is bogus, there will be unproven propositions. I don't know which ones these are but zero in on some mentally for me. Now I ask "are these propositions actually axioms for something else or something else entirely?"

We can't prove axioms

Thanks.

Gödel offered a proof that math is either inconsistent or incomplete — Gregory

Indeed, that's a generalisation of the first theorem. In a given formal system complex enough to do arithmetic there are statements which can neither be proved nor disproved. Assuming the system is consistent (surely not an unreasonable thing to do?) then it must be incomplete - it must contain unproven statement.

SO you asked

Can it (some unprovable proposition) be *understood* intuitively like axioms are and be taken as axioms? — Gregory

I just don't know what to make of this. An axiom in a formal system is a statement that is true within the system, and hence is not understood intuitively but in the terms given by the system. Given some set of axioms sufficient to the task of arithmetic, there will be some statements that are neither among the axioms nor among the theorems. But these cannot be taken as axioms without changing the formal system.

Perhaps you could set me right here?

Given the existence as uttered forth in the public works of Puncher and Wattmann of a personal God quaquaquaqua with white beard quaquaquaqua it is established beyond all doubt. Hope that clears it up.

We can't prove axioms — Gregory

Another axiom?

His hat!

So we have the set of propositions that can be proved and are therefore true. We have the set of propositions that are not true. And we have the set that their truth value is undecidable. And we have a set of propositions which are true (do we know from intuition of axioms?) but unprovable.

Is it being said that within this infinity of unprovable propositions in math each proposition can be analyzed and their meanings understood? What prevents someone from finding the golden thread going thru such a proposition?

Please, just do some reading. Find out about your topic.

But no head.

I don't seem to be able . . . (long hesitation) . . . to depart.

Such is wife.

Ok then

Gödel offered a proof that math is either inconsistent or incomplete and that the dilemma is undecidable. — Gregory

That's not Godel's theorem. You don't know what Godel's theorem is.

Previously asked:

Gödel was trying to find a way to make a line in between what can be known and what can not
— Gregory

Where did you read that? — TonesInDeepFreeze

Gödel was trying to find a way to make a line in between what can be known and what can not — Gregory

According to Roger Penrose Godel was a “very strong”mathematical platonist, so even if proof leads to an infinite regress, you can read Godol’s theorem as perfectly compatible with an absolute god-given grounding for. math.

Well if all I've studied on this is wrong then we have a similar situation as with Bell's theorem where there is no consensus whatsoever of what theyre about. I've watched all the videos I could find on it, read about it in books, and discussed it with people who have computer science degrees. What you are saying is that there is massive misinformation on this but then why haven't you written a couple paragraphs here saying what Gödel really did. I don't believe in self reference in math or logic but maybe you can make a presentation of it will be interesting and fruitful

You did all that and managed still not to know what Godel's theorem is.

why haven't you written a couple paragraphs here saying what Gödel really did — Gregory

It's not required for pointing out that you don't know what Godel's theorem is.

But I will indulge you:

The Godel-Rosser theorem may be given a modern statement as:

If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T.

I don't believe in self reference in math or logic — Gregory

You don't know the actual nature of the "self-reference" in Godel's proof. The proof may be formulated in finite combinatorial arithmetic. If you have a problem with the proof, then you have a problem with finite combinatorial arithmetic.

Imagine some unprovable proposition. Can it be *understood* intuitively like axioms are and be taken as axioms? — Gregory

That is a question that could be asked only by someone unfamiliar with the basics of this subject.

If P is a closed formula, then there is a system S such that P is an axiom for S.

Bertrand Russell was famous for his mathematical ideas. But his paradox is false. Group items together, make a circle around them, and you have a set. A set containing itself is just bizarre, coming from a desire for exotic knowledge, and yes mathematicians aren't perfect. What I said about Gödel was based on what the majority of people have said about from what I personally have seen. Someone needs a really good background in math to read his actual papers so most of us are getting our ideas from second hand sources. Anyway, you can't prove that a set can contain itself from math itself, so rejecting Russell's paradox is a good way to start in approaching Godel. A set containing itself IS self reference

But [Russell's] paradox is false. — Gregory

Every contradiction is false in every model. So what?

Meanwhile, it is a theorem of first order logic that there is not an x such that for all y, y bears relation R to x if and only if y does not bear relation R to y.

A set containing itself is just bizarre — Gregory

So what? Neither Russell nor Godel depended on a claim that there is a set that is a member of itself.

What I said about Gödel was based on what the majority of people have said — Gregory

As Seinfeld put it, "Who are these people?" Whatever "majority of people" you talked with, your conversations did not supply you with a even a fraction of a decent understanding of Godel's theorem.

Someone needs a really good background in math to read his actual papers — Gregory

His original papers are rather old-fashioned in their notation. More recent textbooks have pedagogically supplanted the original papers.

so most of us are getting our ideas from second hand sources — Gregory

Since I don't know your sources, I can't say whether the fault is in the sources or in your misunderstanding of them.

you can't prove that a set can contain itself from math itself — Gregory

In set theory, we prove that there is not a set that is a member of itself.

However, with set theory without the axiom of regularity, there is not a proof that there is not a set that is a member of itself.

And, dropping regularity, but adding a different axiom, there is a proof that there is a set that is a member of itself.

so rejecting Russell's paradox is a good way to start in approaching Godel. A set containing itself IS self reference — Gregory

Those two sentences alone are proof that you are completely mixed up and ignorant of what Godel's theorem is.

/

To understand this subject properly, one should learn basic symbolic logic, then a small amount of basic set theory, then an introductory course in mathematical logic - either in a class or by self-study.

Meanwhile, the best book about Godel's theorem for everyday readers:

Godel's Theorem: An Incomplete Guide To Its Use and Abuse - Torkel Franzen.

That book will disabuse you of your confusions.

To repeat, since you skipped this:

I don't believe in self reference in math or logic
— Gregory

You don't know the actual nature of the "self-reference" in Godel's proof. The proof may be formulated in finite combinatorial arithmetic. If you have a problem with the proof, then you have a problem with finite combinatorial arithmetic. — TonesInDeepFreeze

Apparently even for you Gödel's theorem is hard to put into words. You can't provide what the theorem says, since you say I don't know it properly, in a few paragraphs. As I said, I've seen sources on this for years. I watched the Veratasium video two weeks ago and it said what I've heard everywhere else. My point in this thread is that if there are unprovable propositions, they don't exist in weird loopy ways but have a straightforward reason for why they are closer to axioms than from what is probable. When I studied Euclidean geometry in college our teacher kept telling us to see the golden thread in each proposition and how it runs from the first to the last. Russell's paradox is a different species of thinking. All it takes is a conversation to reveal what someone means if they present you with a paradox like that. It was a linguistic problem, not a logical one

even for you Gödel's theorem is hard to put into words — Gregory

What? It's not at all hard for me. I did it a few posts ago!

If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T. — TonesInDeepFreeze

Please let me know that you see it now so that I may know that I'm not posting to an insane person. Then I'll see about correcting yet more of your ignorant confusion in your post above.

So you are saying that Godel's examples of things that are unprovable do not require a loop in them? As I see it, unprovable things can be 1) axoims which we understand intuitively as unprovable but which make sense ("common sense" comes in) as the basis of a system, or 2) propositions that are unprovable but which can be understood by intuition (thus knowledge is fully knowable), or 3) loopy statements like Russell's paradox that are really fallacious logically.

I am always willing to learn new things, but you wrote:

If P is a closed formula, then there is a system S such that P is an axiom for S. — TonesInDeepFreeze

Couldn't you just have said "systems have axioms"? That is all that says! This is my problem with the whole symbolic logic stuff. They get into problems and call things paradoxes because they don't converse with adult conversation language. We should be truly speaking about truths, not fitting them into structures which confuses matters. We have crazy people try to PROVE there is a God from modal logic ("ontological argument"). It's just ridiculous that people would even consider trying to do this. I think very fluidly and I don't get a pleasant sensation from a paradox that just reverts back on itself. And you say:

If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T. — TonesInDeepFreeze

In real human language, you are saying that a theory has a part of it is and is not a part of it. Again, key word is "recursive". I don't understand why anyone would want to think about logic eating itself like a snake eating its tail. That kind of stuff gives me a headache. It's not cool

You have so many misconceptions. But let's take one thing at a time, starting here: Do you recognize that I did state Godel's theorem?