Being interested in different forms of relativism, I am interested in logical paradoxes and tonight watched a video on Godel. — Gregory

From the rest of your post I gather that you either misunderstood a sensible video or else believed a nonsensical one. Can you please link the video so I can determine which is the case? Nothing in your post remotely corresponds to Gödel's incompleteness theorems.

From what I know, his incompleteness proofs are very dependent on the use of human language. — Gregory

Not so. Gödel's incompleteness theorems are works of pure mathematical logic. They could be, and have been, presented purely symbolically. They could be valdated by a computer. Can you explain why you erroneously believe what you said? Perhaps you saw a video that compared incompleteness to the liar paradox. This is a handwavy and inaccurate description. The incompleteness theorem is simply about the limitations of axiomatic systems in determining mathematical truth.

I am not sure there can't be an intelligent life form, following what Quine wrote on that, which couldn't, if it thought purely in numbers, prove to itself all of mathematics. — Gregory

Can you explain what Quine wrote about intelligent life forms violating known results in mathematical logic? What would it mean to think purely in numbers? Are we doing sci-fi speculation? Kirk and Spock encounter an alien race that thinks purely in numbers, and Kirk must find a way to make it confuse itself before it destroys the Enterprise? While putting the moves on Yeoman Rand? Quine wrote about this? Reference please. What you said truly makes no sense in the context of incompleteness.

Such an ideal might be impossible for a human, but if Spinoza were right God is closer us than we are to ourselves, — Gregory

Anything has distance zero from itself, and no distance culd be less than that. It's difficult to know what Spinoza could have meant here unless he's speaking metaphorically. What exactly are you talking about and what does this have to do with a technical result in mathematical logic?

But you did say something interesting that gets to the heart of your misunderstanding of the first incompleteness theorem.

ergo if we are God we could prove all of mathematics. — Gregory

No. Let us assume, for sake of discussion, that:

* There is a Platonic realm of mathematics in which every well-formed mathematical statement has a definite truth value. For example the Continuum hypothesis (CH) is either true or false. There either is or isn't a set whose cardinality lies strictly between that of the natural numbers and the real numbers.

* Let us further assume that God has access to this Platonic realm.

* If you will grant me that God, for all his or her powers, is nevertheless bound by the laws of reason and logic; then God knows the answer to CH, but God can NOT prove CH from the axioms of ZFC; nor from any axiom system whatever that doesn't essentially posit either the affirmation or denial of CH. And even then, that new axiomatic system must itself still be incomplete, unless it is inconsistent.

What the first incompleteness theorem says is that no axiomatic system complex enough to model the arithmetic of the natural numbers, can be both complete and consistent. Even God must be bound by this result as long as God is contrained by logic and reason.

It may well be true that God knows the truth value of CH; but God can not PROVE CH from any axiomatic system that does not already include a new axiom (beyond ZFC) that implies CH. And that new system must itself be either inconsistent or incomplete.

Let me state this again. Incompleteness is about the limitation of axiomatic reasoning to determine mathematical truth. It says that there are always truths outside the reach of any axiomatic system (for sufficiently complex axiomatic systems). That's all it says. God would be bound by it even if God knew all mathematical truths. I hope I've made this clear.

Whatever video you watched either grossly mis-stated incompleteness, or you grossly misunderstood it.

At least that is how I see it, because relativism for me is such that the relative is in Becoming, never in the Absolute. — Gregory

You're simply misunderstanding incompleteness. It says nothing about relativism or absoluteness. Gödel himself was a Platonist and believed that CH had a definite truth value. All he did was show the limitations of axiomatic reasoning to determine mathematical truth.

Now as for sets, each one can be considered nominally as a succession of units — Gregory

You are confusing sets with well-ordered sets. And even well-ordered sets are not entirely built from succession, but also by taking upward limits. Your concept of sets is greatly at odds with how mathematical sets are understood. What you said here can't be made to correspond with mathematical set theory at all.

But if a set is a succession of units, how can you describe the set of real numbers in this manner?

or taken as a whole so that it includes itself. — Gregory

A set that contains itself violates the axiom of regularity. No set contains itself in standard set theory.

It is true that mathematicians study non well-founded set theory, but this is off to the side of mainstream set theory.

I think of the latter as including the nature of the set. — Gregory

If you prefer non well-founded set theory that's fine, but I suspect you are just making up your own concept of sets.

The set of all sets that do not contain themselves would be a series, maybe infinite, of individual units. — Gregory

Sets have no inherent order. Once again you are making up your own concept of sets that have nothing to do with sets as commonly understood.

The"set of all sets that do not contain themselves" is not a valid set because it is a set specification that does not conform to the standard rules of set formation; in this case, the axiom schema of specification. Specification says that to form a set from a predicate, we have to first start with a known set and then cut it down via the predicate. We can't just state the predicate without any enclosing set, without creating a contradiction. That's what Russell's paradox shows.

The set of all sets that do contain themselves likewise does not require that a set contain and not contain itself. — Gregory

True enough, if only by accident. The set of all sets that contain themselves is the empty set, in the presence of the axiom of regularity.

It would have merely all the individuals in addition to their groupings. — Gregory

I'm afraid that doesn't mean anything. What are groupings in this context?

But I am open to relativism in mathematics and if someone has a proof of it i'd be interested. — Gregory

All these proofs are on Wikipedia. I can't imagine what video you saw that gave you these wildly inaccurate ideas, which can best be characterized as "not even wrong."

I get that you're sincere, but your ideas are not in accord with mathematics nor do you seem to understand what incompleteness is about.

Godel seemed to mesh language with numbers so tightly that Becoming seemed to enter mathematics. — Gregory

You have stated this twice without context, argument, justification, or evidence. And it's not true. The burden is on you to explain yourself.

However, the Absolute in my opinion must be consistent with proving everything logically and mathematically. — Gregory

Why? Does the Absolute get its marching orders from misunderstood Youtube videos? The fact is that axiomatics are not sufficient to determine all mathematical truth.

Maybe I am entering Leibnizian territory, — Gregory

Can you be more specific? Leibniz did a lot of different things.

but I find this topic to taste like steel and I like it. — Gregory

What? What does this even mean? What does steel taste like? What does it mean to enjoy the taste of steel? What does that have to do with anything? I enjoy the taste of burritos, but that doesn't imply any particular result in set theory.

Unless you mean set theorist John Steel. I like puns too, were you testing your readers?

If we understand ambiguity in language, I don't see how Godel or Russell could prove that contradiction lies in the heart of numbers — Gregory

But nobody claims they did any such thing. Russell showed that naive set theory, in particular unrestricted comprehension, leads to a contradiction. And Gödel showed that no axiomatic system complex enough to include the natural numbers can be both consistent and complete.

Neither result bears on "contradictions in the heart of numubers." That's something you're making up; either as a result of misunderstanding a video, or believing a bad video, or both.

Incompleteness simply says that axiomatic systems (of sufficient complexity) are not capable of determining all mathematical truth, unless they're inconsistent.

The set of all sets that do contain themselves likewise does not require that a set contain and not contain itself. It would have merely all the individuals in addition to their groupings. — Gregory

The 'set' of all sets...is a contradiction since it is not a set. But it can be an infinite collection of sets. The paradox is superficial and only exists because it is assumed that this set exists when it doesn't.

Thanks for the detailed response. The videos I've seen on Godel is the In Our Time one from philosophyoverdose and the ones from actualized.org, all on Youtube. I was under the impression that Godel used the ambiguity of the Liar Paradox to formulate codes in mathematics. This reminded me of Russell's paradox. In common language "The set of all sets that do not contain themselves" would contain itself if it didn't contain itself. A set containing itself would be very strange though and I don't think this paradox is a contradiction in numbers but in language. IF Godel's theorems were based on language, then they could be revised like Russells theorem. A fault in language should have no effect on numbers. I did say God (whether we are God or not) should be able to prove everything in mathematics. Self reference might be an illegal move in mathematics and could possibly be godel's problem. I don't know. Actualized.org had a video on relativism and mentions a Quine paper from the 20's where he discusses talking with aliens. Godel, being a Platonist, is setting up a whole theory he thinks is true for all species and divinities for all time. But an alien might have a numbering system wherein there is no self reference. As a genuine question, is it possible Godel put too much of human language into mathematics?

I think it is common knowledge that Leibniz envisioned a machine that could prove everything in mathematics and logic. I don't know what work he wrote this in. I was just looking through that video I mention for the Quine paper, but haven't found it yet. But if we say aliens can prove the consistency of mathematics and all of it, or God, or us in God-mode, it doesn't really matter. My point was that we need to strip math of all language (Becoming) and let it stand naked as the Absolute. Russels paradox shows how distorting language can be to numbers and maybe even the idea of set is based on language instead of numbers. I used sets as a "group of units" which is legit because that's it's definition. The set seen apart from the units is like 6 seen not as 1 1 1 1 1 and 1. A set containing itself seems like language encroaching on numbers, which you might agree with. I don't see why we need the word set at all or self reference either. Math should work seamlessly and I hope it can prove itself someday

Godel attempted to prove two things:

1) you can't prove that math is true and consistent

2) there are parts of math you can never prove from any group of axioms

This is the minimum of what he attempted to prove from everyone Ive asked and from all the sources I've seen

The comment I made about steel goes back to my reading of a book called LSD and the Search for God. It mentioned someone on acid who said he could taste the categorical imperative and that it tasted like steel (that is, as absolute). When I read this I thought "that is how I feel about mathematics!" and I went on to read a book about Post Modernism and Mathematics, and now I am in the middle of another book called Why is There a Philosophy of Mathematics at All. I am much more interested in the philosophy of mathematics than in the equations, I admit, and would rather read Heidegger of the foundations of mathematics than study equations. Having said that, I see that Russell's paradox refutes the very
idea of having sets UNLESS we give do credit to how language works. Post modernism has all along said that mathematics is based on relative language use, and although I somewhat agree with them on this, I do believe there could be a way to get to an objective mathematics that could show its truth and perhaps prove everything that is provable in its field. I have not misused what a set is because it's not complicated. Everyone says a set is a collection of members. 6 is 1, 1, 1, 1, 1, and 1 taken together as a whole (I said nature but I should have said whole to be clearer). It's a set of 6 1's. The first 10 whole numbers can be taken as a group and can be a set. For a set to include itself, I agreed, is weird. That is why I asked if it can be shown that a set can include itself since i find this very awkward but also interesting. The Liar paradox is not identical to Godel's theorems. I know that. But I sense that there is something in language, like in Russell's paradox, that is tripping people up and encouraging them to hold Godel's position. If someone did math an LSD, for example, the situation of language's use in mathematics might completely change. We have to be very careful with language. That's all I am basically saying

When I read this I thought "that is how I feel about mathematics!" and I went on to read a book about Post Modernism and Mathematics, and now I am in the middle of another book called Why is There a Philosophy of Mathematics at All. — Gregory

When you finish this last book, let us know the answer, please. :roll:

Thanks for the detailed response. The videos I've seen on Godel is the In Our Time one from philosophyoverdose and the ones from actualized.org, all on Youtube. I was under the impression that Godel used the ambiguity of the Liar Paradox to formulate codes in mathematics. This reminded me of Russell's paradox. In common language "The set of all sets that do not contain themselves" would contain itself if it didn't contain itself. A set containing itself would be very strange though and I don't think this paradox is a contradiction in numbers but in language. IF Godel's theorems were based on language, then they could be revised like Russells theorem. A fault in language should have no effect on numbers. I did say God (whether we are God or not) should be able to prove everything in mathematics. Self reference might be an illegal move in mathematics and could possibly be godel's problem. I don't know. Actualized.org had a video on relativism and mentions a Quine paper from the 20's where he discusses talking with aliens. Godel, being a Platonist, is setting up a whole theory he thinks is true for all species and divinities for all time. But an alien might have a numbering system wherein there is no self reference. As a genuine question, is it possible Godel put too much of human language into mathematics? — Gregory

I want to say something about all this (and your other two posts) but it seems too difficult to me to respond point by point and the more I mull it all over, the farther back all this becomes. So let me just write down a couple of thoughts that are on my mind and maybe something will resonate. This isn't meant to be a comprehensive response to everything (or anything) you wrote; it's just a few thoughts that I might as well write down now to avoid the risk of never writing anything at all.

1. Why should man-made proof be able to approach what God knows about math? What is a proof? A proof is a finite-length string of symbols. Just as "The Tao that can be told is not the eternal Tao," the truths that can be proved are not all of the truths. God knows all the mathematical truths; and humans are restricted to finite-length strings of symbols. Why on earth would you or any human arrogantly expect that we can reach God's knowledge with finite strings of symbols that we ourselves made up?

I am arguing here that your expectation that we should be able to prove everything that God knows, is unrealistic and unreasonable. After all, we should not expect to know everything that God knows using our puny human minds; any more than a caterpillar or a bat can know everything God knows.

Icarus couldn't fly with wings made of feathers and wax; and we can not know what God knows using only finite strings of symbols.

2. Of more interest to our discussion than Gödel's first incompleteness theorem, is Tarski's undefinability theorem. Tarski proved (and in fact Gödel proved on his way to proving his own theorems) that no system can define or characterize all its truths.

Let me give a paraphrase of the idea. Consider the natural numbers. God surely knows every true fact about them. In fact God keeps a list of all true facts about the natural numbers: 2 + 2 = 4, 3 x 5 = 15, "there is no natural number solution to $x^n + y^n = z^n$ for n > 2," and so forth.

Each of those truths can be written as a formal statement in the language of math; and each formal statement of math can be assigned a positive integer, its Gödel number. I'll omit the details but just ask if you're curious. It not difficult to show that every formal mathematical expression can be uniquely assigned to some positive integer.

So God knows a set of positive integers that encode all mathematical truths and nothing else.

What Tarski proved is that there is no mathematical statement or formula or algorithm that can crank out or describe that set of numbers! That is, mathematical truth is essentially random.

This, I think, gets at the heart of what you're saying better than banging on Gödel's incompleteness theorem, about which there is already way too much misunderstanding in the world. We cannot write down a formula that describes all the truths of a system. God knows those truths, but they lie beyond any formulaic enumeration. That's a fact, not a mistake or a quirk of language.

3. This bit about the liar paradox. The liar paradox is about semantics. incompleteness is about syntax. Incompleteness is about the limits of what we can do by pushing formal symbols according to formal rules. Analogizing it to the liar paradox is on the one hand helpful, because it's easier to visualize. But it's also very much NOT helpful, because syntax is not semantics and incompleteness is NOT the liar paradox.

The Wiki article on Gödel's incompleteness theorems explicitly mentions this point:

"Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section ..." My bolding. It's a bit like the rubber sheet and bowling ball visualization of Einsteinian gravitation. On the one hand it's a nice visualization because it illustrates how mass distorts spacetime in a way that can be understood by a child. On the other hand, it's profoundly wrong. What makes the bowling ball push down on the rubber sheet in the first place? Meta-gravity? Of course not. The bowling ball and rubber sheet is a popularized analogy that falls apart if you think about it much.

Likewise, the liar paradox is an analogy. It's not the incompleteness theorem. The former is semantic; the later, syntactic. The Wiki article expands on this point:

"It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, Alfred Tarski."

In short, your belief that incompleteness is just the liar paradox is erroneous. The liar paradox is a conceptual aid to understanding, but it's not what incompleteness is about. You're confusing the bowling ball and rubber sheet model with Einstein's actual theory of general relativity.

4. There are other versions of the incompleteness theorem that don't even involve the liar paradox as an analogy; they're completely different approaches. Among these are Rosser's trick and Chaitin's proof of the incompleteness theorem. From the Wiki article on incompleteness:

"Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any system that can represent enough arithmetic, there is an upper bound c such that no specific number can be proved in that system to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox."

These are technical matters that I'm not qualified to discuss and that I don't mean to throw at you. I only mean to say that incompleteness is a deep truth about formal systems that has been approached and proved in several different ways, and not all by analogy with the liar paradox.

To sum up:

* Your claim that incompleteness is a word game is simply false. You're confusing syntax with semantics, and helpful analogies with the more complicated ideas they analogize. I daresay the authors of the videos you viewed may well have themselves been confused on this point.

* Why the heck should man have any hope of knowing what God knows, from just using our pitiful finite strings of symbols? God is not nearly so limited.

ps -- I didn't touch on your misunderstandings of sets. Briefly, what we tell people about sets in high school are not sets. Set is an undefined term. I better say this again so that I'm perfectly clear. There is no definition of what a set is. It's helpful to think of a set as a collection of objects, but you have to realize that this is only an intuitive approximation. Another bowling ball on a rubber sheet.

Sets are characterized by the axioms that say what sets do. Sort of like the Supreme Court's famous definition of pornography, that they know it when they see it. Mathematicians know a set when they see one. But there's actually no definition at all. I'm sure that must come as a suprise but it's true. Sets are not "well-defined collections," or groups of discrete elements lined up like soldiers, 1, 2, 3, etc. Your ideas about sets aren't directly related to incompleteness so I'll leave that for another time. Let me just add that Wikipedia says that "a set is a well-defined collection of distinct objects ..." and that is just factually false. Even their disclaimer that they're talking about naive versus axiomatic sets is a little off the mark. They're really talking about high school sets and those are of no use to us at all.

Thank you, you cleared up my confusion on all this.