Sorry I just gasped at the comments about "Cantor's program". I meant Hilbert's program. Such a dumb and stupid mistake... — Wallows

Better to have stayed w/Cantor. Hilbert's program got demolished by Gödel.

Because first, Gödelian incompleteness does not apply to physical theories. — fishfry

Even though I formally agree with your views on the matter, I still want to point out what Stephen Hawking wrote on the subject:

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted. — Stephen Hawking, Gödel and the end of physics

Hawking may indeed be overstepping boundaries there. What he says, is also not genuine syntactic entailment, but rather some kind of wholesale "intuition". Furthermore, Hawking is also not respectful of the plethora of fine print surrounding Gödel's incompleteness theorems.

But then again, I was just pointing out that more or less "serious" opposite views also exist!

It applies (loosely speaking) to axiomatic systems of a particular logical structure, that support mathematical induction. Secondly, incompleteness is not a statement about mathematical truth. It's a statement about axiomatic theories. — fishfry

Yes, that is indeed some of the fine print. Gödel's incompleteness is a statement that is even only about Peano-Arithmetic-like (PA) first-order axiomatic theories, and not even about all first-order theories.

There are weaker first-order theories that are complete (and consistent):

Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete. Gödel's theorem applies to the theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic. — Wikipedia on the completeness of arithmetic

Within the confines created by a lot of fine print, stronger, second-order theories can also be complete (and consistent):

As mentioned above, Henkin proved that the standard deductive system for first-order logic is sound, complete, and effective for second-order logic with Henkin semantics, and the deductive system with comprehension and choice principles is sound, complete, and effective for Henkin semantics using only models that satisfy these principles. — Wikipedia on the possibility of completeness in the context of second-order logic

So, yes, agreed, it is quite easy to overstep the boundaries of Gödel's incompleteness theorems by incorrectly applying them where they do not apply. One really has to read the fine print!

General question, does Godel assume negation or cancelling precedes positing? He ends up saying there are incorrect equations that can be used properly noneleless, bringing monsters to the math party

Hilbert's program got demolished by Gödel. — fishfry

Well, no, disagreed. Hilbert merely received a negative answer to one of his many questions ...

Some of these axioms will remain unprovable as long as the theory is incomplete. — Wallows

Oh boy, here we go again. Might want to reconsider; it's theorems not axioms that could possibly be true, but not provable. :roll:

As long as the theory is consistent, then, one can always add new axioms to the theory to expand its power and magnitude. — Wallows

Adding axioms does not necessarily increase the power of a theory, but it pretty much always increases the amount of trust that the theory requires.

For example, number theory can "see" all the Gödelian numbers representing the theorems and their proofs in set theory. So, number theory "knows" all theorems in set theory, but there are quite a few of these theorems that it does not trust. So, set theory is not necessarily more powerful than number theory. It could just be more gullible!

If Godel is a Platonist, is he sabotaging it with math\logic? How is his approach different from the " indentification problem"

Benacerraf thought platonic objects were like a cathedral, where there has to a certain pattern. However patterns spiral into uncountable infinities, and the ultimate mathematical truth would be a beatific vision of... what?

We make models of reality, for example mathematical formulas that portray some aspect of the complex reality around us. — ssu

If you make a model of reality, then you are engaged in an empirical discipline (such as science). Such model cannot possibly be an exercise in mathematics any more, because the model-theoretic model for a theory in mathematics is NEVER the real world. Such mathematical real-world theory would be the elusive theory of everything (ToE) to which we do not have access, and do not even expect to ever have access.

The problem is truly about crossing boundaries between mathematics and downstream user disciplines of mathematics (such as science), which merely use mathematics to maintain consistency in their use of language. These downstream disciplines talk about something else. They do not talk about mathematics. They merely use mathematics.

Furthermore, such empirical discipline always requires its own regulatory framework that duly constrains what exactly it is willing to talk about. They do not use just mathematics either. There is always also a completely native bureaucracy of rules.

Beauty trumps Godel

Or Godel is beautiful. I bet the Godel supporters get hung up reconciling Wittgenstein with linguistic relativism. Reading GEB..

Oh boy, here we go again. Might want to reconsider; it's theorems not axioms that could possibly be true, but not provable. :roll: — jgill

Well, I tend to think that axioms can be theoretical in nature? Have you ever encountered such a sentiment in your line of work?

We make models of reality, for example mathematical formulas that portray some aspect of the complex reality around us. Fine, but the problem of subjectivity comes with when that model itself has an impact on what it's modelling. Then it has to model itself into the model. Now you might argue that this can be still modeled and in many cases it surely can be, but not when the 'correct' answer is something that the model doesn't give. — ssu

Hence inter-subjectivity? The observer effect seems to play a role here.

But, that aside, I can see the point of utilizing some ideas from Godel to justify the need for us, as a species, to slow down, as there doesn't seem to be a light at the end of the tunnel. You can thank Godel for that.

If you make a model of reality, then you are engaged in an empirical discipline (such as science). — alcontali

Or social sciences, like economics.

Such model cannot possibly be an exercise in mathematics any more, because the model-theoretic model for a theory in mathematics is NEVER the real world. — alcontali

Uh, the math used in the models have to be correct. Yes, math used as a tool (as you point yourself also), but just as all tools, you have to use it correctly.

Furthermore, such empirical discipline always requires its own regulatory framework that duly constrains what exactly it is willing to talk about. They do not use just mathematics either. There is always also a completely native bureaucracy of rules. — alcontali

Say that to an economist.

No really, math is a very useful tool. It really is. We have to be logical in every field of study. Math has a huge advantage in being logical. Something like statistics, which one can argue isn't pure math, is truly an inherent part of many fields of study. Think about how much we use a thing called a mean average.

I think that something as profound like the incompleteness results of Gödel (and others) has also huge implications to math as a tool and our use of mathematical models in picturing reality. Put it another way, if you have problems to mathematically model some phenomenon, some event, and cannot make a mathematical function y=f(x) out of it, perhaps the problem lies is that the correct model would be something that falls into the category of the incompleteness results, in a way is a Gödel number the Gödels incompleteness theorem (is it the first theorem?) talks about.

This might be confusing to understand, but what I mean that there would be something that we have extreme difficulties in modelling with a mathematical formula. Then what would we do?

We do have an unmathematical method. And what is that?

We use narrative: "First happened this, which then lead to that". How we got from this to that and onward cannot be explained by a variable and made into a function (used in it's broadest definition possible). We tell a story. Best example of this is the field of History. There hasn't been much use of math, except statistics, in history and the field of cliometrics hasn't given much to the field. And many don't even consider history as a science. Yet history is one of the most important things for social sciences and our understanding of ourselves and our societies is firmly based on history.

So I personally believe that the incompleteness results have far more to give to our understanding of the World than now is thought.

Hence inter-subjectivity? The observer effect seems to play a role here. — Wallows

Yes, the observer plays the dominant role.

The observer cannot simply observe without interaction in this case. Hence the inescapable subjectivity. When the correct answer is what the observer doesn't give, how could the observer give then the correct answer? In the end we are a part of the universe, hence the idea of a Laplace's Demon is utterly wrong: not everything can be extrapolated from all knowledge and understanding of the present, if the observer is part of the universe. And the remarkable thing here is that this isn't in conflict with determinism. There is a correct extrapolation of the future (that Laplace's demon ought to know). The basic line, which is not understood now, is that this is because of mathematics, the incompleteness results.

In 2008 a guy at NASA (now at Santa Fe Institute) called David H Wolpert showed that the "problem" (it isn't a problem, you know) was incompleteness results, or in the most simple way, Cantor's diagonalization. Wolpert had written about it even far earlier. Once the observer is part of the universe, it cannot avoid this problem. The problem is that this is not understood or it simply hasn't broken ground in the wider scientific and mathematical community.

For example, economist and game theorist like Oscar Morgenstern understood the problem in economic forecasting and gave his example of Holmes and Moriarty (in 1928), where he showed that there's no possible way for Holmes to reason what Moriarty will do. A counterargument given by another future Nobel-laureate to Morgenstern noted, quite correctly, that there had to be a correct forecast, but then didn't grasp that here was the real issue, that there is a correct answer, but it's uncomputable.

Another ignorant answer is to avoid the problem by assuming premises that give you a min-max game theoretic answer.

In all, this is really a field which should be studied and understood. Perhaps even this meager forum can do something about it.

But, that aside, I can see the point of utilizing some ideas from Godel to justify the need for us, as a species, to slow down, as there doesn't seem to be a light at the end of the tunnel. You can thank Godel for that. — Wallows

Or perhaps to speed up, improve our understanding of reality.

You see, talking about this subject gets people simply to the defensive. If you go and say: "Hey, these incompleteness result have a HUGE ROLE in things!" it comes of as anti-science or something. And the first idea is that this person is showing a problem. As if to attack against science. And science has to be defended. Hence the emphasis goes to avoid it, go past. Try to show that the "problems" can be overcome. And then what is left out is that here is an extremely profound mathematical insight that we ought to understand.

perhaps the problem lies is that the correct model would be something that falls into the category of the incompleteness results, in a way is a Gödel number the Gödels incompleteness theorem (is it the first theorem?) talks about. — ssu

It would mean that all sentences that are provable in the theory can also be confirmed to be true in its standard model-universe-world (=semantic completeness) but that there are also facts in that standard model-universe-world that are true but that are not provable from the theory (=syntactic incompleteness).

So, the situation with natural-number theory (PA) is:

• In theory provable means in practice true. (semantic completeness)
• In practice true does not necessarily mean in theory provable (syntactic incompleteness)

The main problem will be that an empirical theory will not even be semantically complete. Concerning physics, Hawking said the following in that regard:

But we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are describing. Thus a physical theory is self referencing, like in Godel’s theorem. One might therefore expect it to be either inconsistent or incomplete. The theories we have so far are both inconsistent and incomplete. — Hawking in Gödel and the end of physics

In my opinion, just like Hawking said, even semantic completeness is already an unattainable ideal for an empirical discipline.

By the way, the proof for Carnap's diagonal lemma does indeed use a self-referencing expression:

Let f: N→N be the function defined by:

f(#(θ)) = #(θ(°#(θ))) — Wikipedia, start of the proof for Carnap's diagonal lemma

However, Carnap's diagonal lemma is itself not self-referencing, and Gödel's first incompleteness theorem is neither. The entire self-referencing thing is just a hack to get the proof going. I don't understand Hawking's obsession with the "self-referencing" thing. Alan Turing uses a similar hack to get the proof for the impossibility to solve his Halting Problem going. It is not that the Halting Problem itself would be self-referencing.

It's more like: You have a function that computes something about any function. Ok, let it now compute this about itself. Look, it explodes when it does that. So, the overly generalized claim that says x,y,z ... is wrong about what it computes.

That strategy is much more of a hack than anything else ...

However, Carnap's diagonal lemma is itself not self-referencing, and Gödel's first incompleteness theorem is neither. The entire self-referencing thing is just a hack to get the proof going. I don't understand Hawking's obsession with the "self-referencing" thing. Alan Turing uses a similar hack to get the proof for the impossibility to solve his Halting Problem going. It is not that the Halting Problem itself would be self-referencing. — alcontali

As I earlier said, many don't see the subtle difference between Russell's paradox and Gödels (or Turings) finding. Wittgenstein accused Gödel of finding the paradox again and didn't hit it so well with Turing either. It's basically indirect reference.

Just as the real number that Cantor shows cannot be in any list. Of course, we do see the relation between the real number and the list of real numbers. Same thing. Not a paradox.

Hence the issue is about the limitations of computing, modelling, giving proofs. Not that there would be a correct model or proof. And the observation that non-computable mathematical objects can be very important to us. Even if we cannot compute them, they still are mathematical.

So math rules, I guess.

As I earlier said, many don't see the subtle difference between Russell's paradox and Gödels (or Turings) finding. — ssu

Yes, agreed. Russell's sentence actually is self-referencing.

Just as the real number that Cantor shows cannot be in any list. Of course, we do see the relation between the real number and the list of real numbers. Same thing. Not a paradox. — ssu

Yes, agreed. As far as I am concerned, there is indeed nothing paradoxical about Cantor's theorem (countable versus uncountable infinite cardinalities).

His proof strategy, diagonalization, is certainly an interesting approach. I usually like it. For example, the diagonalization of the term "heterological" is really intriguing. The argument in the video below (just 3 minutes) is incredibly simple, pure math, and leads to a rather unexpected (surprising) result:



Diagonalization does not have to be hard (but, of course, sometimes it is ...)

Diagonalization does not have to be hard (but, of course, sometimes it is ...) — alcontali

EXACTLY!!!

You see, going through Gödel's Incompleteness Theorems are hard. Turings proof with a Turing Machine isn't also the most easiest thing. Even Gödel had to really think about it before he agreed that Turing's findings were equivalent with his theorems. But luckily in math things can be simplified.

But you are totally right, one should talk about in general about diagonalization. Referring to Gödel's incompleteness Theorem is confusing, because the theorems aren't easy to understand.

And of course, from Cantor's diagonal argument we next get to the Continuum Hypothesis, which is, well, a kind of Holy Grail in Math.

Thank you for that link. That was a brilliant explanation.

Who proved the rule that if there is an uncountable infinity in addition to another countable infimity, it is greater than another countable infinity. Is that what Cantor proved diagonallly? Someone can say infinity is just infinity still

Oh boy, here we go again. Might want to reconsider; it's theorems not axioms that could possibly be true, but not provable. :roll: — jgill

Well, I tend to think that axioms can be theoretical in nature? Have you ever encountered such a sentiment in your line of work? — Wallows

To be more specific, it's certain mathematical statements that could be undecidable, including theorems. There are a few examples in combinatorics and number theory, but I'm not aware of such things occurring in complex analysis, for example. If you come across something let me know. As for axioms themselves, I don't think Godel applies. Again, let me know if you come across a contrary opinion.

In the actual work of mathematical research outside of set theory it seems very rare that one might encounter incompleteness. But I am familiar only with my own interests. :cool:

Why limit Godel? Why not let it run wild throughout math? Pi does not have repetition in it, so is ill-ordered in a sense. Since it starts on the left side always in the same way, going off into infinity, it might be a pattern, if you believe in such things. it can be controlled. Russell's paradox is not a paradox . Whether we say "the sets of all sets that ARE members of themselves" or "the set of all sets that are NOT members of themselves" is null. A set being in containing itself is what it is about, and this is possible. IF the set is wellordered. And sets don't fall in a necessary way says the Platonic identification problem, so there are options here in containing Godel too

Since it starts on the left side always in the same way, going off into infinity, it might be a pattern, if you believe in such things. — Gregory

There is no repeating pattern, yet we can instantly notice it in it's decimal form 3,14159 26535 89793.... And you have even ways to calculate it.

If it would be repeating, then it would be a rational number and then you could square the circle (if I'm correct). Transcendental numbers are interesting. In the real numbers they are the numbers that with you get an infinity that cannot be mapped 1-to-1 to the natural numbers. Algebraic numbers you still can.

Is that a coherent explanation for why we still go on or should we stop at some point and realize some deeper truth? — Wallows

Yes, that seems to me to be a fair characterisation of why Godel's incompleteness theorem is no barrier to progress in mathematics in most areas of interest and use. The only maths I know of it having killed was Hilbert's project to try to prove maths to be complete and consistent.

If math is infinite, it could be understood by an infinite intelligence. Godel is claiming that even God could not prove math is self consistent. What a strange creation of God then, like randomness. Instead of 4 for fire, 6 for earth, 8 for air, and 20 for water, we have random spontaneity for all four. Dasein. Theoretical physics is actually philosophy. So that was Hawkings point. He was the best philosopher at physics ever!

For Plato, the pattern that holds 4, 6, 8, and 20 together is 12. Sounds right to me. Those thinkers get confused with 7's versus triangles. But the 12 thing feels syllogistic. I don't think you can prove irrationality from logic structures and math equations alone. Who shaves the barber who shaves all and only those who shaves themselves. Through out the misleading "all" and walls the barber can't shave himself

So, yes, agreed, it is quite easy to overstep the boundaries of Gödel's incompleteness theorems by incorrectly applying them where they do not apply. — alcontali

There's a book devoted to exactly that. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzen. I have a copy, I should leaf through it.

Adding axioms does not necessarily increase the power of a theory, but it pretty much always increases the amount of trust that the theory requires. — alcontali

Adding any axiom that is independent of the other ones always increases the power of a theory. By definition that means it can prove more theorems. It can prove all the original theorems plus itself, as proof.

For example, number theory can "see" all the Gödelian numbers representing the theorems and their proofs in set theory. So, number theory "knows" all theorems in set theory, but there are quite a few of these theorems that it does not trust. — alcontali

Are you saying that number theory "knows" what theorems are true; but it does not necessarily have proofs for all of them? If so then I agree.

There's a system that contains exactly all the true theorems of number theory. It's called true arithmetic. Tarski's theorem says that this class is not arithmetically definable, which is why it can sneak under Gödel. But conceptually the class exists and is a consistent and complete account of arithmetic. As I understand it, anyway ... I don't know any more about it than the Wiki page.

Hilbert's program got demolished by Gödel.
— fishfry

Well, no, disagreed. Hilbert merely received a negative answer to one of his many questions ... — alcontali

@andrewk says killed and that's good enough for me! :-)

The only maths I know of it having killed was Hilbert's project to try to prove maths to be complete and consistent. — andrewk

Thanks! I always confuse Hilbert's program with Hilbert's 23 questions anyway.

Even though I formally agree with your views on the matter, I still want to point out what Stephen Hawking wrote on the subject:

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.
— Stephen Hawking, Gödel and the end of physics — alcontali

I am constantly struck by the fact that physicists in general but particularly many celebrity physicists with TED talks and Youtube channels, don't know the first effing thing about math. Hawking, rest his soul and give him all the credit for all his amazing accomplishments, doesn't know the first thing about this subject.

First, he says: "whether we can formulate the theory of the universe in terms of a finite number of principles?" So he does not know that ZFC has infinitely many axioms, and that axiomatic systems with only finitely many axioms are barely of interest as far as I know. He's mixing up physical theories and mathematical logic into a mishmash to suit his confused purposes.

Secondly, when is says, that "... a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted," he is leaping from:

* A physical theory is a mathematical model;
* Therefore since SOME mathematical models are subject to incompleteness -- [and surely not physical models!!] -- therefore a physical theory must be subject to incompleteness.

This is a leap of bad logic and bad thinking. Most likely any physical axiomatic theory would not be subject to incompleteness. You would not accept this fallacious syllogism from an undergrad in logic 101.

If Hawking thinks there's a physical theory subject to incompleteness, he must think that this theory can represent the infinite set of natural numbers. Hawking is claiming that an actual infinity exists in the physical world. Else incompleteness doesn't apply.

To say this another way: In any finite domain, to determine whether a given proposition is true we just go enumerate every possibility and look. All finite systems are complete. We have a proof by examining cases for any question we could ask.

The only way to have incompleteness is if your domain is infinite. And if Hawking is making that claim, he ought to give evidence and consider the implications; not just write a bad syllogism to sell books.

If there is an actual infinity in the world: Why aren't physics postdocs applying for grants to study the physical truth or falsity of the Continuum hypothesis? Until that happens I know that physicists are talking through their hats about infinity, as Hawking is doing here.

What strikes me is that an amateur forum poster like me who studied math back in the day, knows far more about this subject than the great Stephen Hawking. And that this happens to me all the time when I watch celebrity physicist videos. I've seen many physicists make egregious math errors especially as regards set theoretic matters and the nature of the real numbers. Physicists I respect. Sean Carroll did it the other day.

So, uh, sorry for the rant there but that's what popped into my head when I read that quote.

Aristotle to Aquinas to Scotus thought an eternity in the future to be impossible to complete, — Gregory

So then you agree that the physical world does NOT instantiate an actual infinity. Didn't you imply otherwise earlier, or did I misunderstand?

Good points fishfry. An eternally falling piano with only it in existence and with no beginning.. is possible! Math doesn't work this way though

My understanding of Brouwer is that only those things that are capable of an infinity of proof are properly mathematical. The intuition part comes into play for the "observer"