Gödel demonstrated that mathematics is limited in scope, who's to say that physics is in that scope?

I thought that Godel demonstrated that there are theorems in a formal system that cannot be proved in that system. Physics is not expressed with the formalism that is reality, and there are different choices of formalism that can be used to express the concepts of physics. I don't think that Godel showed that there were statements that can't be proven in any system.

> I don't think that Godel showed that there were statements that can't be proven in any system.

He definitely didn't, since we know of systems that are capable of proving any statement. It's sufficient for the system to be internally inconsistent.

Outside of that, we'd need to be a little more clear about what we mean by "there are statements that can't be proven in any system". I would bet that it is not the case that there is a single special statement with a fixed meaning which all consistent systems are incapable of proving. In the general case, I would expect that a statement, taken by itself, would constitute a consistent (if uninteresting) system. That system would be easily able to prove its only axiom.

In my mind, the incompleteness theorem says that a consistent system of "sufficient power" cannot prove its own consistency. We could reify that into a statement that looks like "This system is consistent"; that's what I'm trying to rule out above by referring to "a statement with a fixed meaning". "This system" would refer to different things within different systems.

Wikipedia says this:

> The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers.

> For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.

> The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

This allows for a single statement (say, a result about the natural numbers) that no sufficiently powerful consistent system can prove, but it doesn't require it. The sets of true-but-unprovable statements might be disjoint between two particular systems.

> I would bet that it is not the case that there is a single special statement with a fixed meaning which all consistent systems are incapable of proving.

A typical example -- now that I think about it, the only example I have seen -- is the statement "this statement cannot be proved".

Technically, it is not a single statement, because for different axiomatic systems A1, A2... you get different statements "this statement cannot be proved in A1", "this statement cannot be proved in A2" etc.

For any given axiomatic system A, if the system can prove "this statement cannot be proved in A", it contains a contradiction: it just proved something that is not true. But if the system cannot prove "this statement cannot be proved in A", we have an example of a statement that is true but unprovable (in A).

Gödel's trick was inventing a way how a statement can talk about itself. Like, if you say "it is forbidden for a statement to talk about itself", Gödel asks "but surely it is okay for a statement to talk about another statement that it can define unambiguously?" and then creates a statement that sounds like "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A", but when you actually unpack "7461287618276312876873", you get "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A" again. So it talks about itself without actually using the words "this statement".

What exactly this all means... I wish I would grok, but I don't. But I am pretty sure it means none of the usual "human brain has mystical magical quantum powers beyond mere computation".

A proposition that cannot be proved in a theory T that

axiomatizes its own provability is the following:

      Theorems of T can be computationally enumerated.

[Church 1934] presented the following simple proof that T

cannot computationally enumerate its own theorems:

    In order to obtain a contradiction, hypothesize theorems
    of a foundational theory T are computationally enumerable. 
    Then procedures that are provably total in T are 
    computationally by a procedure that is provably total in 
    T. Consider the Boolean procedure Diagonal defined on 
    natural number input n to be the result of the complement 
    of the result of the ith provably total procedure on input 
    i. The procedure Diagonal differs from every procedure in 
    the enumeration of provably total procedures in T. 
    However, by construction, diagonal is a provably total 
    procedure in T, which is a contradiction.

See the following article for additional information:

https://papers.ssrn.com/abstract=3603021

For a theory T in which proof-checking is computationally

decidable that formalizes its own provability, the following

proposition is true but unprovable:

      Theorems of T are computationally enumerable.

There is a proof the above proposition is true but

unprovable here:

https://papers.ssrn.com/abstract=3603021

Note that the above result is more powerful than the one

quoted in the post above because theorems of T might not

computationally enumerable, which is the case for the

Dedekind axiomatization of Natural Numbers.

That's a really good framing. I like it.

Assume consistency. Take the collections of all true statements about the standard model of the natural numbers and make these statements your axioms. Now every true statement about the standard model of the natural numbers is provable. It’s a useless axiomatic system since there is no procedure for knowing if a statement is an axiom of not (that works for every statement). The second order Peano axioms are categorical and assuming consistency every true statement is ‘provable’.

Higher-order Dedekind/Peano axioms are categorical in the

sense that they characterize Natural Numbers up to a unique

isomorphism. However, there is still a proposition about

Natural Numbers that true but unprovable using the

Dedekind/Peano axioms. Further information can be found here:

"Physical Indeterminacy in Digital Computation"

https://papers.ssrn.com/abstract=3459566

So the last sentence I made is false. The rest is correct, right?

Yes, the last statement that you made is indeed false, that is,

it is not the case that every true proposition is provable in

Dededkind's theory.

In math you can prove statements given a set of axioms. But in physics you can just validate things with experiments.

Odd you were downvoted ! I think that this is a good insight because it underlines physics!=math!=physics.

Gödel showed that a consistent, recursively enumerable axiomatic system capable of producing the standard model of natural numbers will have statements in which that statement is true in some models but not in other models.

In the Dedekind axiomatization of Natural Numbers, there are

no standard models :-)

I was using the first order Peano axioms in my comment. I'm not familiar with the Dedekind axiomatization. Didn't even know such a thing existed but the statement I made holds I believe. The standard model under the first order Peano Axioms exists and I'm assuming it is also a model of the Dedekind axiomatization. The incompleteness theorems thus do apply to the Dedekind axiomatization, right?

There is just one model of Dedekind's axioms up to a unique

isomorphism, which is indeed the standard model. However,

Gödel's incompleteness proofs do not work for Dedekind's theory

because there are uncountable axiom instances. (Dedekind's

theory is nevertheless effective because proof checking is

algorithmically decidable.)

Consequently, the first incompleteness result (inferentially

undecidable) has been proved by other means. See the

following for a proof:

https://papers.ssrn.com/abstract=3603021

The above article explains that Gödel's second

incompleteness result is false because the theory can prove

its own consistency.

When non-specialists use "set" they usually mean "finite set" or "finitely schematized set" or maybe at most "recursively enumerable set."

There's no guarantee the universe is reducible to any of these (although in the likely case the universe is finite in space and time there will be some degenerate case of formulas for everything which "happened" - imo less satisfying than even an approximation via some small set of axiom schemata).

> The goal of a unified theory is to have a single formula.

No, this is ridiculous. Not even electromagnetism is a single formula

Expecting we end up with "a single formula" is a ridiculous strawman.

What we need is explain GR with a foundation in QM (or with a common base, which might - but probably won't - be String Theory)

>Not even electromagnetism is a single formula

"dF = 0" represents all of Maxwell's (homogeneous) equations in terms of "differential forms".

With the right pattern recognition, the right definitions, and the right paradigm shift, you can compress multiple equations into a single equation.

With the right assumptions, ignoring not a few significant details, to a useful order of precision, as long as you don't look too hard.

Maxwell's equations are a nice continuous approximation to whatever it is that really happens.

But it's not a good idea to confuse the map with the territory.

This is more akin to lossless compression than the transcription of a territory onto a map.

A single formula is entirely possible. Simple rules in principle can generate the complexity of the universe we see. Look into the Wolfram Science Project to see how.

Yes and all of his CA's are at least 3 conditional statements, but if you're inclined to count that as one formula with split definitions then anything can be bashed into "a single formula"

> Look into the Wolfram Science Project

Ah yes that "theory" that ignores the complexity and quirks of reality

Would it? As a counter point - consider a universe where all possible things exist. We know from algorithmic information theory that most things are incompressible. Who is to say we don't live in an area within the totality of all possible things that happens to be in part entirely incompressible?

If algortihmic infomarion theory applies to the universe:

1. All possible things won't exist

2. Entirely incompressible areas wont exist becuse that would mean they are completely noise

3. The univese willl have the simplest possbile "generating function", that is, fundamental theory.

1: why? 2: why? 3: why?

Your answer feels dismissive and yet at the same time naive.

1. Due to entropy our universe won't contain "all possible things"

2. It's the definition of "incompressible"

3. It's the fundamental principle of algorithmic information theory

Ok I see what you mean. I agree.

But you have also misunderstood what I was saying wrt 'universe'

Consider that this universe could be an instance of the structure that is formed by the algorithm that you are talking about. Consider that all possible alternate structures may also exist. Now consider that there exist 'Garden states' that are unreachable because they are incompressible. So might it not be possible to be conscious in a universe that has no generating function? The mathematical structure representing this universe exist in the abstract, timeless and static. Your life is a slice through this giant structure.

Infinitely many of those structures exist, is there one out there that contains conscious beings that believe they inhabit an orderly universe but they actually inhabit one where the starting state of their universe was a garden state, containing a vast un-generatable structure that has informed the evolution of the universe from afar.

But then it would still have a minimal function. Bugger. What if that universe there existed a capricious demon that used the information in the initial state to... Nope that doesn't work either. Ok you win :(

Just because the levels we have got through are comprehensible to (some) humans and accessible to mathematics (some) humans can devise is no guarantee that the next level down, at higher energy and finer scale, will be, too. The bits we understand are accessible at energies we can muster, but there is no reason to expect we will evade extinction long enough to be able even to test string theory.

We have been lucky thus far, but the fundamental nature of luck is that it runs out.

Physics departments are already beginning to turn toward theology. In another two generations will there still be experiments in fundamental physics? What we can be certain of is that they won't close up shop and go home. They will still publish as many papers as required, and graduate post-docs. Ultimately, academic forms do not require factual correction.

There are two separate things here:

1. whether we will find a unified theory.

2. whether it is in theory possible to find a unified theory.

The comment you replied to addresses #2. They're saying basically that if the universe is constrained to some set of knowable rules, then it can be reduced to a set of formulae, which is pretty much the same as saying if the rules are knowable, they're expressible as a set of formulae.

Whether this set of rules is simple enough that we are able to is a separate issue.

Ultimately, the two are indistinguishable without a unified theory in hand.

There is no guarantee that if a unified theory is possible, it is expressible in mathematics; or even necessarily in any form smaller than the universe itself. We have been lucky for a long time.

You're right in one sense, in that if #2 is true, then #1 is true in at least some sense.

That is, if the universe is not ineffable - that is, unless there are unknowable components to the universe - then a "worst case" unified theory can be defined as itself. That is, the universe itself is an expression of itself.

That we can express it in mathematics is a given in that case because mathematics is malleable; all you need to do - though it wouldn't be very useful - is to invent a symbol that represents the universe-as-is that by definition acts how the universe would act in a given context.

Obviously that is not very interesting, but the point it makes is that if the universe is not ineffable, then the question is not if a unified theory is possible, because in that case the universe itself is proof that it is, but how much we can compress such a theory, and whether we can reduce it to something simple.

As such whether or not a theory is possible may seem like it doesn't matter, but it's an important point to make, because it means that anyone making the argument that a unified theory is not possible rather than arguing that a simple unified theory is not possible is in effect arguing the universe is ineffable.

This is exactly what I mean when I say physics departments are turning toward theology.

Well, it's exactly the opposite - it is pointing out that rejecting the possibility of a unified theory borders on the religious because it requires the universe to have supernatural qualities.

Nobody asserts that a unified theory is impossible. Obviously, no one can have any way to know such a thing.

But anyone (e.g., I) can fail to discover one, even if given infinite time trying. It is obviously impossible to distinguish (1) the condition of a conveniently-finite unified theory not being possible from (2) simply failing, collectively, to discover one. It is also impossible, in practice, to distinguish (3) a possible universe-sized theory from, e.g., (4) a necessarily infinite-sized one, having neither. We know that the universe itself has always succeeded at discovering a next state, but do not know whether it might at some time, e.g., find the next state incomputable, and stop evolving.

Insisting that a unified theory conveniently smaller than the universe must be possible, having failed to discover one, is religion. Insisting you have actually got that theory, despite being utterly unable to check its correctness, is religion. That is the religion physics departments are flirting with already.

> Nobody asserts that a unified theory is impossible. Obviously, no one can have any way to know such a thing.

The very article being discussed claims the very idea that there "may be a deeper, more unified theory" has become completely discredited. It does not call it impossible, but calling the very idea completely discredited skirts very close to it.

And the message you originally replied to criticised a comment above that called the idea a "false idol" and questioned the very possibility of it.

It seems quite clear that while these people may not entirely and completely dismiss the possibility, they see it as unlikely or not worthwhile. My first reply to you was explicitly to point out that the comment you replied to discussed specifically the possibility, not whether or not we'll find one.

> We know that the universe itself has always succeeded at discovering a next state, but do not know whether it might at some time, e.g., find the next state incomputable, and stop evolving.

If so, then this end-state is part of the computation itself that would itself be part of the model. If the universe is not ineffable, such an end-state must be discoverable from the universe itself, and so does not impact the possibility of a unified theory in any sense.

> Insisting that a unified theory conveniently smaller than the universe must be possible, having failed to discover one, is religion.

With the caveat of the assumption that the universe is not ineffable, such an argument certainly can be put forward in a "religious" way, but there are also a whole lot of approaches that could be taken to try to get an idea of an upper bound without knowing the precise details, and while we certainly can not make that claim in the strongest possible interpretation of "must", we can make that claim in the sense that we have strong enough indicators of compressibility to assign it a high probability.

Consider that "all" we're seeking to do is to unify the forces, and depending on who you ask also explain physical constants, within a single framework, not creating a precise simulation of the universe. As such, we can "subtract" any actual randomness - if it exists - and any constant values. We're looking for a model, not an instantiation.

For us to be unable to compress a unified theory pretty much all matter and energy in the universe would need to be integral to the functioning of the universe itself in a way that can't be abstracted away with less complexity.

E.g. "freezing" the universe and moving a few particles around and "resuming" would need to change not just what happens next but how the universe functions, and would need to do so in an irreducible manner - that is, the relationship could not be constant and universal, because if the positioning matters but is following a consistent set of rules that is in any way reducible, then the positions themselves do not need to be included in the universal theory. In that case we can reduce that part of the theory to "here's how to predict what moving mass around will do to e.g. universal constants or relationships between forces".

Now I will accept that it may be possible that a universe that is not ineffable might be possible where a unified theory would be so complex that the shitty fallback of declaring the universe itself the definition would be the smallest option, in about the same sense I'll concede the possibility of Russel's teapot, but while a general compression algorithm capable of compressing every input is impossible, once you take genuinely random output out of a huge data set, a data set that is not compressible becomes increasingly improbably as the size of the data set increases.

In effect, a universal theory smaller than the universe is possible if we can sufficiently for whatever value of precision you want your model to have, compress any information about the universe including the data needed to specify how to integrate it back into the universe (I'm assuming we agree that any compression here needs to include the complexity of the decompressor, as otherwise a special purpose decompressor that includes the universe as a constant can compress the universe to nothing, which would make the discussion pointless).

Now, of course, the question is how much do we need to be able to compress before we'd agree it was "conveniently smaller". But to me at least the extent to which we can e.g. compress information about gravity alone suggests sufficient compressibility that it would seem to be a truly extraordinary claim that we will not be able to achieve similar compressibility in other areas.

But like Russell's teapot, it's possible.

> Insisting you have actually got that theory, despite being utterly unable to check its correctness, is religion. That is the religion physics departments are flirting with already.

We can maybe agree that many people do have strong feelings of which theories will get there and are invested in them more than they might have justification for, that is fair enough. But I've at least never seen anyone suggest they can know for sure, and a lot of large advances have come from people getting extremely invested in untested theories. As long as people don't all get invested in the same untested theories, we'll keep seeing advances.

Thank you for taking time to explain yourself.

My interpretation of their "discredited" is very different from yours. I did not take it as saying anything at all about the nature of the universe, as such, but only about the purely human activity of hustling Theories of Everything.

Ineffability is an interesting notion. If you have a theory that would work given complete information, but you can never get complete information, even in principle, is that ineffable? If you would need complete information to complete the theory (say, to choose among 10^500 possible variations) is that ineffable? Or just too hard? Can you really call it a difference?