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What are the flaws in the following reasoning?

By Gödel's theorem, for any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that cannot be derived from this formal system.

If we extend it to math in general (math is a formal system, that certainly includes ordinary arithmetic; we'll mark the whole math with $M$ here): in math ($M$) there exist true statements $G_i(M)$ that cannot be derived by any formal rules that currently exist in math. In other words, math in general cannot contain within itself an algorithmic procedure that will discover all possible true statements in math ($G_i(M)$).

Which basically means that there are true statements in math that need an external explorer to discover and prove them. We, people, are actually such explorers and we have the capability of finding and proving some of $G_i(M)$. (Well, it can be shown by using Gödel's theorem itself, but I'll omit this proof for brievity).

And finally, the above means that the world we live in cannot be described by math in all it's entirety. Because, if it could, our human mind (that is certainly a part of this world) would be also fully described by math, which would include an algorithmic description of our capability to find and prove $G_i(M)$, which is a contradiction.

UPDATE 1:

Second take on the above reasoning (trying to be a bit more formal):

  1. By Gödel's theorem, in any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that can be formulated in terms of $F$ but cannot be proven with $F$.

  2. All current math (marked with $M$) is a formal system, to which Gödel's theorem is applicable. Hence there exist $G(M)$ (or multiple $G_i(M)$) that can be formulated with $M$, but cannot be proven by $M$.

  3. Also, $M$ does not contain within itself an algorithmic procedure to even suggest $G(M)$, because the existence of this algorithmic procedure within $M$ would actually be a proof of $G(M)$. (More rigorous proof is required here, I guess.)

  4. Humans (mathematicians) can certainly suggest some $G_i(F)$ and prove them afterwards using rules/approaches additional to $F$, thus forming a new formal system $F'$, where those $G_i(F)$ are proven to be true. (I need to provide some good examples here). Same holds for $M$.

  5. Finally, currently known math $M$ cannot describe the world in all it's entirety, because it at least cannot describe our human capability to suggest $G_i(M)$. After we prove $G_i(M)$, we come up with new math $M'$. But $M'$ now contains $G_i(M')$ by Gödel's theorem, and we come back to step 3. This can go on infinitely many times, and neither $M$, nor $M'$, nor $M''$ etc. would be able to fully describe the world and, in particular, our mathematical thinking and the way we do math.

So, what are the flaws/mistakes in the above reasoning? We already have some good answers here, but maybe some additions/corrections?

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    $\begingroup$ if a theory fails add more axioms $\endgroup$ – janmarqz Jul 24 '18 at 18:55
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    $\begingroup$ How do we know that statement is true (or untrue) in some system of axioms if it can not be derived from that system of axioms? $\endgroup$ – Maria Mazur Jul 24 '18 at 18:57
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    $\begingroup$ It's very obscure to me what you're trying to say with your last two paragraphs, I don't see what your logic is at all. I would say the opposite is more likely. It's plausible that the universe can be described by a formal system $U$, and if that's true, it seems like that would imply that there are things we can never prove (not even by adding axioms to our mathematics, since the process of coming up with those axioms would be describable within $U$). $\endgroup$ – Jack M Jul 24 '18 at 19:00
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    $\begingroup$ @janmarqz If the theory fails because of Gödel, adding enough axioms to fix that will just make your theory inconsistent. $\endgroup$ – Ray Jul 24 '18 at 20:13
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    $\begingroup$ The simple answer here is really "No, you're making what can only be called a 'general talking point' out of a firmly precise mathematical concept." It would be like saying .. oh .. "We know there is no largest integer; this makes me think love is infinite." Your view is essentially poetic, it's not in the same realm as the math in question. $\endgroup$ – Fattie Jul 24 '18 at 21:34
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The major flaw is in your presumption that humans can find and prove statements that are true but not provable within math. Think about what exactly you mean by "math": you presumably mean something like "all math people do". In other words, $M$ would be a set of axioms so that everything mathematicians have ever proven follows from $M$. If that's so, what's your evidence that a human can prove something not proven by $M$? By definition, that has never happened before! And if they could, what would such a proof look like? It would have to invoke axioms or steps of reasoning that aren't in $M$ - but that means it would be assuming something that we don't actually know is true. So how could we believe that "proof"?

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    $\begingroup$ To my understanding, a formal system is not only a system of axioms, but a system of axioms + derivation rules. And it's exactly the derivation rules that F may lack to derive G(F). But we, humans, have the capability to go beyond this set of derivation rules to see and prove that G(F) is true. A good example, I guess, is proving that there is no such Turing machine H(m) that tests if a given Turing machine m stops. To prove this, we use the Kantor's diagonal process approach, that's certainly beyond the scope of Turing machines theory, but does not contradict it's axioms. $\endgroup$ – weekens Jul 25 '18 at 6:29
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    $\begingroup$ Going on with the above, M is axioms + derivation rules. Each time we discover G(M), we have to use some new approach to prove it, effectively creating new math M'. But when we get it, there now exists G(M'), and the process goes on. But does it follow from Gödel's theorem, that neither of all these possible math's (M, M', M'' and so on) would be ever capable to fully describe the Universe (in particular, our human mind that can prove G(M))? $\endgroup$ – weekens Jul 25 '18 at 6:34
  • $\begingroup$ @weekens Gödel's theorem can say something about each of M, M', M'' etc. But It can not say anything about the limit of the sequence of these systems, because this limit is not a formal system. $\endgroup$ – Taemyr Jul 25 '18 at 9:12
  • $\begingroup$ @weekens what is a "derivation rule" and how does it differ from an axiom? $\endgroup$ – Shufflepants Jul 25 '18 at 14:03
  • $\begingroup$ @weekens I think you're quite deeply mistaken, there. "We, humans, have the capability to go beyond this set of derivation rules to see and prove that G(F) is true" - No, we don't. The "example" that you give is incorrect. You claim that the halting problem proof is "certainly beyond the scope of Turing machines theory"; that's not an accurate statement because there's no such thing as "Turing machines theory". The halting problem can be proved in ZFC, which was invented in 1922, whereas Turing machines were invented in 1936. No new derivation rules were invented to prove the halting problem. $\endgroup$ – Tanner Swett Jul 25 '18 at 14:22
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Godel's theorem is about axiomatic formal systems. Reality is not axiomatic; no humanly postulated premises are necessary for its existence, completeness, or self-consistency. So what is is part of a non-axiomatic system, raising the hope that it can be described fully in the context of its own system (whatever that might be). Math, which is an axiomatic system, can describe a larger universe of entities than exist in physical reality (for example, there is no physical correlate to Fermat's Last Theorem). Some of those entities apparently include truths that are not provable within given axiomatic formal systems. All of that neither establishes nor disproves that every actual thing can be represented by math. Whether or not the world is fully describable by math might be, but is not necessarily, commented on by Godel's theorem.

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I would like to add a different perspective. One problem with the reasoning is the assumption that we need to capture reality in just one mathematical system. Note that Gödel's incompleteness theorems generate, for each "good" (i.e. sufficiently similar to ordinary arithmetic) formal system $F$, a Gödel sentence $G$ which depends on $F$. So, at best, this would rule out having one big formal system which globally represented "reality". But it does not rule out having many local systems which represented aspects of reality and which jointly exhausted it.

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    $\begingroup$ Unless by "many", you mean "infinitely many", it does more or less rule out having many "local" systems that jointly exhaustively formalize "reality". You could just stick such systems together into one formal system. If this combined system is inconsistent, this suggests that your "local" systems can make contradictory empirical claims. On the other hand, there is little to nothing that suggests that reality isn't able to be captured in a single formal system. Certainly ZFC is way more powerful than any of our current physical theories require. $\endgroup$ – Derek Elkins Jul 24 '18 at 21:37
  • $\begingroup$ @DerekElkins - Yes, "many" Can be interpreted as also including the "infinitely many" in the above. $\endgroup$ – Nagase Jul 24 '18 at 21:49
  • $\begingroup$ Your argument is completely valid, if these local systems are independent, i.e. one local system cannot be used to prove statements in another. But is it true for math? $\endgroup$ – weekens Jul 25 '18 at 6:39
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    $\begingroup$ You can stick infinitely many local systems together as well, so long as you have an algorithm for identifying what is a local system. It is a trivial matter, incidentally, to have local systems that exhaust all facts about reality; for each fact $T$, simply take the local system "$T$ is true". $\endgroup$ – Hurkyl Jul 25 '18 at 8:55
  • $\begingroup$ @weekens What formal systems are you aware of that are not subject to math? $\endgroup$ – Cubic Jul 25 '18 at 10:16
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You have to be careful with what you mean by "true". In Gödel's incompleteness theorem for instance, "truth" refers to truth in a specific model of arithmetic ($\mathbb{N}$, to be specific- the so-called standard model of arithmetic).

By contrast, Gödel's completeness theorem that anything which is true in any model (understand "any world" if you don't know any model theory) is provable: in other words, what is true is provable. Of course this doesn't contradict the version of the incompleteness theorem you stated.

So, with that in mind, what do we mean by "true" ? For instance, for set theory (e.g. ZFC) it doesn't make sense to refer to the standard model of set theory, because there is no such thing. In particular, the version of Gödel's theorem you quoted makes no sense for theories that cannot be interpreted in $\mathbb{N}$ (or some "standard model") !

In particular, you cannot apply the version "there are true statements that are unprovable" of Gödel's theorem to the mathematical universe (and there's little hope in applying the theorem to the physical universe, anyway)

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    $\begingroup$ "it doesn't make sense to refer to the standard model of set theory, because there is no such thing." - Well, that somewhat depends on your point of view. $\endgroup$ – Kevin Jul 25 '18 at 1:44
  • $\begingroup$ @Kevin : does anyone actually refer to minimal models as standard models ? $\endgroup$ – Max Jul 25 '18 at 9:20
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    $\begingroup$ @Kevin: To answer Max's rebuttal. No, while the minimal transitive model of set theory is a point of interest, and it is useful in certain situations (e.g. pointwise definable models are fun for confusing people about internal/external definability), it is a far cry from being "the standard model of set theory". For example, it satisfies "there are no standard models", which is something I guess most people would expect to be true. It has no large cardinals, which are also axioms many people expect to be "true" (or at least not inconsistent). So, no, the minimal model is not enough. $\endgroup$ – Asaf Karagila Jul 25 '18 at 11:07
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There's an unjustified assumption in there that the human mind and human thought process is a part of this world.

...our human mind (that is certainly a part of this world)...

Not to get religious on the math stack exchange, but strictly logically, you must proceed from basic assumptions only to conclusions based on your assumptions. You can't make hidden assumptions and get a universally valid conclusion.

Taking the same set of assumptions you have, it could still be possible that mathematics could be used to describe everything about the world we live in, except the human mind.

You've also assumed that the human mind's ability to resolve problems only proceeds along lines that could be algorithmically described. There are plenty of counterexamples, such as ESP, telepathy, prescience, etc., though of course due to their very nature they are usually ignored by scientists seeking solid provable evidence about the nature of the world. If you only consider the behavior of human minds that are behaving like Turing Machines, then of course it will appear to you that minds operate like Turing Machines (a sneaky sort of selection bias).

The nature of life is the key to your question, in actual fact. If you are only a Turing Machine (which is utterly unproven), then your reasoning holds. Otherwise, it doesn't.

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  • $\begingroup$ None of ESP, telepathy, or prescience require human minds to be unrepresentable by Turing machines nor "supernatural". The first two pretty explicitly suggest merely that there is a scientifically undiscover(ed/able) input to the mind, while the third could be viewed that way or could even be reduced to "you can compute real fast". There is a general issue with positing "super Turing" anything, similar to Reese's answer. Basically, it's a claim that some person/process produces "answers" that cannot be verified by definition. $\endgroup$ – Derek Elkins Jul 24 '18 at 22:12
  • $\begingroup$ @DerekElkins, not so. Your comment only holds true if you take "cannot be verified" in the frame of reference of Turing Machines. But if you're not a Turing Machine, then you could know something that a Turing Machine would be unable to verify—but other human minds could. $\endgroup$ – Wildcard Jul 24 '18 at 22:23
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    $\begingroup$ @DerekElkins, actually, it seemed obvious to me this philosophic underpinning is what the OP is really talking about from his statement: "Which basically means that there are true statements in math that need an external explorer to discover and prove them. We, people, are actually such explorers and we have [that] capability...." Math can't prove what math can't prove, but obviously if the human mind can do things that math can't, those capabilities would be outside the capability of math to methodically demonstrate. $\endgroup$ – Wildcard Jul 24 '18 at 23:23
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    $\begingroup$ If you allow the "super Turing" abilities to vary with time and person, then you are right back at the beginning where one person is making unverifiable claims as far as another person is concerned. At any rate, my downvote is based on my original comment which is all the phenomena you mention, even if real, in no way require the mind to be "super Turing" and so are a non-sequitur. The rest is just an argument that even if they were reflections of "super Turing" abilities, there's no way to verify that except by assuming "super Turing" abilities which they were supposed to be evidence for. $\endgroup$ – Derek Elkins Jul 24 '18 at 23:50
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    $\begingroup$ I agree, stating that "human mind is a part of the universe" is a somewhat weak point (it just looks so obvious!). And no, I'm not saying we make derivations using the algorithmic procedures. I agree with you that the opposite is true. And I guess it's our way of proving the theorems that cannot be mathematically described. $\endgroup$ – weekens Jul 25 '18 at 6:52
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Maybe the theory does not need to be complete in Gödel sense to describe the world well enough for our purposes. Maybe it just needs to be good enough.

  1. It is not sure the propositions describable but undecidable in the theory will be observable in the world. If they are not observable, then why should we care about them?
  2. Then we won't have any practical need to add neither them nor their negations to the theory.
  3. It might be that propositions can become observable because of advances in technology and measurement equipment. Then there is nothing that stops us from expanding our theory by adding the proposition or it's negation as a theorem. That is the nice thing about mathematics never being possible to finish (in Gödel sense). We can always expand on it when we learn more about the world.
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I've looked at the "bit more formal" version of your question. I think the problem is with point 3.

First, let me quote your first two points:

  1. By Gödel's theorem, in any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that can be formulated in terms of $F$ but cannot be proven with $F$.

  2. All current math (marked with $M$) is a formal system, to which Gödel's theorem is applicable. Hence there exist $G(M)$ (or multiple $G_i(M)$) that can be formulated with $M$, but cannot be proven by $M$.

I'm going to assume that "all current math" means ZFC. Certainly, some people (including me!) are going to disagree with the notion that ZFC satisfactorily captures "all current math", but my objection to your argument is going to be pretty much the same regardless of exactly which system is chosen for $M$.

  1. Also, $M$ does not contain within itself an algorithmic procedure to even suggest $G(M)$, because the existence of this algorithmic procedure within $M$ would actually be a proof of $G(M)$. (More rigorous proof is required here, I guess.)

This isn't true at all.

It is, in fact, possible to define ZFC inside of ZFC. If you're doing mathematics inside of ZFC, then you could define a set $z$ which represents ZFC itself.

It's also possible, inside of ZFC, to define a function $f$ which takes any formal system $F$ and returns the corresponding statement $G(F)$. The function $f$ can even be defined as a particular algorithmic procedure.

And so, now that you've defined the function $f$ and $z$, the expression $f(z)$ refers to the Gödel statement of ZFC, $G(\text{ZFC})$.

You say that the existence of $f$ "would actually be a proof of $G(M)$". As a matter of fact, $f$ does exist, but it doesn't prove $G(M)$ at all.

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  • $\begingroup$ Thanks for your answer! I wonder, how would this $f$ look like. $\endgroup$ – weekens Jul 31 '18 at 20:02

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