Q: Does Gödel’s Incompleteness Theorem imply that it’s impossible to be logical?

The original question was:
The question is in context with Kurt Gödel’s Incompleteness Theorem.

Question: If no formal logic can ever be, complete and consistent, does that mean, humans are illogical beings?

Further elaboration of the question:  Imagine a mathematician seeing a statement and intuitively knowing that the statement is true.  But the statement is unprovable.

Conclusion:  If the mathematician was a logical being, he would not be able to conclude that the statement is true.  Because it is unprovable.  But since he knows it is true, it must mean that the mathematician is an illogical being.  Because the mathematician doesn’t use logic to conclude – he use intuition.

Further clarification:  Since maths is based on axioms – which are statements that are true but unprovable.  Does this mean that the creator of the axioms is an illogical being?  Because the axioms would never exist if the creator was a logical being.

To take the question further:  Are humans constrained by logic in the sense that we could model the universe by a big equation, then calculate for a later given time, what the state of the universe is in?  So our actions would just be a product of a predefined mathematical path, which we don’t know (or might never know).

Or are humans not limited by logic so it would be impossible to model the universe by a big equation.
Because the equations ‘foundation’ is based on something which cannot ‘contain’ an equation for an illogical being?

I’m trying to hint on the question about: free will vs. fate.

Mathematician: Human beings are very, very far from perfectly logical beings (by which I mean that our thinking is often at odds with logic, and we routinely draw false conclusions by misprocessing information). There is no need to reference the incompleteness theorem to show that. Wikipedia’s list of fallacies does a nice job of cataloging many of the most common errors in thinking that humans make. I think it is very safe to assume that most people in the world have made at least a handful of the errors in this list, and everyone has made at least one of these errors at some point.

But, the fact that we are frequently illogical says nothing about whether human behavior could be predicted by applying the laws of physics to a detailed description of a human’s current state. It may be possible to make such a prediction in theory (though accuracy will be inherently limited by Heisenberg’s uncertainty principle, imperfections in our measurement tools, and other factors). In practice, however, predicting a person’s behavior using physics is absurdly difficult to do and it is possible that we will never come close to being able to do so. It is not inconceivable though that one day we will be able to model approximate human behavior in the short term by using a very detailed simulation of a person’s brain. At this point in time though, we can only speculate.

Physicist: The most frustrating thing about the Incompleteness theorem is not just that it shows that there are an infinite number of true and unprovable statements, but that it provides no means to figure out which statements those are.  We have a lot of statements that we assume (or define) to be true without proof, like “you can’t split a point”.  To make it sound as though there’s more to it than that, we call them “axioms”.  It’s not that they’re intuitively true, they’re defined to be true, and what we call logic follows afterward.  When a mathematician says something is “true” it’s always “true, given some set of axioms”.
As far as modeling the universe goes: The universe is, on one level, entirely deterministic.  If you had access to the total quantum wave function of the universe you could roll it forward in time (deterministically), no problem.  However, this isn’t how we experience the universe.  We only experience a tiny fraction of this wave function.

Basically, different versions of you experience every possible outcome of every event, so it no longer makes any sense to ask “which will happen?”.  They all happen (all that are possible), but each version only experiences one.  That isn’t terribly clear, but there’s a post about almost exactly this question here.

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28 Responses to Q: Does Gödel’s Incompleteness Theorem imply that it’s impossible to be logical?

  1. It’d be cool if you guys do an intro specifically to Gödel’s incompleteness theorem, its statement and implications in layman’s terms. It’s been mentioned around here a few times and seems to be a great source of inspiration/imagination but also easily misunderstood.

  2. Johnny says:

    He he I posted that question… I thank you for both of your answers. But I guess after sometime I was still contemplating about that subject… I guess I didn’t really know what I was thinking =) I guess I was more interested in the last part of my question about:

    [Slightly rephrased]
    “…model the mind by a big equation, then simulate for a later given time, what the state of our mind is in?”

    This of course meaning deterministic within the bounds of probability. This also meaning ‘fate’ because the future being pre-determined.

    But after some time, I was going through youtube videos and happened to saw an interview with Sir Roger Penrose (http://www.youtube.com/watch?v=yFbrnFzUc0U)

    Here is a snippet of the conversation:
    Mathematical understanding depends upon consciousness.
    But mathematical understanding is not something of a purely computational character

    There is something else which has to come into that

    So that’s… The mathematics only comes in, to demonstrate that there is some part of our conscious thinking which you cannot simulate in a computer.

    [Woman]Well then, what is that ‘something else’ that comes into play?

    Well if you believe as I do, that whatever controls our actions in our brains and so on, is the physics of the world then that tells you that there has to be something in the physics of the world which is not controlled computationally. Does not mean its not mathematical but its not computational.

    As it seems to indicate is, I had to distinguish the ‘mathematical’ part from the ‘computational’ part…. So I concluded in my own mind that it might mean that we can (in theory) model the human mind but may not be able to compute it (simulate it?)

  3. Johnny says:

    I might add that the event that triggered my questions is the BBC program (Dangerous minds) about Cantor – Gödel and Boltzmann.

    The thing that brought up the original question might have come from the following, where the narrator in the mentioned program says:

    Google-video, link: http://video.google.com/videoplay?docid=-8492625684649921614#docid=-1663091361786740235. About 27 minutes in…
    “… (Alan Türing) understood that if our minds were computers, then incompleteness would apply to us. And the limitation of logic would be our limitations and we would be incapable of leaps of imagination beyond logic.”

    I guess my thought process was something like this:
    – If we are computers, then we could predict/simulate/compute the state/situation in the future.
    – But since computers are logical, and humans are not. It must mean humans aren’t computers.
    — So incompleteness does not apply to us.

    (Then I went too far)

    — So since incompleteness is a mathematical statement, it must mean we are not mathematical (logical) beings.
    — So we can’t predict / simulate/compute what a person would do in the future.

    And then I went to write a question which was posted here =)

  4. Christian Luca says:

    The responses given by both the mathematician and physicist are interesting, and it seems the mathematician treated the original question of whether humans are illogical beings, relating to Gödel’s Incompleteness Theorem, while the physicist’s answer treated more the question of free will vs. fate.

    Based on the answer given by the physicist with regard to the universe on one level being entirely deterministic, whereas we only experience a tiny fraction of the total quantum wave function of the universe, seems to suggest that from the human perspective there is free will; however, from the perspective of the universe, reality is deterministic. Would this be fair to say based on the response given by the physicist?

  5. The Physicist Physicist says:

    If you assume free will exists if and only if the universe is non-deterministic, then yes. That sounds right.

  6. Christian Luca says:

    Thank you for your response, Physicist!

  7. create14all says:

    Gödel’s Incompleteness Theorem simply proves that it is wrong. The ultimate Russel Paradox.

  8. Jim Ward says:

    Gödel’s Incompleteness Theorem shows we can’t prove everything. Is there anything interesting we can’t prove?

  9. The Physicist The Physicist says:

    So much…
    One of the spookiest is the Goldbach conjecture, which says that every even number (above 2) can be expressed as the sum of two primes.

    This has been tested for ridiculously huge numbers, but no one has ever written a proof.

  10. Peter says:

    I find the Physicist responses interesting: I thought the wierdness of QM was a result of the fact that what happens at the QM level was non-deterministic and that Heisenberg’s Uncertainty principle says that it is impossible to know the “total wave function” as you call?

    Perhaps a follow-up question might be is there some equivalence between Godel’s Theorem and Heisenberg’s Uncertainty principle.

  11. The Physicist The Physicist says:

    Wave functions can be explicitly known, without giving you any details about the exact positions or momentums involved. The problem is analogous to knowing that the probability of getting either a heads or a tails on a (fair) coin is 0.5. Those probabilities are like your “wave function”, but they don’t tell you which way the coin will land. Heisenberg’s Uncertainty principle is merely a statement about what kind of probability distributions you can expect.
    As awesome as a “Unified Godel-Heisenberg theorem” would be, unfortunately they’re talking about completely different things.
    I guess you could combine them under the umbrella of the “Universal Pessimism Theorem”?

  12. Johannes says:

    Hello Physicist…

    I would like to respond to a post of you…

    The Physicist: If you assume free will exists if and only if the universe is non-deterministic, then yes. That sounds right.

    If we make such a claim then we should be able to verify the claim by experiment. Let a moment be written as A and let the next moment be written as B. For example A=”I am at home” and B=”I go to the cinema”. When A is given – is B then determined? An experiment allows us to measure both A and B. But we can do the experiment only once – not twice.

    We can do the experiment and we store the results – say in a computer. Suppose that we have a device that allows us to create any given moment – then we might be able to re-create the moment A and repeat the experiment. However – the moment A does not contain the outcome of the experiment – so even when we would be able to re-create the moment A – the outcome of the experiment would be lost. If A would contain the outcome of the experiment – then the moment would be A’ not A and that would not be the same experiment. So every experiment is unique – two identical experiments do not exist (in general two and identical are contradictions) – so what is the purpose to speak about non-determinism – if we cannot do any experiment that shows non-determinism?

  13. The Physicist The Physicist says:

    I’m not sure I follow the logic. What’s the problem with A containing the outcome? Why wouldn’t the experiment be repeatable?

  14. Johannes says:

    Hello Physicist,

    (I do not know how te reply directly so I just added a reply)

    I will try to explain.

    Why wouldn’t the experiment be repeatable?

    The experiment is not repeatable because we want to do the SAME experiment. If we speak about two experiments then there has to be at least one property or parameter by which we can distinguish the experiments, otherwise we cannot speak about two experiments. But since two experiments have at least one different property or parameter – they are not the same. We cannot do the same experiment twice.

    What’s the problem with A containing the outcome?

    We may consider the outcome as information and we store this information. However stored information interacts – for if not – we would not be able to re-read the information. Since stored information interacts -the given moments A and A+outcome are two different moments. Two different inputs allow an experiment to have two different results – so we cannot draw any conclusion from that about deterministic or non-deterministic. So A+outcome does not serve the experiment.

    Regards – Johannes

  15. I’m not certain that it is meaningful to ask about the implications of Gödel’s theorem for free will. The reason for this is simple: the theorem is, as has been pointed out above, a statement which is true with respect to some set of axioms. Axioms could be constructed which would not imply the theorem, but more importantly, outside of the context of some particular model, the theorem has no implications at all. Here I mean “implication” in the strict sense, the only sense which makes any sense at all when discussing a mathematical entity. This view is, of course, not shared by all mathematicians.

    The theorem does suggest something sort of interesting (but also obvious) within epistemology: that human beings obtain knowledge from sources which are not explicitly deductive.

    I enjoyed this question, and the resulting discussion, and will return to this site for more!

    Also, @Physicist: is the Goldbach conjecture something which cannot be proven because of incompleteness, or is it simply that no proof has yet been found?

  16. The Physicist The Physicist says:

    No proof has been found.
    It just gives me an “incompleteness vibe”.

  17. Ellen says:

    I find the mathematician’s answer beyond unsatisfactory. Asking if human beings are inherently illogical is a much deeper question than if humans are sometimes wrong, but other people can set them straight. Common misconceptions are provable and logical, so it dodges the question entirely.

    This is like saying that Heisenberg’s Uncertainty principle is true because error measurements in labs are common.

  18. The Mathematician The Mathematician says:

    I’m not sure what you mean by “inherently illogical”. How are you defining that? Our brains did not evolve in such a way that we can carry out logical steps perfectly without error. But I’m not sure that addresses what you mean by “inherently illogical.”

  19. Ellen says:

    Since Godel’s Incompleteness Theorem implies that something can be true and not provable, also that the foundations of math (set theory) are shaky (this is the best way I can write it for the sake of brevity), math is built sort of from the top down and lacks a foundation. What this may imply is that there’s a fundamental inability for humans to find a basis for logic, not that sometimes they are illogical. I guess that’s the best way I can think of to explain it and I think that’s more along the lines of what the reader was asking.

    Inherent means intrinsic. Something that we can’t get beyond. Kind of like Heisenburg’s Uncertainty principle and the quantum measurement problem. It’s not that it’s difficult to take measurements, but more that there’s an eery fundamental wall between “us” (humans) and a foundational knowledge. (Read Law without Law by Wheeler).

  20. Dan says:

    Somebody asked for a layman introduction to Godel’s Incompleteness Theorem. I very recently wrote just such an article on my blog, so I hope you’ll allow me to engage in a little bit of some shameless self-promotion.


    Hope it helps!

  21. I think it is important to take into account the way in which Godel’s proof actually works. He considers a particular logical system and constructs a special statement that is not provable within that system. His result is important because it is general: a similar proof can be given for any logic that can encode its own formulas (e.g., using arithmetic).

    Douglas Hofstadter gives a nice analogy in his book “Godel, Escher, Bach”. One person tries to build a perfect record-player and another person tries to construct disks that it cannot play. (When the book was written, most readers were familiar with such machines.) Give a particular record-player P, for example, you might find frequencies at which it resonates and record exactly those frequencies on a disk, D. When P plays D, it shakes itself to pieces. The point is to distinguish between the claims “this disk cannot be played” and “for each player P of sufficient quality, we can create a disk D that it cannot play”. There is a dependency between D and P.

    Here’s my answer as to whether mathematicians (or even ordinary people) have Godelian limitations. Imagine a kind of super MRI machine that can monitor the state of each one of your brain cells and thus allow you to inspect your own brain. There would probably be statements of the form “Cell 12345 is not firing now” which you could see to be true but could not say, because saying them would make them false. This is how Godel’s proof works: it uses the system to operate against itself.

    In Computer Science, the “Halting Theorem” is a result similar to Godelian incompleteness. (Many people actually consider HT and GI to be the same.) It says that there is no algorithm (or, equivalently, computer program) that can decide whether a given program terminates (“halts”) in a finite time. Again, it is proved by carefully constructing a program to defeat the algorithm. Here’s an informal description of the proof:

    Let H be a program that decides termination. Specifically, H(P) yields “true” if program P halts and “false” otherwise. Now consider this program:
    Q = if H(Q) then loop for ever else halt
    From this program, and assuming H works as advertised, we conclude that Q terminates if and only if it doesn’t terminate. Since this is a contradiction, we infer that there is no such program H. Once again, the proof depends on a very special program, one designed to defeat the proposed algorithm.

    Godel demonstrated the existence of just one unprovable statement for a given system. In fact, there will be many, but the others are harder to find. The work of Chaitin and others suggests that, in some sense, most statements are unprovable. The success of mathematics, however, suggests that there are enough interesting and provable statements to keep us going for a while.

  22. Puiu Botan says:

    Physicist, according to you, free will is not possible in a deterministic Universe. I agree.
    But how can it be possible in a non deterministic one ?
    If my life would depend on the outcome of an experiment which has 60% chance to detect the spin of a particle “up” and 40% to detect it “down”, how could my free will help, given that I drop dead if the spin is “down”?
    How free is my choice, since the non deterministic nature of things prevents me to predict with certainty the result of my decision ?
    Free will is not possible in a non deterministic Universe either.
    Actually, this old issue of free will is as a heritage of the good old religion, a smart concept invented in order to save the prestige of the Almighty, who wouldn’t have created the evil Himself, but gave “the free will” to humans, to make them responsible for evil happening.

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  24. When the physicist says if we had access to the total wave function of the universe we can roll it forward and theoreticale to predict all future action. This is a nonsensical statement and is in direct violation of information theory and an understanding of quantum theory.

    This is a problem of language and relating to physics.

    There is no way too to have knowledge of a complete wave function because everything is based on interActions.

    There is nO complete function of the entire universe this is the same as saying a round square it’s a meaningless nonsensical statement

  25. Antonio Carlos motta says:

    Is interesting that all the paradoxes as of Turing or godet are relationed with the ideas of large infinity(continuum) and the small infinitesimal as placed by Newton and Leibniz ,but think that the idea of Newton of the fluxions contained the idea of the discrete space and time.It is his thought in the conceit of discontinuity,where the non commutative property gave the conceit of motion to the space and time.was some dynamics,while the of Leibniz was static. But the o paradoxes of yodel and Turing is connected at the Infinity,as Cantor’s sets that implies that the proper infinities has different story potential.the infinities of classQ has different potential than the continuum C.then some infinities are finities in theirs potential.but exist some class greater than Q and minor than C.think that exist ‘holes” in the space that can explain those completed infinities.that are associated that the curves has a potential of “holes’ and the straight lines has not.then the variable of space and time is intrinsically linked to the discontinuities generated by the movements.
    The transcendent transformations of left handed to right handed and from right handed demonstrate that there is strongest violation of symmetry in the 3dimensional manifold,the new.there is number also infinity to that transformations of left handed into right handed and viceversa ,that is generated by the violation of pt only could to be conjugated in the string called transfinite.for that reason the asymmetry of spacetime that is generated in the differentiation odd parity and even parity,that feline the metrics of theirs by complex numbers or quaternions or more amplied a sphere S3 with 8 dimensions more 4 dimension resulting 12 dimensions,that simultaneously generate space time from left handed(rotations) to right handed(rotations) and vice eras generating all the space time contoninuos as particles and antiparticle.i think that the Aphrodite of cantor show a new order mathematics.between the rational numbers to the continuum.there are others alephs

  26. Antonio Carlos motta says:

    The yodel and Turing theorems go much beyond of ours number theories.the cardinal and ordinal numbers of g cantor that demonstrate a new concept of dimensions and number theory as the metrics n dimensional of banach And banachtarskiparadoxes

  27. Antonio Carlos mota says:

    Is really correct the thought of hawking to respect of the godel’s theorem that if valid the everything theory is incorrect

  28. Dez Szeredi says:

    Hawking was quite correct in his interpretation presented in his essay “Godel and the end of Physics”. However, I would be overjoyed to see a paper that discusses whether or not the “Godel Therom” is applicable to the “Godel Therom”; And If the answer is yes, why and what does imply? If the answer is no, why, and where should we go from here. I hope that one can see that if the first question is answered yes, it answer’s where we should go next. Now for the second question, that is a little bit twisty.

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