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  • Goedel’s Theorem Extended

    Posted by David Foster on August 8th, 2020 (All posts by )

    In 1931, the mathematician Kurt Goedel showed that for any consistent  formal system of logic of logic (of at least a certain degree of complexity), there will always be some true statements that cannot be proved, and some false statements that cannot be disproved within the system.  No matter how many axioms you add to the system, there will still be statements that cannot be proved or disproved within it.

    I was reminded of Goedel’s Theorem by some of the more far-out accusations of racism, sexism, etc that have been made against individuals lately, and was thinking that there should be an analogous theorem:  No matter how an individual chooses to act and speak in a way that will shield him from accusations of X-ism, there will always be a way that someone can build a case that he is in fact an X-ist.

    But before I could post about that extension to the theorem, along comes this post by a physician, talking about some of the ways his patients have managed to misinterpret the instructions for using birth control pills–leading to a need to specify more and more detail when giving such instructions.

    But adding more detail probably like adding more axioms to one of Goedel’s formal systems…so the additional analogous theorem is: No matter how detailed the instructions for doing something may be, there will always be a way for someone to interpret them incorrectly.




    20 Responses to “Goedel’s Theorem Extended”

    1. James the lesser Says:

      Perhaps a systems of laws and regulations can be treated as such a formal system. Whack-a-mole is the the name of the game..

    2. Kirk Says:

      It is axiomatic that no matter how hard you strive to make something proof against idiots and their misunderstandings/misconceptions of reality… The universe will, loathing a state where idiots are unable to be idiots, throw up a new and improved idiot so as to make your efforts irrelevant and entirely useless.

      Frankly, I think it’s far past time we just abandon the entire idea, take the safety railings of life off to storage, and allow all and sundry to suffer the effects of their idiocies. Dealing with accusations of racism? I say bring back dueling–You accuse, you then risk death.

      Watch what happens. Too much of our social infrastructure has gone, over the years, to enable bad behavior and conduct that would, in a less padded world, lead to some salutary lessons. Some of which would remove many from the gene pool.

      And, observing the world around me, this fine day? I am not sure that that would not be a good thing…

    3. Mike K Says:

      It is pretty well known that around 50% of patients do not take medications as instructed or do not fill the prescription at all. There is an axiom in Decision Theory that offering more choices reduces the probability that any choice will be made. It’s the reason a car salesman is reluctant to show you more than one car.

    4. MCS Says:

      My first inclination was to respond with a rather uncivil duh. No one that’s ever tried to write instructions or specifications would disagree.

      The systems Goedel was dealing with were things like Euclidean Geometry that could be expressed in a couple of pages and then elaborated endlessly.

      The U.S. Constitution is a miracle of concision. Compare it to the proposed European Constitution that was at, I think, 1,200 pages before the whole thing rather predictably collapsed, or any recent mega-bill. The length alone insures that it is so riven with self-contradiction as to meaningless.

      I can remember science fiction stories where proposed legislation would have to undergo a quasi-mathematical “semantic analysis” as a step before being voted on. Can you imagine the present political establishment going along with this? Assume that any law proposed will be drafted in a deliberately misleading way and you’ll on the way to figuring out what it is actually intended to do. Or short circuit the whole process by assuming it is intended to enrich, first the legislators proposing it and then whoever is backing them. Notice I didn’t mention either Democrats or Republicans. Diogenes would have his work cut out for him in Washington or any state capital.

    5. David Foster Says:

      As I’ve noted before, the only intelligent thing I’ve ever heard Nancy Pelosi say was, ‘We have to pass the bill to find out what’s in it”….(although the way in which it was intelligent was probably not the way she meant the statement.)

      You can’t set up a complicated algorithm dealing with human behavior and accurately predict what it will do in advance. The Founders did about as well as possible, and that could be achieved only because they developed a limited and relatively simple framework. (and also, they drew on things that they already knew from experience were workable, such trial by jury….there was much less of an ab initio component than there was in the French Revolution, or even the Napoleonic Code.

    6. Kirk Says:

      The only way around the inherent flaws in any system is… Not to have a system in the first damn place.

      There are parallels with all this in the martial arts and Eastern philosophy–The concept of mushin no shin, or “the mind without mind” is what I’m getting at, in terms of organization and social structures. The ideal is that society and organizations within that society should function as organic outgrowths of the participants in it–There should be no need to resort to enforcement measures when actions need to be taken. The members of an organization should come together, reach a consensus about what needs to be done, and then do it cooperatively.

      Examples? Amish barn-raisings come to mind. You occasionally see it in emergency situations, as well. But, all too often, we utterly fail to meet the mark in these regards, and wait until some authority figure tells us what to do.

      If you want real civilization, one that springs forth without coercion, the roots of it have to be set firmly in the minds of the participants. You can’t build something lasting via coercion and force, because as soon as the coercion is taken away by whatever means, the entire house of cards will collapse. If you want lasting civilization and proper cooperative behavior, that has to be inculcated with mother’s milk and made a part of every members upbringing. Attempting to enforce “civilization” via institutions and agencies of your state-structure are not going to be enough, and will not work over the long haul. We’re witnessing the failure of these structures right now, in real time.

    7. Lucretius Says:

      Gödel himself provided the following extension to his theorem: “A completely unfree society (i.e., one proceeding in everything by strict rules of “conformity”) will, in its behavior, be either inconsistent or incomplete, i.e., unable to solve certain problems, perhaps of vital importance. Both, of course, may jeopardize its survival in a difficult situation. A similar remark would also apply to individual human beings.”

    8. David Foster Says:

      Lucretius…how interesting, didn’t realize he’d said that.

      Someone should tell the EU.

    9. Lucretius Says:

      I suspect Gödel had in mind the completely unfree societies of Soviet Russia and Nazi Germany (he was Austrian but as far as I know never returned to Austria after the Anschluss).

    10. drjim Says:

      “Murphy’s Law” in action…..

    11. Christopher Says:

      I engineering this is known as “idiot proofing”, except you will always find a bigger idiot who circumvent your efforts.

    12. Christopher Says:

      In engineering this is known as “idiot proofing”, except you will always find a bigger idiot who circumvent your efforts.

    13. Christopher B Says:

      As I said on another thread, the documentation never matches how the system actually runs. This is one of the reasons. You can never fully describe the behavior of a system under all input/output conditions. You can get all of the trivial ones, and some of the complex ones, but not all of them, and any general statement about the operation of the system will be untrue in at least some of the conditions.

    14. OBloodyHell Says:

      }}} But adding more detail probably like adding more axioms to one of Goedel’s formal systems…so the additional analogous theorem is: No matter how detailed the instructions for doing something may be, there will always be a way for someone to interpret them incorrectly.

      You cannot make something foolproof, because fools are so ingenious.


      If you make something foolproof, only a fool will want to use it.


      One outgrowth of fool-proofing is the that the applicable Darwinian process will then be forced to invent a better fool.

    15. David Foster Says:

      While ‘idiot proofing’ is not possible, ‘mistake proofing’…a misnomer, should really be called ‘mistake resistance’ or something like that…is an important activity that we need more of.

      Here’s an example…two packagings of the same medicine, one of 10 units and the other of 10,000 units, looked a lot alike. Disasters happened. The packaging was changed…different colors, different shapes and sizes. The hospital chose to keep using the old packaging they had in inventory, rather than throwing it away.

      Guess what happened?

      (That’s why it should be called ‘mistake resistance’ rather than ‘mistake proofing’)

      An example in aviation is the different shapes which are given to the landing gear control and the flap control…the gear control is shaped like a wheel, the flap control like a flap.

    16. Anonymous Says:

      The more detailed and lengthy instructions are the less likely people are to read them, understand them or remember them. Common prior related experience helps by allowing for eliminating understood ideas or processes. Repetition in performing the task reduces the need to refer to the instructions as often.

      When instructions are written by lawyers to ensure that the idiot has everything spelled out, the normal person ignores the instructions as too tedious. The idiot won’t understand that much info. Performance generally falls, but the lawyer is pleased that the written document contained every possible detail and due diligence has been met.

      Don’t put your foot under the running power mower!


    17. MCS Says:

      I was looking at the manual for something or other Friday. There was about 15 pages of different warnings that I completely ignored. This means that if there was something that wasn’t completely obvious, I missed it. I’ll have to trust that all my years of experience after which I still have all my fingers, toes, both eyes and eye brows will see me through. As far as communicating anything, null.

      I was also looking at a Safety Data Sheet to see how we could ship some material. Occasionally someone will mistakenly put useful information in one, not this time. My favorite is the normal statement on disposal that tells you you follow local laws without, of course, giving enough information that you can tell which might be applicable.

      The joke is on the supplier of the SDS since this means we will have to choose the most expensive shipping option and the slowest. As in $1K+ and 4-8 weeks.

    18. Assistant Village Idiot Says:

      We counsel about psychiatric medications and legal requirements for conditional discharges. I have very specific directions to read these to patients word-for-word, or “Hearing Officers” – many of whom resent that they are not judges – will overturn a patient’s being sent back to the hospital. I am semi-retired and old, and I don’t care if they fire me. I would rather my patients understood me and so explain it to them in plain, sometimes blunt English. Younger people feeding families usually don’t have that flexibility.

    19. Mike K Says:

      Here’s an example…two packagings of the same medicine, one of 10 units and the other of 10,000 units, looked a lot alike. Disasters happened. The packaging was changed…different colors, different shapes and sizes. The hospital chose to keep using the old packaging they had in inventory, rather than throwing it away.

      I have a similar story in my book. The County Hospital used to make their own IV solutions. One solution was distilled water. The bottle had a pink label and a warning, “Must be made isotonic before use.” The hospital changed suppliers and the pink label went away. They killed a patient of mine by using the new bottle and not adding glucose or salt to make it isotonic. Small print is not enough.

    20. David Foster Says:

      Former Soviet officer ‘Victor Suvorov’ provided an interesting example of mistake possibilities and improved mistake resistance:

      The calibre of the standard Soviet infantry weapon is 7.62mm. In 1930, a 7.62mm `TT’ pistol was brought into service, in addition to the existing rifles and machine-guns of this calibre. Although their calibre is the same, the rounds for this pistol cannot, of course, be used in either rifles or machine-guns.
      In wartime, when everything is collapsing, when whole Armies and Groups of Armies find themselves encircled, when Guderian and his tank Army are charging around behind your own lines, when one division is fighting to the death for a small patch of ground, and others are taking to their heels at the first shot, when deafened switchboard operators, who have not slept for several nights, have to shout someone else’s incomprehensible orders into telephones–in this sort of situation absolutely anything can happen. Imagine that, at a moment such as this, a division receives ten truckloads of 7.62mm cartridges. Suddenly, to his horror, the commander realises that the consignment consists entirely of pistol ammunition. There is nothing for his division’s thousands of rifles and machine-guns and a quite unbelievable amount of ammunition for the few hundred pistols with which his officers are armed.
      I do not know whether such a situation actually arose during the war, but once it was over the `TT’ pistol–though not at all a bad weapon–was quickly withdrawn from service. The designers were told to produce a pistol with a different calibre. Since then Soviet pistols have all been of 9mm calibre. Why standardise calibres if this could result in fatally dangerous misunderstanding?
      Ever since then, each time an entirely new type of projectile has been introduced, it has been given a new calibre.