Incompletion

When we talk about mathematics, we talk about logic, conciseness, completeness and abstraction. So that means, everything in this world, can be described in terms of mathematics? Even religion? Atleast thats a common belief of a lot of atheists that if something cant be proven through logic or lacks empirical evidence, it shouldnt be believed. Have they tried proving mathematics to be correct? Hmmm…so thats where the problem starts because of Gödel’s incompleteness theorems.

In 1931, the Czech-born mathematician Kurt Gödel demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules. He proved it impossible to establish the internal logical consistency of a very large class of deductive systems – elementary arithmetic, for example – unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves.

The proof of Gödel’s Incompleteness Theorem is very simple, a lil confusing but surely makes it clear that even logic can be illogical. His basic procedure is as follows:

  1. For instance there is a machine, UTM, that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
  2. Lets call the program P(UTM) for Program of the Universal Truth Machine.
  3. Gödel writes out the following sentence: “The machine constructed on the basis of the program P(UTM) will never say that this sentence is true.” Call this sentence G. Note that G is equivalent to: “UTM will never say G is true.”
  4. Now the UTM is asked whether G is true or not.
  5. If UTM says G is true, then “UTM will never say G is true” is false. If “UTM will never say G is true” is false, then G is false (remember G = “UTM will never say G is true”, point 3). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
  6. We have established that UTM will never say G is true. So “UTM will never say G is true” is in fact a true statement. So G is true (since G = “UTM will never say G is true”).
  7. So we finally we know something that although G can be true, it cannot be universally true!

G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics!

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13 thoughts on “Incompletion”

  1. Er, you lost me somewhere in point no. 5, even though the proof you have written down is much simpler than the one present in that Wikipedia article. (Probably I shouldn’t have tried understanding it after a tiring day!)

    Those articles mention the liar paradox, though, and it was exactly what came to my mind while I was reading point no. 5. Also, those silly riddles of having the choice of asking a single (but same) question to two people: one always speaks the truth, the other always lies, but you don’t know who the liar is, and you need to know which path to take, blah blah blah.

    But anyway, this stuff is compelling. I’ll come back to it when I am not feeling tired and hungry!

  2. hmmm
    lets go a lil deeper then.
    now if you have G=UTM will never say G is true
    if UTM says G is true, G becomes false automatically as UTM is not capable of saying G is true and the UTM has made a satement which is an example of the liar paradox. Now UTM cannot even saythat G is false because that would mean that UTM is saying NOT(UTM will never say G is true ). (The two NOTs would actually nullify each other) The resultant would again mean that UTM is saying that G is true.
    ummm…i hope i just made some sense..its too confusing 😛

  3. Okay, too much mathematical jargon for a social science person … but for what it’s worth … faith can’t be proved by mathematics. No. Not even explained.

  4. allow me to let you in on a lil secret my friend! This Gobel fella had a sense of humor. Here’s what he was prolly thinking when he wrote down his seven steps:
    1. Lets write some bullshit.
    2. a lil more detail will make it look sophisticated!
    3. Lets go crazy!
    4. We need to check if people are actually buying this!
    5. Yes they are, whos going to stop me?!
    6. Damn, still not figured it out!
    7. Humanity is doomed!

    hahaha i hope this does not offend the gods of mathematics! 😛

  5. @ Mahwash
    well ill tell u wat prompted me to study the theory of incompletion. i was having a discussion with two mathematicians about how the younger generation is always looking for logic. As in it becoming very ‘trendy’ to say that ‘oh, how can i simply believe in that just coz its written in the Qur’an, i need a logical explanation.’ so this is wen one of those two guys told me about this theory of incompletion. It was quite intriguing to know that even logic itself can be proven to be illogical. So basically in other words, there’s nothing like ‘logic’ that exists and all the laws of physics and mathematics are merely assumptions (for example, there is nothing like time and space in reality but its an assumption that we have made to make our lives simpler and to explain various things which would otherwise have been impossible).

    @farooqk
    lol..i like your proof more than Gödel’s 😉

  6. @karachiwalikhatoon

    Dont tell anyone i told u, i dont want to serve as a labrat for american scientists trying to discover where pure genius comes from! 😀

    @Saadat

    try searching ‘Kimbo Slice’ on you tube! I promise you a whole lot more entertainment! 😛

  7. There is a similar concept we learnt in our university… Although it was initially proposed by Einstein, “Everything is relative”, In our universe everything is indeed relative which includes truths and lies.

    Then we have the Heisenberg uncertainty principle which if ported to the topic wil sound something like “At the basic level, There is a chance/probability of any truth to be false.” So basically if you take a closer look at the atheism’s principles, there is always a chance of everything they know to be proven wrong, and since we don’t know the probability of this event, it can very much be so already.

    Lastly, any doctrine or set of beliefs believed and incorporated to the point of being rigid and inflexible tending towards idealism if not extremism, even if it is as ‘logical’ as atheism is bound to stumble and crash onto itself.

  8. just as one of my friends says..’Everything is relative, except for one absolute: Allah’.
    Khair, you are quite right in the sense that even truths and lies are relative. A truth for one might be a lie for another and that reminds me my favourite definition for a terrorist: A terrorist for one is the freedom fighter for another. I guess, its our perception, our point of view that makes the difference

  9. The quote was by Thomas Jefferson I think. And if you track the record of science, The theories and Laws that it proposes have been in a state of constant flux. First there were the Newtonian Laws then came in relativity and then the Grand Unification Theory. So if you take it from an atheist point of view, the logic they propose is always changing and orthogonally so.

    And come to think of it, the most hailed scientist of all, Einstein, was a spiritual, if not religious, man.

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