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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18 Rules between Men and Women

Got this in email today from a friend at work…had to share this!

1. The Female always makes THE RULES.

2. THE RULES are subject to change without notice.

3. No Male can possibly know all THE RULES.

4. If the Female suspects the Male knows all THE RULES, she must immediately change some or all of THE RULES.

5. The Female is never wrong.

6. If it appears the Female is wrong, it is because of a flagrant misunderstanding caused by something the Male did or said wrong.

7. If Rule #6 applies, the Male must apologize immediately for causing the misunderstanding.

8. The Female can change her mind at any given time.

9. The Male must never change his mind without the express, written consent of The Female.

10. The Female has every right to be angry or upset at any time.

11. The Male must remain calm at all times, unless the Female wants him to be angry or upset.

12. The Female must, under no circumstances, let the Male know whether she wants him to be angry or upset.

13. The Male is expected to read the mind of the Female at all times.

14. At all times, what is important is what the Female meant, not what she said.

15. If the Male doesn’t abide by THE RULES, it is because he can’t take the heat, lacks ackbone, and is a wimp.

16. If the Female has PMS, all THE RULES are null and void and the Male must cater to her every whim.

17. Any attempt to document THE RULES could result in bodily harm.

18. If the Male, at any time, believes he is right, he must refer to Rule #5