Posts tagged with kurt Gödel

Brought to you by into the continuum:

To paraphrase Boolos paraphrasing Gödel paraphrasing himself:

It is true that 2 plus 2 is 4. I hope you can at least believe that much. Of course, it can be proved ⊢ that 2 plus 2 is 4. It can also be proved that it can be proved ⊢⊢ that 2 plus 2 is 4.

Moreover, it can be proved that it can be proved that it can be proved ⊢⊢⊢ that 2 plus 2 is 4. We could continue in this seemingly trivial manner, but what we are hoping for is that 2 plus 2 is not 5. Fortunately, it can be proved ⊢ that 2 plus 2 is not 5, and that too can be proved ⊢⊢.

Now just to be on the safe side what we really should ask is if it can be proved that it can’t be proved ⊢⊬ that 2 plus 2 is 5…. It can’t! ⊬ ⊢⊬ In fact, no claim of the form “claim X can’t be proved” can be proved.∀X:(⊢⊬X)→⊥ or ∀X:⊬X

So thanks to Gödel … it can be proved that if it can be proved that it can’t be proved that 2 plus 2 is 5, then it can be proved that 2 plus 2 is 5. ⊢(⊢⊬X→⊢X) [Which is a contradiction.] I would rather take an incomplete theory over [an unquestionably wrong, self-]inconsistent one any day. Wouldn’t you?

… pretty tanpura drones in the pitches of C# and A, respectively (which I suggest listening to with eyes closed and nice headphones).

If on reading that you thought: “the ⊢⊬⊢⊢ symbols…could you do math with those operators?” then here’s some more at the SEP.




  1. The world is rational.
  2. Human reason can, in principle, be developed more highly (through certain techniques).
  3. There are systematic methods for the solution of all problems (also art, etc.).
  4. There are other worlds and rational beings of a different and higher kind.
  5. The world in which we live is not the only one in which we shall live or have lived.
  6. There is incomparably more knowable a priori than is currently known.
  7. The development of human thought since the Renaissance is thoroughly intelligible (durchaus einsichtige).
  8. Reason in mankind will be developed in every direction.
  9. Formal rights comprise a real science.
  10. Materialism is false.
  11. The higher beings are connected to the others by analogy, not by composition.
  12. Concepts have an objective existence.
  13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
  14. Religions are, for the most part, bad— but religion is not.