Topics Mathematics

Question Hello philosophers, I have yet another question. This time it's on the fundamental foundations of mathematics. I would like to know what Gödel's incompleteness theorem and inconsistency theorem actually stated. Intuitively, math seems logical, in the physical world, if you have two inanimate objects say two pencils laying on the table is it not logical that if you take one away you are only left with one on the table? An ex- professor of mine once told us in mathematics that ZF math was inconsistant and if we could prove that math does not work not only would we win a Fields Prize but we would also be the Herod of children all over the world ( assuming kids don't like to learn fundamental mathematics). Thank You again, Dale G.

Accepted:
May 8, 2014

Comments

You asked what Goedel's incompleteness and inconsistency theorems state.

Goedel proved two theorems known as his incompleteness theorems; I don't know of any called an "inconsistency" theorem (of course, he proved many other theorems, too!):

Informally, the first one--perhaps it is also the most famous one--says that any formal system that is based on first-order logic plus Peano's axioms for arithmetic is such that:

if it is consistent

(that is, if no contradiction can be proved in it),

then it is incomplete

(that is, there is some proposition P in the language of the system such that neither P nor not-P can be proved in the system;

presumably, only one of P and not-P is true; hence, there is some proposition in the language of the system that is true but unprovable in the system).

Even more informally, an English-language version of the true-but-unprovable sentence can be expressed thus:

This sentence is not provable.

(If it is false, then it is provable, hence true. So it can't be false. Hence it is true. Hence it is unprovable.)

Informally, the second theorem that Goedel proved says that if the system mentioned above is consistent, then its consistency cannot be proved in that system.

(You might be able to prove its consistency in some other system, however.)

You ask if it is logical that only one pencil would be left on a table if you had two pencils and removed one. Yes, it is. I'm not sure what that has to do with Goedel.

I have also never heard that ZF is inconsistent. I suppose that, if you could prove that it was, you might win the Fields medal. (But, even if you could prove that and did win the medal, I'm not sure that being "the Herod of children all over the world" would be a good thing, though being a hero of children might be :-)

For good introductions to Goedel's theorems, see Douglas Hofstadter's wonderful book Goedel Escher Bach, or Torkel Franzen's equally wonderful--but more technical--book Goedel's Theorem: An Incomplete Guide to Its Use and Abuse.