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Real Complete Fields, Natural Numbers, and Higher-Order Logics

April 28, 2013

When encountering Gödel’s incompleteness theorems, usually towards the end of an introduction,  we find statements that explain that the real numbers are complete, and Gödel’s incompleteness theorems do not apply.  Incompleteness of the natural numbers is then argued to not contradict this since the natural numbers are not contained in the real numbers. And this is it, often with no embellishment. Confusion ensues.

Natural numbers not contained in the real numbers? Surely the natural numbers are a subset of the real numbers? Well, yes they are, and this is close to the point of the matter. Gödel’s theorems are a result in first-order logic, and as such, we can quantify only over the objects in a theory. Second-order logics allow us instead to quantify over functions, predicates, and most relevantly here, over sets of numbers. The natural numbers are contained in the real numbers in a second-order theory. The containment here is not a set-theoretic containment, but a containment in the sense of being definable in a first-order theory. Mathematicians should beware when reading logic texts.

“So? Why not just work in second-order logic?”, a practical but impatient person bellows from the back. The answer depends on what kind of mathematician you are. You may think since number theorists are concerned with properties of integers, that they would be particularly aggrieved by incompleteness; some of them are, but often, usually analytic number theorists, actually work over the complex numbers. Think analytic continuations of zeta functions, and such. Here, one can do mathematics without giving Gödel a thought.

Complications from incompleteness arise in arithmetic geometry, however. Maybe I will write more on this later, but in algebraic geometry, the methods developed are roughly the same for every dimension, and only get more complicated. This has to be the most understated use of “only” ever recorded; 3-folds are significantly more difficult to classify than algebraic surfaces. For arithmetic geometry, the methods developed depend essentially on the dimension.

So, are there any concrete examples where incompleteness is manifest? This will be the subject of the next post, and will discuss the Paris-Harrington theorem.


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