In principle a statement that involves only finite sets of natural numbers can be proved by exhaustive calculation. A statement involving infinity in an essential way cannot. This means there is a gap in certitude between finite and infinite mathematics. There's many ways to describe this divide. For example, an arithmetic statement involving unbounded quantifiers. You can measure how much such a statement fails to be finite by counting the alternation of universal and existential unbounded quantifiers. This is a measurement of the naive logical complexity of a statement.

Godels incompleteness theorems can be seen as a statement about the gap between finite and infinite mathematics. Decidability, semidecidability and undecidability can be seen as the relationship between boundedly quantified arithmetic statements, statements with one unbounded existential quantifier, and statements with one unbounded universal quantifier.

Another avenue of exploring the gap between finite and infinite mathematics is via linear logic. There the thesis is that contraction, the logical reuse of variables, is where infinity creeps into logical reasoning. Indeed logic without contraction is quite tame. Logic with unlimited contraction is wild. Surprisingly there are logics with an intermediate strength of contraction: so-called light linear logics. These can classify reasoning that embodies polynomial time computation or elementary time computation. So in another sense infinity can be measured by algorithmic complexity.