Page "Reverse mathematics" Paragraph 8

Another effect of using second-order arithmetic is the need to restrict general mathematical theorems to forms that can be expressed within arithmetic.

For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".

In practical terms, this means that theorems of algebra and combinatorics are restricted to countable structures, while theorems of analysis and topology are restricted to separable spaces.

Many principles that imply the axiom of choice in their general form ( such as " Every vector space has a basis ") become provable in weak subsystems of second-order arithmetic when they are restricted.

For example, " every field has an algebraic closure " is not provable in ZF set theory, but the restricted form " every countable field has an algebraic closure " is provable in RCA < sub > 0 </ sub >, the weakest system typically employed in reverse mathematics.