Page "Presentation of a group" Paragraph 34

Every finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented.

However a theorem of Graham Higman states that a finitely generated group has a recursive presentation if and only if it can be embedded in a finitely presented group.

From this we can deduce that there are ( up to isomorphism ) only countably many finitely generated recursively presented groups.

Bernhard Neumann has shown that there are uncountably many non-isomorphic two generator groups.

Therefore there are finitely generated groups that cannot be recursively presented.