Final answer:
The correct option is 3). Kurt Gödel proved with his Incompleteness Theorems that no consistent formal system capable of arithmetic can be both complete and prove its own consistency. His work indicates the existence of mathematical truths that cannot be proven within such a system, which has profound implications in the realm of formal logic and mathematics.
Step-by-step explanation:
Kurt Gödel showed that it's not possible to find a consistent system with only the statements Gödel's Incompleteness Theorems. His theorems demonstrate the inherent limitations of every formal axiomatic system capable of modeling basic arithmetic. Gödel's first incompleteness theorem states that any consistent formal system that is rich enough to have within it a certain amount of arithmetic cannot be both complete and consistent. This means that there will always be truths in the system that cannot be proven within the system. The second theorem extends upon this, showing that no such system can demonstrate its own consistency. These findings have significant implications in mathematics, logic, and philosophy, as they imply that mathematical truths can exist that are beyond the reach of human derivation within a formal system.
It is important to distinguish Gödel's work from other statements such as the Axiom of Choice, the Continuum Hypothesis, and the Zermelo-Fraenkel Set Theory, each of which addresses other foundational aspects of mathematics, and are not by themselves indicative of the consistency or completeness of a system. The Axiom of Choice is an important principle in set theory that allows the construction of sets in a specific manner, while the Continuum Hypothesis deals with the possible sizes of infinite sets, and Zermelo-Fraenkel Set Theory is a foundational system for much of mathematics, excluding the Axiom of Choice unless specified (ZFC).