for any formal, consistent system there will always be statements about the natural numbers that are true, but that are unprovable within the system.

arithmetical truth cannot be defined in arithmetic.

truth in the standard model of the system cannot be defined within the system

it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run.

there is a fundamental limit to the precision with which complementary pairs of physical properties of a particle can be known.

simply observing a situation or phenomenon necessarily changes that phenomenon

See also

• Wikipedia: Gödel's incompleteness theorems
• Wikipedia: Undecidable problem
• Wikipedia: Uncertainty principle
• Wikipedia: Tarski's undefinability theorem
• Wikipedia: Entscheidungsproblem
• Wikipedia: Observer Effect