Example of an Outcome of Gödel’s Incompleteness Theorems

Question

If you look up Gödel’s Incompleteness Theorems, you will often read that a consequence of them is that some recognizable truths about arithmetic are unprovable given the set of axioms of arithmetic (the Peano Axioms?) Could anyone give me an example of such “recognizable truths?”

Answer 1

There are actually two incompleteness theorems. The first states there are undecidable statements in a consistent system that proves the Peano axioms. The second states the system includes, but cannot prove, the statement that it's consistent.

For more directly arithmetical statements that are undecidable in the Peano axioms, see here and here.