Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.
Gödel's Second Incompleteness Theorem
See also
Gödel's Completeness Theorem, Gödel's First Incompleteness TheoremExplore with Wolfram|Alpha
References
Gödel, K. "Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme, I." Monatshefte für Math. u. Physik 38, 173-198, 1931.Gödel, K. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. New York: Dover, 1992.Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 17, 1989.Rucker, R. Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1995.Cite this as:
Weisstein, Eric W. "Gödel's Second Incompleteness Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoedelsSecondIncompletenessTheorem.html