See also
Consistency,
Gödel's Completeness Theorem,
Gödel's
Incompleteness Theorems,
Gödel's
Second Incompleteness Theorem,
Goodstein's
Theorem,
Hilbert's Problems,
Kreisel
Conjecture,
Natural Independence
Phenomenon,
Number Theory,
Paris-Harrington
Theorem,
Peano Arithmetic,
Presburger
Arithmetic,
Recursively Enumerable Set,
Richardson's Theorem,
Undecidable
Explore with Wolfram|Alpha
References
Barrow, J. D. Pi in the Sky: Counting, Thinking, and Being. Oxford, England: Clarendon Press,
p. 121, 1993.Erickson, G. W. and Fossa, J. A. Dictionary
of Paradox. Lanham, MD: University Press of America, pp. 74-75, 1998.Franzén,
T. "Gödel on the Net." http://www.sm.luth.se/~torkel/eget/godel.html.Gödel,
K. "Über Formal Unentscheidbare Sätze der Principia Mathematica
und Verwandter Systeme, I." Monatshefte für Math. u. Physik 38,
173-198, 1931. https://doi.org/10.1007/BF01700692.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.Kolata, G. "Does Gödel's Theorem Matter to Mathematics?"
Science 218, 779-780, 1982.Rucker, R. Infinity
and the Mind: The Science and Philosophy of the Infinite. Princeton, NJ:
Princeton University Press, 1995.Smullyan, R. M. Gödel's
Incompleteness Theorems. New York: Oxford University Press, 1992.Whitehead,
A. N. and Russell, B. Principia
Mathematica. New York: Cambridge University Press, 1927.Wolfram,
S. A
New Kind of Science. Champaign, IL: Wolfram Media, pp. 782,
2002.Referenced on Wolfram|Alpha
Gödel's First
Incompleteness Theorem
Cite this as:
Weisstein, Eric W. "Gödel's First Incompleteness
Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html
Subject classifications
is a consistent axiomatic
system whose axioms form a recursively
enumerable set and which is strong enough to formalize elementary arithmetic,
then 
is incomplete. In other words, there is a proposition 
such that neither 
nor its negation can be proved
in 
.
