Two results, published together in 1931, that set a permanent limit on what any sufficiently powerful formal axiomatic system can achieve. The first shows that any consistent system able to express basic arithmetic must contain statements that are true but unprovable within the system; the second, a corollary, shows that such a system cannot prove its own consistency using only its own resources. Together they showed that David Hilbert's program to place all of mathematics on one finite, provably consistent axiomatic foundation could not succeed in the most literal, most ambitious form Hilbert had proposed.
Facts
StatementFirst incompleteness theorem: any consistent formal axiomatic system powerful enough to encode basic arithmetic contains true statements that cannot be proved within the system. Second incompleteness theorem: no such system can prove its own consistency using only resources available within the system. 1 Classification
Statement Form FAQs
Does Gödel's second incompleteness theorem mean a formal system can never be shown to be consistent?
It says such a system cannot prove its own consistency, assuming it is consistent.
Not in that sweeping form. The theorem says that a formal system with enough elementary arithmetic cannot prove that the system itself is consistent, on the assumption that it is in fact consistent. It is a statement about what a system can prove about itself.
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Associated With
Formal System, Concepts Godel's theorems state that any consistent formal system able to encode elementary arithmetic is incomplete; the theorem is about a formal system exactly, not an axiom, which is why the sibling edges lane refused to substitute the axiom concept for this edge.
Additional Source Formal System (Wikipedia)Lead section
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Source The Stanford Encyclopedia of Philosophy
Additional Source Wikipedia: Godel's Incompleteness TheoremsLead section
Additional Source Wikipedia: Godel's Incompleteness TheoremsImplications for consistency proofs section
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Source The Stanford Encyclopedia of Philosophy
Additional Source Wikipedia: Godel's Incompleteness TheoremsLead section
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Also in Philosophy Atlas: which discusses this subject.
Source The Stanford Encyclopedia of Philosophytier 1
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1. The Stanford Encyclopedia of Philosophy
Center for the Study of Language and Information, Stanford UniversityFirst incompleteness theorem
Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
Second incompleteness theorem
For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself.
View the Source Godel (Wiktionary)
Wikimedia FoundationPronunciation section, GermanQuote, Pronunciation section, German
/ˈɡøːdəl/, [ˈɡøː.dl̩]
Wikipedia: Godel's Incompleteness Theorems
Wikimedia FoundationProved By: Kurt Gödel, Lead section
These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in philosophy of mathematics.
In Branch: Logic and Foundations, Lead section
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories.
In Branch: Proof Theory, Implications for consistency proofs section
Gentzen's theorem spurred the development of ordinal analysis in proof theory.
In Group: Mathematical Logic and Model Theory, Lead section
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability
View the Source Formal System (Wikipedia)
Wikimedia FoundationAssociated With: Formal System, Lead sectionQuote, Associated With: Formal System, Lead section
A formal system (or deductive system) is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
Dissenting Readings (1 dissenting reading)
Description
Godel's incompleteness theorems show more than a limit internal to formal systems: because any consistent formalization of arithmetic contains truths it cannot prove, and a human mind can nonetheless recognize those truths as true, no formal system, and therefore no Turing machine, can fully capture what a human mind can do when it reasons about arithmetic. Human minds are therefore not equivalent to formal systems in the relevant sense, contrary to a purely mechanist view of mind. This reading, first argued by Lucas in 1961 and later revived and extended by Roger Penrose, is rejected by most logicians and philosophers of mind working on the theorems today, who hold that the argument equivocates on what it would mean for a mind to know its own consistency and that no version of it has been made rigorous enough to draw the conclusion it wants.
A dissenting reading, from J. R. LucasThe Stanford Encyclopedia of Philosophy, Center for the Study of Language and Information, Stanford University
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