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In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: There is no non-trivial elementary embedding of the universe V into itself.
Gödel showed that any such theory also including a statement of its own consistency is inconsistent. A key element of the proof is the use of Gödel numbering to construct a "Gödel sentence" for the theory, which encodes a statement of its own incompleteness: "This theory can't prove this statement"; or "I am not provable in this system".
Gödel's incompleteness theorems show that Hilbert's program cannot be realized: if a consistent computably enumerable theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strong enough to model a weak fragment of arithmetic (Robinson arithmetic suffices), then the theory cannot prove its own ...
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theorem, whose exact values are independent of ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal between ℵ 1 and 2 ℵ 0 .
Sometimes called a "budget letter" or proof of income letter, the benefit verification statement from Social Security is used for several different instances where proof of your status or income is...
Your Social Security statement can also be used as proof of income. See: Your Social Security Statement Is About To Get a Lot Easier To Read Find: ...
This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system. The proof of the second incompleteness theorem is obtained by formalizing the proof of the first incompleteness theorem within the system F itself.