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The real numbers can be defined synthetically as an ordered field satisfying some version of the completeness axiom.Different versions of this axiom are all equivalent in the sense that any ordered field that satisfies one form of completeness satisfies all of them, apart from Cauchy completeness and nested intervals theorem, which are strictly weaker in that there are non Archimedean fields ...
In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers , the complex numbers , and complete valued fields (such as the p -adic numbers ).
For example, the sequence of powers of two (1, 2, 4, 8, ...), the basis of the binary numeral system, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 100101 2 = 1 + 4 + 32). This sequence is minimal, since no value ...
The seldom-considered dual notion to a dcpo is the filtered-complete poset. Dcpos with a least element ("pointed dcpos") are one of the possible meanings of the phrase complete partial order (cpo). If every subset that has some upper bound has also a least upper bound, then the respective poset is called bounded complete. The term is used ...
Semantic completeness is the converse of soundness for formal systems. A formal system is complete with respect to tautologousness or "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system ...
This works with all perfect numbers () with odd prime p and, in fact, with all numbers of the form () for odd integer (not necessarily prime) m. Owing to their form, 2 p − 1 ( 2 p − 1 ) , {\displaystyle 2^{p-1}(2^{p}-1),} every even perfect number is represented in binary form as p ones followed by p − 1 zeros; for example:
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Please help improve this article by introducing more precise citations. ( January 2025 ) ( Learn how and when to remove this message ) In mathematical logic , a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable.