While we have shown that every loop that terminates has a variant, this does not mean that the well- foundedness of the loop iteration can be proven.
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The concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917) cf. and . Mirimanoff called a set x "regular" (French: "ordinaire") if every descending chain x ∋ x1 ∋ x2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets; in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology).
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In 1917, Dmitry Mirimanoff introduced the concept of well-foundedness of a set: :A set, x0, is well-founded if it has no infinite descending membership sequence \cdots \in x_2 \in x_1 \in x_0. In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity. In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC− (that is, ZFC without the axiom of regularity) that well-foundedness implies regularity. In variants of ZFC without the axiom of regularity, the possibility of non- well-founded sets with set-like ∈-chains arises.
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Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well- foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non- well-founded sets; he did not regard well-foundedness as an axiom. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until Peter Aczel’s hyperset theory in 1988.
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Well-foundedness fails specifically for rank-into-rank extenders; but Itay Neeman showed in 2004 that it holds for all weaker types of extender. The Mitchell rank of a measure is the ordertype of its predecessors under ◅; since ◅ is well-founded this is always an ordinal. A cardinal which has measures of Mitchell rank α for each α < β is said to be β-measurable.
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Mirimanoff in a 1917 paper introduced the concept of well-founded set and the notion of rank of a set.cf. and Mirimanoff called a set x "regular" (French: "ordinaire") if every descending chain x ∋ x1 ∋ x2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (well-foundedness) as an axiom to be observed by all sets; in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology).
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The homomorphism F is an isomorphism if and only if R is extensional. The well- foundedness assumption of the Mostowski lemma can be alleviated or dropped in non-well-founded set theories. In Boffa's set theory, every set-like extensional relation is isomorphic to set-membership on a (non-unique) transitive class. In set theory with Aczel's anti-foundation axiom, every set- like relation is bisimilar to set-membership on a unique transitive class, hence every bisimulation-minimal set-like relation is isomorphic to a unique transitive class.
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The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal). The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic). Many larger epsilon numbers can be defined using the Veblen function.
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In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating ARS is confluent precisely when it is locally confluent.Franz Baader, Tobias Nipkow, (1998) Term Rewriting and All That, Cambridge University Press Equivalently, for every binary relation with no decreasing infinite chains and satisfying a weak version of the diamond property, there is a unique minimal element in every connected component of the relation considered as a graph. Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Gérard Huet in 1980.
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