Extensionality

Extensionality

Extensionality refers to the principle or axiom that states that two sets are equal if and only if they have the same elements. This means that the identity of a set is determined solely by its members, and not by any other properties or characteristics. The Axiom of Extensionality is a fundamental principle in classical set theory that guarantees the uniqueness of a set defined using the Comprehension Axiom Schema. It allows us to determine a subset of a set based on a given predicate, and ensures that this subset is unique and well-defined. Extensionality is closely related to the concept of mereological pluralism, which acknowledges that objects may have systematic differences with respect to their whole or their parts, but still share a common identity based on their extension or set of members.From: Engineering Systems Integration [2016], Set-theoretic relations for metasets [2022]

Preliminaries

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Published in Aliakbar Montazer Haghighi, Indika Wickramasinghe, Probability, Statistics, and Stochastic Processes for Engineers and Scientists, 2020

Aliakbar Montazer Haghighi, Indika Wickramasinghe

The axiom of extensionality speaks about the type of things sets are. The elements of each set distinguish one set from another if they are to be different; otherwise, they are the same. So, when there is an element in one that is not in the other, the two sets are not equal.

Set-theoretic relations for metasets

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Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022

Bartłomiej Starosta

The inclusion of metasets may be characterised in terms of the conditional membership relation. Proposition 5.11. Let, be arbitrary metasets and. Proof. See proof of Theorem 5.8.The following relationship between conditional equality and conditional inclusion, similar to the one known from classical set theory, holds as well.Corollary 5.12. Let, be arbitrary metasets and let. Proof. Refer to interpretations.The Axiom of Extensionality (33) in the classical set theory guarantees that a set defined with Comprehension Axiom Schema (39) is unique: given a set and a predicate , we can determine a subset whose members are precisely the members of that satisfy : In order to formulate a corresponding property for metasets, we need to distinguish a class of metasets with non-binary membership on the first level of membership hierarchy at most.Definition 5.13. A metaset whose domain consists of canonical metasets is called a first-order metaset.

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Preliminaries

Set-theoretic relations for metasets