Axiomatic set theory

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.

Well-foundedness and hypersets

In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness:

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: a set, x0, is well-founded iff it has no infinite descending membership sequence:

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:: · · · in x_2 in x_1 in x_0.

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In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is,

Related Topics:
Axiom of regularity

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ZFC without the axiom of regularity) that well-foundedness implies regularity.

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In variants of ZFC without the axiom of regularity, the possibility of non-well-founded sets arises. When working in such a system, a set that is not necessarily well-founded is called a hyperset. Clearly, if A ∈ A, then A is a non-well-founded hyperset.

Related Topics:
Axiom of regularity - Non-well-founded sets - Hyperset

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The theory of hypersets has been applied in computer science (process algebra and final semantics), linguistics (situation theory), and philosophy (work on the Liar Paradox).

Related Topics:
Process algebra - Final semantics - Situation theory - Liar Paradox

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Three distinct anti-foundation axioms are well-known:

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• AFA (‘Anti-Foundation Axiom’) — due to M. Forti and F. Honsell;
• FAFA (‘Finsler’s AFA’) — due to P. Finsler;
• SAFA (‘Scott’s AFA’) — due to Dana Scott.

The first of these, AFA, is based on accessible pointed graphs (apg) and states that two hypersets are equal if and only if they can be pictured by the same apg. Within this framework, it can be shown that the so-called Quine atom, formally defined by Q={Q}, exists and is unique.

Related Topics:
Accessible pointed graph - Quine

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It is worth emphasizing that hyperset theory is an extension of classical set theory rather than a replacement: the well-founded sets within a hyperset domain conform to classical set theory.

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