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'axiom of foundation'
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| Title of object: |
axiom of foundation |
| Canonical Name: |
AxiomOfFoundation |
| Type: |
Definition |
| Created on: |
2002-09-28 19:26:46 |
| Modified on: |
2002-09-28 19:53:45 |
| Classification: |
msc:03C99 |
| Synonyms: |
axiom of foundation=foundaton
axiom of foundation=regularity
axiom of foundation=axiom of regularity |
Revision comment (for changes between this and next version):
| Changes for correction #4179 ('artinian sets'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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%\PMlinkescapeword{theory} |
Content:
The \emph{axiom of foundation} (also called the \emph{axiom of regularity}) is an axiom of ZF set theory prohibiting circular sets and sets with infinite levels of containment. Intuitively, it states that every set can be built up from the empty set. There are several equivalent formulations, for instance:
For any nonempty set $X$ there is some $y\in X$ such that $y\cap X=\emptyset$.
For any set $X$, there is no function $f$ from $\omega$ to the transitive closure of $X$ such that $f(n+1)\in f(n)$.
For any formula $\phi$, if there is any set $x$ such that $\phi(x)$ then there is some $X$ such that $\phi(X)$ but there is no $y\in X$ such that $\phi(y)$. |
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