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'axiom of infinity'
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| Title of object: |
axiom of infinity |
| Canonical Name: |
AxiomOfInfinity |
| Type: |
Axiom |
| Created on: |
2003-07-03 11:52:04 |
| Modified on: |
2003-07-03 11:52:04 |
| Classification: |
msc:03E30 |
| Synonyms: |
axiom of infinity=infinity |
Revision comment (for changes between this and next version):
| Changes for correction #2130 ('missing }'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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Content:
There exists an infinite set.
The Axiom of Infinity is an axiom of Zermelo-Fraenkel set theory.
At first glance, this axiom seems to be ill-defined. How are we to know what
constitutes an infinite set when we have not yet defined the notion of a
finite set? However, once we have a theory of ordinal numbers in hand, the axiom makes sense.
Meanwhile, we can give a definition of finiteness that does not rely upon
the concept of number. We do this by introducing the notion of an inductive
set. A set $S$ is said to be inductive if $\emptyset \in S$
and for every $x \in S$, $x \cup \{ x \} \in S$. We may then state the
Axiom of Infinity as follows:
There exists an inductive set.
In symbols:
\exists S [\emptyset \in S \land (\forall x \in S)[x \cup \{ x \} \in S]]
We shall then be able to prove that the following conditions are equivalent:
\begin{enumerate}
\item There exists an inductive set.
\item There exists an infinite set.
\item The least nonzero limit ordinal, $\omega$, is a set.
\end{enumerate |
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