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'transfinite induction'
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| Title of object: |
transfinite induction |
| Canonical Name: |
TransfiniteInduction |
| Type: |
Theorem |
| Created on: |
2002-02-25 18:55:24-05 |
| Modified on: |
2002-02-25 19:35:40-05 |
| Classification: |
msc:03F07, msc:03B48 |
| Keywords: |
well ordered set, well-ordering principle |
| Synonyms: |
transfinite induction=principle of transfinite induction |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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Content:
Given a well ordered set $A$ and a proposition $P(a)$ formulated for every $a \in A$, the principle of \emph{transfinite induction} states that $P(a)$ is true for all $a \in A$ if the following criteria are met:
\begin{enumerate}
\item $P(a)$ is true for the least element of $A$.
\item If $P(a)$ is true for all $a < b$, then $P(b)$ is true.
\end{enumerate}
The principle of transfinite induction is very similar to the principle of finite induction, except that the set of natural numbers has been replaced with an arbitrary well ordered set. Note that since any set can be well ordered (by the well-ordering principle), it is possible to apply transfinite induction to any set whatsoever. |
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