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| \section*{Definitions} |
\section*{Definitions} |
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| Let $(P,\leq)$ be a poset. A subset $A\subseteq P$ is said to be \emph{cofinal} in $P$ if for every $x\in P$ there is a $y\in A$ such that $x\le y$. |
Let $(P,\leq)$ be a poset. A subset $A\subseteq P$ is said to be \emph{cofinal} in $P$ if for every $x\in P$ there is a $y\in A$ such that $x\le y$. |
| A function $f\colon X\to P$ is said to be \emph{cofinal} if $f(X)$ is cofinal in $P$. |
A function $f\colon X\to P$ is said to be \emph{cofinal} if $f(X)$ is cofinal in $P$. |
| The least cardinality of a cofinal set of $P$ is called the \emph{cofinality} of $P$. |
The least cardinality of a cofinal set of $P$ is called the \emph{cofinality} of $P$. |
| Equivalently, the cofinality of $P$ is the least \PMlinkid{ordinal}{2787} $\alpha$ such that there is a cofinal function $f\colon\alpha\to P$. |
Equivalently, the cofinality of $P$ is the least \PMlinkid{ordinal}{2787} $\alpha$ such that there is a cofinal function $f\colon\alpha\to P$. |
| The cofinality of $P$ is written $\cf{P}$, or $\cof{P}$. |
The cofinality of $P$ is written $\cf{P}$, or $\cof{P}$. |
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| \section*{Cofinality of totally ordered sets} |
\section*{Cofinality of totally ordered sets} |
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| If $(T,\leq)$ is a totally ordered set, then it must contain a well-ordered cofinal subset which is order-isomorphic to $\cf{T}$. |
If $(T,\leq)$ is a totally ordered set, then it must contain a well-ordered cofinal subset which is order-isomorphic to $\cf{T}$. |
| Or, put another way, there is a cofinal function $f\colon\cf{T}\to T$ with the property that $f(x)<f(y)$ whenever $x<y$. |
Or, put another way, there is a cofinal function $f\colon\cf{T}\to T$ with the property that $f(x)<f(y)$ whenever $x<y$. |
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| For any ordinal $\beta$ we must have $\cf{\beta}\leq\beta$, because the identity map on $\beta$ is cofinal. |
For any ordinal $\beta$ we must have $\cf{\beta}\leq\beta$, because the identity map on $\beta$ is cofinal. |
| In particular, this is true for cardinals, so any cardinal $\kappa$ either satisfies $\cf{\kappa}=\kappa$, in which case it is said to be \emph{regular}, or it satisfies $\cf{\kappa}<\kappa$, in which case it is said to be \emph{singular}. |
In particular, this is true for cardinals, so any cardinal $\kappa$ either satisfies $\cf{\kappa}=\kappa$, in which case it is said to be \emph{regular}, or it satisfies $\cf{\kappa}<\kappa$, in which case it is said to be \emph{singular}. |
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The cofinality of any totally ordered set is necessarily a regular cardinal.
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The cofinality of any totally ordered set is a necessarily a regular cardinal.
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| \section*{Cofinality of cardinals} |
\section*{Cofinality of cardinals} |
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| $0$ and $1$ are regular cardinals. All other finite cardinals have cofinality $1$ and are therefore singular. |
$0$ and $1$ are regular cardinals. All other finite cardinals have cofinality $1$ and are therefore singular. |
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| It is easy to see that $\cf{\aleph_0}=\aleph_0$, so $\aleph_0$ is regular. |
It is easy to see that $\cf{\aleph_0}=\aleph_0$, so $\aleph_0$ is regular. |
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| $\aleph_1$ is regular, because the union of countably many countable sets is countable. |
$\aleph_1$ is regular, because the union of countably many countable sets is countable. |
| More generally, all infinite successor cardinals are regular. |
More generally, all infinite successor cardinals are regular. |
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| The smallest infinite singular cardinal is $\aleph_{\omega}$. |
The smallest infinite singular cardinal is $\aleph_{\omega}$. |
| In fact, the function $f\colon\omega\to\aleph_{\omega}$ given by $f(n)=\omega_n$ is cofinal, so $\cf{\aleph_\omega}=\aleph_0$. |
In fact, the function $f\colon\omega\to\aleph_{\omega}$ given by $f(n)=\omega_n$ is cofinal, so $\cf{\aleph_\omega}=\aleph_0$. |
| More generally, for any nonzero limit ordinal $\delta$, the function $f\colon\delta\to\aleph_\delta$ given by $f(\alpha)=\omega_\alpha$ is cofinal, and this can be used to show that $\cf{\aleph_\delta}=\cf{\delta}$. |
More generally, for any nonzero limit ordinal $\delta$, the function $f\colon\delta\to\aleph_\delta$ given by $f(\alpha)=\omega_\alpha$ is cofinal, and this can be used to show that $\cf{\aleph_\delta}=\cf{\delta}$. |
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| Let $\kappa$ be an infinite cardinal. |
Let $\kappa$ be an infinite cardinal. |
| It can be shown that $\cf{\kappa}$ is |
It can be shown that $\cf{\kappa}$ is |
| the least cardinal $\mu$ such that $\kappa$ is |
the least cardinal $\mu$ such that $\kappa$ is |
| the sum of $\mu$ cardinals each of which is less than $\kappa$. |
the sum of $\mu$ cardinals each of which is less than $\kappa$. |
| This fact together with K\"onig's theorem tells us that |
This fact together with K\"onig's theorem tells us that |
| $\kappa<\kappa^{\cf{\kappa}}$. |
$\kappa<\kappa^{\cf{\kappa}}$. |
| Replacing $\kappa$ by $2^\kappa$ in this inequality |
Replacing $\kappa$ by $2^\kappa$ in this inequality |
| we can further deduce that $\kappa<\cf{2^\kappa}$. |
we can further deduce that $\kappa<\cf{2^\kappa}$. |
| In particular, $\cf{2^{\aleph_0}}>\aleph_0$, from which it follows that $2^{\aleph_0}\neq\aleph_\omega$ (this being the smallest uncountable aleph which is provably not the cardinality of the continuum). |
In particular, $\cf{2^{\aleph_0}}>\aleph_0$, from which it follows that $2^{\aleph_0}\neq\aleph_\omega$ (this being the smallest uncountable aleph which is provably not the cardinality of the continuum). |
