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text: '[summary(brief): The **cardinality** of a [-3jz] is a formalization of the "number of [5xy elements]" in the set.]\n\n[summary: If $A$ is a finite set then the cardinality of $A$, denoted $|A|$, is the number of elements $A$ contains. When $|A| = n$, we say that $A$ is a set of cardinality $n$. There exists a [499 bijection] from any finite set of cardinality $n$ to the set $\\{0, ..., (n-1)\\}$ containing the first $n$ natural numbers.\n\nTwo infinite sets have the same cardinality if there exists a bijection between them. Any set in bijective correspondence with [45h $\\mathbb N$] is called __countably infinite__, while any infinite set that is not in bijective correspondence with $\\mathbb N$ is call **[2w0 uncountably infinite]**. All countably infinite sets have the same cardinality, whereas there are multiple distinct uncountably infinite cardinalities.]\n\n[summary(technical): The cardinality (or size) $|X|$ of a set $X$ is the number of elements in $X.$ For example, letting $X = \\{a, b, c, d\\}, |X|=4.$\n\n[todo: technical summary of infinite cardinality]]\n\nThe **cardinality** of a [-3jz] is a formalization of the "number of elements" in the set.\n\nSet cardinality is an [-53y]. Two sets have the same cardinality if (and only if) there exists a [499 bijection] between them.\n\n## Definition of equivalence classes\n\n### Finite sets\n\nA set $S$ has a cardinality of a [-45h] $n$ if there exists a bijection between $S$ and the set of natural numbers from $1$ to $n$. For example, the set $\\{9, 15, 12, 20\\}$ has a bijection with $\\{1, 2, 3, 4\\}$, which is simply mapping the $m$th element in the first set to $m$; therefore it has a cardinality of $4$.\n\nWe can see that this equivalence class is [ well-defined] — if there exist two sets $S$ and $T$, and there exist bijective functions $f : S \\to \\{1, 2, 3, \\ldots, n\\}$ and $g : \\{1, 2, 3, \\ldots, n\\} \\to T$, then $g \\circ f$ is a bijection between $S$ and $T$, and so the two sets also have the same cardinality as each other, which is $n$.\n\nThe cardinality of a finite set is always a natural number, never a fraction or decimal.\n\n### Infinite sets\n\nAssuming the [69b axiom of choice], the cardinalities of infinite sets are represented by the [aleph_numbers]. A set has a cardinality of $\\aleph_0$ if there exists a bijection between that set and the set of *all* natural numbers. This particular class of sets is also called the class of [-countably_infinite_sets].\n\nLarger infinities (which are [2w0 uncountable]) are represented by higher Aleph numbers, which are $\\aleph_1, \\aleph_2, \\aleph_3,$ and so on through the [ordinal ordinals].\n\n**In the absence of the Axiom of Choice**\n\nWithout the axiom of choice, not every set may be [55r well-ordered], so not every set bijects with an [-ordinal], and so not every set bijects with an aleph.\nInstead, we may use the rather cunning [Scott_trick].\n\n%%todo: Examples and exercises (possibly as lenses) %%\n\n%%todo: Split off a more accessible cardinality page that explains the difference between finite, countably infinite, and uncountably infinite cardinalities without mentioning alephs, ordinals, or the axiom of choice.%%',
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Summary: 'If $A$ is a finite set then the cardinality of $A$, denoted $|A|$, is the number of elements $A$ contains. When $|A| = n$, we say that $A$ is a set of cardinality $n$. There exists a [499 bijection] from any finite set of cardinality $n$ to the set $\\{0, ..., (n-1)\\}$ containing the first $n$ natural numbers.\n\nTwo infinite sets have the same cardinality if there exists a bijection between them. Any set in bijective correspondence with [45h $\\mathbb N$] is called __countably infinite__, while any infinite set that is not in bijective correspondence with $\\mathbb N$ is call **[2w0 uncountably infinite]**. All countably infinite sets have the same cardinality, whereas there are multiple distinct uncountably infinite cardinalities.',
brief: 'The **cardinality** of a [-3jz] is a formalization of the "number of [5xy elements]" in the set.',
technical: 'The cardinality (or size) $|X|$ of a set $X$ is the number of elements in $X.$ For example, letting $X = \\{a, b, c, d\\}, |X|=4.$\n\n[todo: technical summary of infinite cardinality]'
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