Powerset: Difference between revisions
RhubarbJayde (talk | contribs) (Created page with "The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. Cantor's diagonal argument proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is uncountable. The questi...") |
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The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. |
The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. |
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[[Cantor's diagonal argument]] proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is [[Countability|uncountable]]. The question of whether \(\omega_1\), the least uncountable [[cardinal]], and \(|\mathcal{P}(\mathbb{N})|\) have the same size is a natural question and the affirmative is known as the continuum hypothesis. Surprisingly, assuming its consistency, this is neither provable nor disprovable in [[ZFC]]! |
[[Cantor's diagonal argument]] proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is [[Countability|uncountable]]. The question of whether \(\omega_1\), the least uncountable [[cardinal]], and \(|\mathcal{P}(\mathbb{N})|\) have the same size is a natural question and the affirmative is known as the [[continuum hypothesis]]. Surprisingly, assuming its consistency, this is neither provable nor disprovable in [[ZFC]]! |
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The existence of an arbitrary set's powerset is not provable from [[Kripke-Platek set theory|KP]], even with separation and collection extended to arbitrary formulae, and as such the axiom of powerset ("every set has a powerset") is included explicitly as an axiom in ZFC. |
The existence of an arbitrary set's powerset is not provable from [[Kripke-Platek set theory|KP]], even with separation and collection extended to arbitrary formulae, and as such the axiom of powerset ("every set has a powerset") is included explicitly as an axiom in ZFC. |
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Latest revision as of 16:54, 25 March 2024
The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour.
Cantor's diagonal argument proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is uncountable. The question of whether \(\omega_1\), the least uncountable cardinal, and \(|\mathcal{P}(\mathbb{N})|\) have the same size is a natural question and the affirmative is known as the continuum hypothesis. Surprisingly, assuming its consistency, this is neither provable nor disprovable in ZFC!
The existence of an arbitrary set's powerset is not provable from KP, even with separation and collection extended to arbitrary formulae, and as such the axiom of powerset ("every set has a powerset") is included explicitly as an axiom in ZFC.