See axiom of power set in All languages combined, or Wiktionary
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"ref": "1978, Thomas Jech, Set Theory, Academic Press, page 38",
"text": "The axiom of choice differs from other axioms of ZF by stating existence of a set (i.e., a choice function) without defining it (unlike, for instance, the axiom of pairing or the axiom of power set).",
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"ref": "2003, Thomas Forster, Reasoning About Theoretical Entities, World Scientific, page 51",
"text": "Verifying that the axiom of power set is in #x5C;#x7B;#x5C;phi#x5C;in#x5C;mathcal#x7B;L#x7D;#x2A;#x3A;S#x5C;vdash#x5C;mathcal#x7B;I#x7D;(#x5C;phi)#x5C;#x7D; relies on some rudimentary comprehension axioms.",
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"ref": "2011, Adam Rieger, “9: Paradox, ZF, and the Axiom of Foundation”, in David DeVidi, Michael Hallett, Peter Clark, editors, Logic, Mathematics, Philosophy: Vintage Enthusiasms: Essays in Honour of John L. Bell, Springer, page 183",
"text": "But the ZF axioms of which the hierarchy is an intuitive model involve impredicative quantifications. Most striking is the axiom of power set in tandem with the axiom of separation.",
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"text": "2012, A. H. Lightstone, H. B. Enderton (editor), Mathematical Logic: An Introduction to Model Theory, Plenum Press, Softcover, page 292,\nThe Axiom of Power Set asserts that the collection of all subsets of a set is a set. […] Adding the Axiom of Power Set compels the collection empty to be a set."
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"The axiom that the power set of any set exists and is a valid set, which appears in the standard axiomatisation of set theory, ZFC."
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"word": "potenssijoukkoaksiooma"
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"text": "Verifying that the axiom of power set is in #x5C;#x7B;#x5C;phi#x5C;in#x5C;mathcal#x7B;L#x7D;#x2A;#x3A;S#x5C;vdash#x5C;mathcal#x7B;I#x7D;(#x5C;phi)#x5C;#x7D; relies on some rudimentary comprehension axioms.",
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"text": "But the ZF axioms of which the hierarchy is an intuitive model involve impredicative quantifications. Most striking is the axiom of power set in tandem with the axiom of separation.",
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}
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-05-01 from the enwiktionary dump dated 2024-04-21 using wiktextract (f4fd8c9 and c9440ce). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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