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In [[mathematical logic]] and [[set theory]], an '''ordinal collapsing function''' (or '''projection function''') is a technique for defining ([[Ordinal notation|notations]] for) certain [[Recursive ordinal|recursive]] [[large countable ordinal]]s, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even [[Large cardinal property|large cardinals]] (though they can be replaced with [[Large countable ordinal#Beyond admissible ordinals|recursively large ordinals]] at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an [[Impredicativity|impredicative]] manner of naming ordinals.
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