Audrito and Viale introduced the new large cardinal notion of an (α)-uplifting cardinal (for an ordinal α). We shall show that this notion cannot be defined (or expressed) in the standard first-order language of set theory for every tranfinite α.
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I am most grateful to the anonymous referee for their meticulous feedback and helpful and constructive suggestions: in particular, the notion of a fully reflecting ordinal (and its naming) and the use of it to improve the presentation of my proofs are suggested by them. I would also like to thank Kentaro Sato and Philip Welch for fruitful discussions on some topics related to this?article.
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