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  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ Commented Mar 8, 2012 at 10:30
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    $\begingroup$ There are some obvious statements not allowed in this principle--for instance, "the cardinality of every variety is at most that of the continuum" (which is true over $\mathbb{C}$, but not over an algebraically closed field with cardinality greater than that of $\mathbb{C}$). Part of the interest in attempts to formalize the Lefschetz Principle are attempts to formulate exactly what statements should be allowed. $\endgroup$ Commented Mar 8, 2012 at 14:59
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    $\begingroup$ Another comment is that, while there exist formal versions of the Lefschetz Principle, it is often used as a "proof technique" rather than a theorem to be quoted. For instance, proofs can sometimes proceed in the following manner: 1) Show that the statement you care about can, in any given case, be reduced to a statement over a countable extension of $\mathbb{Q}$, by using the "fact" that the statement only involves countably many elements of your field. 2) Any countable extension of $\mathbb{Q}$ can be embedded in $\mathbb{C}$. $\endgroup$ Commented Mar 8, 2012 at 15:05
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    $\begingroup$ @Charles: Isn't this a little bit unfortunate because there is a precise theorem which one might quote (the one by Eklof)? In practice one then has to check only 1), whereas 2) (and 3), namely the reason why this does the job) may be omitted. $\endgroup$ Commented Mar 8, 2012 at 16:03
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    $\begingroup$ [ct'd] Since many algebraic geometers have not taken the time to master the basic techniques for such "translations," applying the precise theorem when the imprecise version can easily be made to work will unnecessarily reduce the accessibility of one's writing. $\endgroup$ Commented Mar 11, 2012 at 16:34