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I find it a little problematic to refer to doing these things "in" ZFC or NFU. ZFC and NFU only prove things; they don't construct or define anything. I've had quite a job explaining this at articles like [[definable number]] (people want to talk about things like "real numbers definable in ZFC", which is nonsense).
: Theories do prove theorems of the form "there is a unique x such that phi", which amounts to proving the existence of "the x such that phi". This can be construed as showing that the theory allows us to work with a certain object... The theories ZFC (and restrictions, extensions) and NFU (with extensions) describe (partially) worlds in which certain things are found, and, since the two theories have the same language, objects "definable" in this sense in one theory may or may not be "definable" in the other and may or may not have analogous properties. That's what I'm talking about. There probably are real numbers which are "definable in ZFC" in this sense but not "definable in PA2" (that is, there are probably formal descriptions of real numbers which can be phrased both in ZFC and in PA2 and which can be proved to be satisfied in ZFC but not in PA2); so while I can imagine why you might not like this phrase, I don't see that it is ''complete'' nonsense. Nor do I dismiss your objection out of hand; I am trying to understand exactly what you are objecting to. What kinds of errors are you worried that naive readers likely to make? [[User:Randall Holmes|Randall Holmes]] 10:33, 23 December 2005 (UTC)
What the article seems to be about is how to define various concepts in the ''language of set theory'' (not ZFC or NFU) in such a way that ZFC (resp. NFU) proves that they behave the way one wants them to. I think that's fine; I just would rather not see this called "doing things in ZFC or NFU". That's a reasonable shorthand when everyone understands each other, but is likely to cause or reinforce misconceptions among neophytes.
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