In this comment on Terry Tao's page about his Analysis I textbook, he writes,

If one wanted to do things by the book, what one should actually do each time one introduces a new mathematical object, one replaces the mathematical language one is working in (and the universe that models that language) by a conservative extension of that language in which the new objects are separate from the previous objects; for instance, one could start with a language and universe without the notion of an integer, and pass to a conservative extension in which a new type of object is introduced, namely the integers, which obey various axioms and interact with sets using the existing axioms of set theory, but in which integers are not sets. The set-theoretic encoding of the integers as equivalence classes of formal differences of natural numbers can be used to show that the extension exists and is conservative, but once this is done, it is somewhat more preferable to return to an agnostic position on whether integers are sets or not. [...]

My question: is the extension going to be different if we assume "integers are not sets", which he initially indicates he is doing, versus taking "an agnostic position on whether integers are sets or not"? And more generally, is introducing a new type of objects, which are assumed to be not sets, going to still produce a conservative extension? I'm a bit new to the meaning of "conservative extension", but am trying to sort out the differences between these extensions.


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