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I am reading Fraenkel's Abstract Set Theory and in the first chapter he says:

"The logical character of the objects called "sets" is of no importance to the mathematical theory of sets – in the same way as the results of arithmetical calculating are independent of what may be, in the view of the calculator, the logical or psychological meaning of number"

Honestly, I still don't get why we don't care about it. Thanks!

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    $\begingroup$ Somebody cares and the others don't. You happen to be reading a book written by some one who doesn't care. $\endgroup$ – Cave Johnson Dec 27 '17 at 13:18
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    $\begingroup$ It is consistent with the title of the book: Abstract Set Theory; it means that it is the mathematical theory of sets, whose valdiity is independent of the (at that time) very complicated discussion about the "philosophical" discussion about the nature of sets, like e.g. extensions of logical concepts (Frege) or "logical fictions" (Russell) or other. $\endgroup$ – Mauro ALLEGRANZA Dec 27 '17 at 13:23
  • $\begingroup$ Thanks @CaveJohnson! Can you anwser the question I made to Mauro?? $\endgroup$ – César D. Vázquez Dec 27 '17 at 13:42
  • $\begingroup$ @MauroALLEGRANZA thanks again for clearing things up for me. I got tired of dogmatic introductions to set theory and I read on math-exchange that a more formal introduction with logical proofs and philosophical discourse would be this book. Do you think there is a better one? I've read already 4 introductions and I have some grounds in first-order logic and natural deduction. $\endgroup$ – César D. Vázquez Dec 27 '17 at 13:44
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    $\begingroup$ The author didn't say "don't care", he said it doesn't matter. The point is that he's doing axiomatic set theory - the theorems are exactly the statements that follow from the axioms; what a set is doesn't matter, all that matters is we have "sets" with a $\in$ relation that satisfies the axioms. Exactly like when you study group theory - what the elements of a group really are doesn't matter. $\endgroup$ – David C. Ullrich Dec 27 '17 at 17:19

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