3 ms·

+1 for the book recommendation alone [1]. I finally see a book on set theory that explicitly states that the term "set" by itself is undefined. Reading the fi

by alok-g 14y ago

+1 for the book recommendation alone [1]. I finally see a book on set theory that explicitly states that the term "set" by itself is undefined. Reading the first chapter preview at Amazon, I already see the need for a human to explain something to me: Two sentences from the book [1]: "A possible relation between sets, more elementary than belonging, is equality." and "Two sets are equal if and only if they have the same elements". These already seem to be contradicting each other -- how can equality be more basic than belonging if the former is defined in terms of the latter. While I agree with OP on the difficulty I face when learning advanced mathematics (notation, generally tacitly assumed by the authors), one thing that also bugs me is use of human language (!) as a part of the proofs. How do I know that the proof is correct and is not impacted by something fundamental about the language itself? Merely via inclusion of a human language in the proof, a lot more axioms "may" have been included than what meets the eye. Probably not, but I often find it hard to convince myself of this. [1] www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387900926

xyzzyz 14y ago
Two sentences from the book [1]: "A possible relation between sets, more elementary than belonging, is equality." and "Two sets are equal if and only if they have the same elements". These already seem to be contradicting each other -- how can equality be more basic than belonging if the former is defined in terms of the latter. Axiom of extensionality states that the sets A and B are equal if and only if they have the same elements. The point here is this: if A and B are the same set, then whenever some x belongs to A, it also belongs to B, because B is just a different name for A. Similarly, if something belongs to B, it also belongs to A. So, the implication "A is B implies A and B have the same elements" is just a logical tautology, axiom of extensionality does not say anything new in this case. Now suppose you have two sets, C and D, such that whenever something belongs to C, it also belongs to D, and when something belongs to D, it also belongs to C -- in other words, they have the same elements. Can we conclude that C is the same set as D? Without extensionality axiom, no. That's why we usually assume axiom of extensionality, which tells exactly that whenever sets the same elements, they are the same, so that we don't end up with a weird situation where we have two sets that are indistinguishable with set theory, yet are different.

by alok-g 14y ago

xyzzyz 14y ago

Two sentences from the book [1]: "A possible relation between sets, more elementary than belonging, is equality." and "Two sets are equal if and only if they have the same elements". These already seem to be contradicting each other -- how can equality be more basic than belonging if the former is defined in terms of the latter. Axiom of extensionality states that the sets A and B are equal if and only if they have the same elements. The point here is this: if A and B are the same set, then whenever some x belongs to A, it also belongs to B, because B is just a different name for A. Similarly, if something belongs to B, it also belongs to A. So, the implication "A is B implies A and B have the same elements" is just a logical tautology, axiom of extensionality does not say anything new in this case. Now suppose you have two sets, C and D, such that whenever something belongs to C, it also belongs to D, and when something belongs to D, it also belongs to C -- in other words, they have the same elements. Can we conclude that C is the same set as D? Without extensionality axiom, no. That's why we usually assume axiom of extensionality, which tells exactly that whenever sets the same elements, they are the same, so that we don't end up with a weird situation where we have two sets that are indistinguishable with set theory, yet are different.