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Results tagged with big-picture
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Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
1
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How does one identify properties of objects with good "inheritance"?
As to the last part of your question: yes, there is a relation between quantifiers and inheritance.
Any property defined only with universal quantifiers is automatically inherited by every subobject. …
- 65.4k
6
votes
Is it necessary that model of theory is a set?
Gerald's answer is quite correct. This began as a comment justifying it, but because of length considerations I'm leaving it as an answer instead.
A model of ZFC is not a perfect replica of the cate …
- 65.4k
23
votes
Why are finiteness conditions important (and how to recognize them)?
The fact that various finiteness conditions lead to good theorems which are manifestly false in their absence seems like a good explanation of why they are important. (In fact, I am having trouble th …
49
votes
Why is it a good idea to study a ring by studying its modules?
I want to answer your question twice: first with a "top-down" approach and second with a "bottom-up" approach. Let me limit myself to the first answer here and see how I do.
I claim the following an …
- 65.4k
9
votes
What precisely Is "Categorification"?
A longer answer is certainly called for (but I teach a class at 8 am). The article
http://en.wikipedia.org/wiki/Categorification
gives a good initial explanation. As you can see from this article, …
- 65.4k
14
votes
What's a groupoid? What's a good example of a groupoid?
I (mildly) disagree with David Brown's assertion that a set is an example of a groupoid. Given any set, you can put a groupoid structure on it, even "canonically", but not uniquely canonically. (By …
18
votes
Accepted
Is the ABC conjecture known to imply the Riemann hypothesis?
I am pretty sure that the answer to the question is no: no two of those big conjectures are known to imply the third. But I feel somewhat sheepish giving this as an answer: what evidence can I bring …
- 65.4k
19
votes
What are interesting families of subsets of a given set?
A filter on a set $X$ is a nonempty family of nonempty subsets of $X$ which is stable under finite intersection and passage to supersets. These are extremely useful in topology, certain branches of a …
- 65.4k
50
votes
Accepted
"Algebraic" topologies like the Zariski topology?
Yes, there are plenty of such things.
[In the following, "compact" implies "locally compact" implies "Hausdorff".]
1) To a Boolean algebra, one associates its Stone space, a compact totally discon …
- 65.4k
55
votes
Fundamental Examples
The Fermat Equation xn + yn - zn = 0.
This has truly been much more than an example in both algebra and number theory: it was one of the main motivations to develop the theory of unique factorizati …