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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.
4
votes
How should I think about concrete functors and in particular about concrete isomorphism?
Let me add little to Todd's answer, because you asked specifically about "the same morphisms". When we have a concrete category $(\mathbf A,U)$ over $\mathbf X$, we can use $U$ to identify morphisms …
- 73.2k
21
votes
Structures that turn out to exhibit a symmetry even though their definition doesn't
Consider the Desargues configuration. It consists of (1) two triangles, say $ABC$ and $A'B'C'$ such that the lines $AA'$, $BB'$, and $CC'$ all meet at a point $P$, and (2) the three points of intersec …
61
votes
Accepted
What is Realistic Mathematics?
When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theo …
- 73.2k
19
votes
What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?
$\DeclareMathOperator\BB{BB}$Philosophical issues, like acceptance (or non-acceptance) of large cardinals, won't affect $\BB(n)$, because the busy beaver function is defined arithmetically and so depe …
- 73.2k
5
votes
Fundamental Examples
Motivated by Amit Kumar Gupta's answer about the continuum hypothesis, let me add an example that is less natural but has inspired an amazing amount of set theory, namely Suslin's Hypothesis. This co …