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Sep 1 |
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Are there any nontrivial, finite subrings of an infinite ring? added 790 characters in body |
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Sep 1 |
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Quotient objects, their universal property and the isomorphism theorems @Theo: Thanks for the references. I've never gotten around to reading Joy of Cats; perhaps I should if it contains such gems! |
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Sep 1 |
answered | Quotient objects, their universal property and the isomorphism theorems |
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Sep 1 |
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Quotient objects, their universal property and the isomorphism theorems @Bruno: There is already a notion of quotient object (it is the dual of a subobject), but the ones you are referring to seem to be coequalisers of some kind. Is this the kind of answer you want? |
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Sep 1 |
answered | Are there any nontrivial, finite subrings of an infinite ring? |
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Sep 1 |
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Is the set of all deducible formulas decidable? But then again, I suppose we are assuming $T$ is consistent, since otherwise the set of its theorems is trivially decidable. |
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Sep 1 |
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Is the set of all deducible formulas decidable? @Ewan: Good point. But I believe the answer is still no: for if $D$ were decidable, then we would know whether or not $\bot$ is a theorem of our theory $T$; in the case that it is a non-theorem, then we could turn it into a proof of the consistency of $T$ within $T$ itself, contradicting Gödel's incompleteness theorem. I'm not sure what we should do if $\bot$ turns out to be a theorem of $T$, however... |
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Sep 1 |
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Is the set of all deducible formulas decidable? No, because if $D$ were decidable, we could solve the halting problem. |
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Sep 1 |
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Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category @Tilo: Draw the pullback square for $q$ and $f$, and note that there is a antidiagonal arrow $h : Y \to A$. |
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Sep 1 |
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Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category typo |
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Aug 31 |
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Descriptions of sets and the Axiom of Choice There are some difficulties with that argument. Some of the answers and comments to this question may be relevant. |
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Aug 31 |
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Prerequisites for Atiyah Macdonald @Amitesh: Well, I was merely offering an opposing opinion, from the point of view of an undergraduate mathematician with a more conventional background. Not everyone has had the benefit of learning so much, whether by their own efforts or otherwise, by the age of 16! |
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Aug 31 |
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Prerequisites for Atiyah Macdonald @Amitesh: I don't think a page of Eisenbud and a page of Atiyah–Macdonald are comparable in any meaningful sense. It takes me longer to read a page of AM because it's so dense! And a page full of exercises takes much much much longer to ‘read’ than a page of proof because one has to fill in the details... |
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Aug 30 |
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Reference request: group theory @D B Lim: I know about linear representation of finite groups. The representation theory of Lie groups and Lie algebras is a different but related beast. |
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Aug 30 |
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Group extensions of cyclic groups Ah, that's more convincing. You should add that to the post. |
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Aug 30 |
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Group extensions of cyclic groups In case 2, your claim $x^i = x^{-i}$ is incorrect. Let us take $n = 6$, $G = \mathbb{Z} \oplus C_3$, $x = (2, 0)$, $y = (1, 1)$. Then $y^6 = (6, 0) \ne (-6, 0)$. Nonetheless, $\langle x, y \rangle / \langle x \rangle \cong C_6$. |
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Aug 30 |
asked | Group extensions of cyclic groups |
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Aug 28 |
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Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category @Tilo: Ah, then yes. But perhaps there's some subtlety I'm missing here, hmmm. |
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Aug 28 |
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How to show $\det(AB) =\det(A)\det(B)$ @Learner: How do you define the determinant of a matrix? The definition affects what properties we may assume in the proof. |
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Aug 28 |
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What Does it Mean Exactly to Claim Logical Theorems (Axioms) Independent? Yes, actually. I'm more used to that notation. At the moment I can't figure out what your question is about because I have to expend considerable effort translating your formulae into expressions I can understand. |
