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Dec
8
comment Another way of saying that algebraic objects are isomorphic
That's easy enough: given a signature $\Sigma$ and a subset $\Sigma' \subseteq \Sigma$, any $\Sigma$-structure gives rise to a $\Sigma'$-structure in a natural way, called a reduct; so what you are talking about is isomorphisms of reducts.
Dec
8
comment Another way of saying that algebraic objects are isomorphic
How do you intend to give rigorous meaning to "encode the same structure" if not by "isomorphism"?
Dec
8
revised Does elementary embedding exist between two elementary equivalent structures?
added 100 characters in body
Dec
8
answered Help with a logic question
Dec
8
revised Help with a logic question
added 12 characters in body; edited tags
Dec
8
answered Does elementary embedding exist between two elementary equivalent structures?
Dec
8
comment Relation between algebraic geometry over a field of characteristic $0$ and that over $\mathbb{C}$
Ah, I never thought of descent theory as a means of changing to a smaller field. Very interesting!
Dec
7
revised For every closed set $C$, $f^{-1}(C)$ is closed. Is $f$ necessarily continuous?
added 2 characters in body; edited title
Dec
7
comment Automorphisms of saturated models
$\textrm{DCL}(X)$ is the set of all elements fixed by all automorphisms of $A$ that fix $X$ pointwise, and $\textrm{dcl}(X)$ is the set of all elements that are definable in $A$ with parameters in $X$; $\textrm{ACL}(X)$ and $\textrm{acl}(X)$ are defined analogously.
Dec
7
answered Why isn't the covariant powerset functor representable?
Dec
7
comment Automorphisms of saturated models
Sure, I've added it now. I checked the corrigenda and there's nothing reported for this exercise...
Dec
7
revised Automorphisms of saturated models
added 610 characters in body
Dec
7
comment Automorphisms of saturated models
Hmmm, I'm confused then. Have I misinterpreted the exercise in Hodges?
Dec
7
comment What's the thorny issue on: “If all $S\in \ell $ are nonempty, does it follow that $\prod_{S\in \ell} S$ is nonempty? when $\ell$ is infinite?”
@GustavoBandeira Let $\phi(X)$ be the proposition "$X$ is a finite set". Obviously every finite set $X$ satisfies $\phi(X)$. So are you saying that all sets $X$ satisfy $\phi(X)$ as well? There's absolutely no reason why that kind of reasoning should be accepted.
Dec
7
asked Automorphisms of saturated models
Dec
6
comment Is this a property of an integral domain that is not a field?
Well, if you have something like that then you can manufacture a strictly increasing sequence of ideals, so if you know you have a noetherian ring you would have a contradiction.
Dec
6
comment Using Zorn's Lemma
You seem to be confused about what Zorn's lemma says and how to use it. It doesn't say that chains have upper bounds – rather, that is a hypothesis!
Dec
6
answered Tautological 1-form on the cotangent bundle
Dec
5
comment What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…
Without the axiom of choice, it is conceivable that there is a cardinal $\kappa$ such that $\aleph_0 < \kappa$ but neither $\aleph_1 \le \kappa$ nor $\kappa \le \aleph_1$.
Dec
5
comment Question on Peano Arthmetic
These are fairly straightforward theorems in ordinary arithmetic, but if you want formal Hilbert-style proofs, you should say so. (I doubt anyone will actually give you one though... it's just too tedious.)