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Aug
3
comment A Turing machine for which halting is outside ZFC
@ThomasAndrews Whether or not $T$ halts depends on the set-theoretic universe. In a model of $\mathrm{ZFC} + \lnot \mathrm{Con}(\mathrm{ZFC})$, $T$ will halt, and in a model of $\mathrm{ZFC} + \mathrm{Con}(\mathrm{ZFC})$, $T$ will not halt. Since both of these theories are consistent if ZFC is, it must be the case that the halting of $T$ is independent of ZFC.
Aug
3
answered A Turing machine for which halting is outside ZFC
Aug
3
comment Definition of equalizer for $\textbf{Sh}(X)$
Yes, that has the correct universal property. The reason why people don't bother defining it explicitly is because the universal property is enough to pin it down up to isomorphism!
Aug
2
comment The “depth” of a set
This is precisely the concept of set-theoretic rank.
Aug
2
answered Mac Lane exercise - Elegant comma category exercise proven by S.A Huq
Aug
2
comment On consistency of axiomatic systems
I've replaced the definition of completeness with another one. Perhaps this one is more to your liking, but it is equivalent to the one I gave before even for intuitionistic logic.
Aug
2
revised On consistency of axiomatic systems
added 30 characters in body
Aug
2
comment On consistency of axiomatic systems
Those are definitions, not statements.
Aug
1
comment Local-global properties (localization): free, projective, injective, flat, torsion-free, etc?
It is not: see here.
Aug
1
answered Local-global properties (localization): free, projective, injective, flat, torsion-free, etc?
Aug
1
comment Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?
Yes on both counts.
Aug
1
answered On consistency of axiomatic systems
Jul
31
awarded  Revival
Jul
31
comment Why is arc length not a differential form?
Because arclength is a function $\{ \text{smooth curves} \} \to \mathbb{R}$ as well, so we can ask whether it is of the form $\gamma \mapsto \int_\gamma \alpha$ for some fixed differential 1-form $\alpha$.
Jul
31
comment Why is arc length not a differential form?
You misunderstand. I said we can recover $\alpha$ as long as we know its integral with respect to all smooth curves.
Jul
31
comment Can curvature be defined in Topos Theory?
Curvature isn't a property of a bare topological space, though. Before asking about toposes you may as well ask about notions of curvature that can be applied to more than just Riemannian manifolds.
Jul
31
comment Can curvature be defined in Topos Theory?
The question confuses two notions: toposes as generalised spaces (as exemplified by sheaf toposes $\mathbf{Sh}(X)$ for a topological space $X$) and toposes as categories of spaces (as exemplified by simplicial sets). Which do you want to think about?
Jul
31
comment What is a noetherian category?
That's correct.
Jul
31
comment What is a noetherian category?
The cardinality of a finitely-generated $R$-module can be bounded in terms of the cardinality of $R$ and $\aleph_0$, so there is only a set of isomorphism classes of finitely-generated $R$-modules.
Jul
31
answered Why is arc length not a differential form?