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Apr
3
comment False(ified) Axioms
There are much better reasons for saying that $(1 = 2) \land (2 = 3) \to (1 = 3)$ is true though: for one thing, it is a special case of the axiom $(x = y) \land (y = z) \to (x = z)$, and substitutions for free variables preserve truth.
Apr
2
comment Can boolean homomorphisms of boolean algebras correspond to ultrafilters?
A boolean algebra homomorphism $B \to \mathbf{2}$ is necessarily non-trivial, because boolean algebra homomorphisms must preserve $0$ and $1$. (In particular, there is no homomorphism from $\mathbf{1}$ to $\mathbf{2}$.) So the kernel of $B \to \mathbf{2}$ is always a prime ideal.
Apr
2
comment I think a definition is wrong in “Model Categories” by Hovey.
The official errata for the book has been posted here.
Apr
2
comment Why do we prefer classical logic over non-classical logic?
I see your system LK sequent calculus and raise by intuitionistic natural deduction!
Apr
2
comment Why do we prefer classical logic over non-classical logic?
I don't find it useful to reserve "paradox" for actual contradictions; we have a perfectly good word for that already: "contradiction"!
Apr
2
comment $F:C\to D, G:D\to E$ are functors, $G$ has a right adj, $F$ is fully faithful, $G$ is faithful, $F$ is “relatively dense”. Does $F$ have an adj?
What is the motivation behind these questions? Is there some concrete problem you are trying to solve?
Apr
1
comment Why is propositional logic not Turing complete?
Yes, but Turing completeness is a theoretical condition, not a practical one. For instance, taking a literal reading of the definitions, the halting problem is decidable for any (idealised) physical computer, because it is a finite state machine!
Apr
1
comment Why is propositional logic not Turing complete?
Physical computers are not Turing complete either, actually: they only have finite memory, unlike a Turing machine.
Mar
31
comment Why is it impossible to define multiplication in Presburger arithmetic?
It is a consequence of induction in Peano arithmetic. There is something nontrivial going on in the recursion theorem!
Mar
30
comment Are there rules in the useage of prepositions in Math?
This is not morphology – this is syntax!
Mar
29
comment Essential geometric morphism seen topologically
Being essential is a weak form of other conditions. For example, a locally connected geometric morphism is essential but not vice versa. Being locally connected is a condition that can be phrased topologically: see [Butz and Moerdijk, Representing topoi by groupoids].
Mar
29
comment Proof that Riemann Roch Space is a Vector Space
The divisor of $0$ is not defined.
Mar
29
comment $M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)
What is true is that an $R$-module homomorphism is an $R$-module isomorphism if and only if it is a bijection, i.e. that the forgetful functor $R\textbf{-Mod} \to \textbf{Set}$ is conservative (and hence the forgetful functor $R\textbf{-Mod} \to \textbf{Ab}$ and restriction of scalars functor etc.). But that is a fact of undergraduate algebra.
Mar
28
comment In Logic is ⇒, →, and ⊃ basically the same symbol?
From what I understand, $\supset$ was used by Russell and Whitehead, $\to$ by Hilbert, and $\Rightarrow$ by Bourbaki.
Mar
27
comment What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces?
There are two ways of formulating a compactness theorem: syntactic and semantic. The semantic version is of course independent of the deductive system.
Mar
26
comment Products of sites
Hmmm. That's a difficult category. However it is conceivable that products exist. For example, if we restrict to "discrete" sites with finite limits, i.e. if we look at the opposite of the category of small finitely-complete categories, there are products (which are the same as coproducts). See Lemma B2.3.14 in the Elephant.
Mar
25
comment Does induction for a functor algebra imply it is initial?
And, of course, the category of algebras for an endofunctor has equalisers if the base category does.
Mar
25
comment $F:\bf C\to\bf D$ a functor with a right adjoint $G$ and $\bf S$ a full subcat of $\bf C$: When does the inclusion have a right adjoint?
What does $F$ have to do with $I$? They appear to be totally unrelated. Anyway, you can always apply the general adjoint functor theorem. I don't think there are many major simplifications arising from $I$ being fully faithful.
Mar
25
comment Natural Transformation between covariant and contravariant functor
Sure, that's what I said in my comments. It's not really a natural transformation, however.
Mar
25
comment Products of sites
What morphisms are you using? The 2-category of sheaf toposes has products and pullbacks. The product of two localic toposes is the sheaf topos on the locale product, but the locale product does not always coincide with the topological space product.