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2d
comment Correspondence between prime and maximal ideals
The formulation of the statement could be misread, so, to be safe, what is wanted is that the natural correspondence restricts to two nice things: (1) a bijection between the prime ideals of $R$ that contain $I$ and the prime ideals of $R/I$. (2) a bijection between the maximal ideals of $R$ that contain $I$ and the maximal ideals of $R/I$. There is no claim about a bijection between prime ideals and maximal ideals.
Jan
26
comment Must complete atomless Boolean algebras of the same cardinality be isomorphic?
@AsafKaragila To finish translating my answer into the language of forcing, the last sentence says that random is $\omega^\omega$-bounding and Cohen isn't.
Jan
25
answered Proof that the minimal inductive set does not contain any more elements than $\{\emptyset, s(\emptyset), s(s(\emptyset)),s(s(s(\emptyset))),…\}$?
Jan
25
comment Turn sentence to predicat formula
@trolkura No, the formula is true whenever you've read at least two books. For example, if you've read all three books of C. S. Lewis's space trilogy, then BrianO's formula is true because you can instantiate $x$ as "Out of the Silent Planet" and $y$ as "Perelandra"; these instantiations make $C(x)$, $C(y)$, and $x\neq y$ true as required. The fact that you've also read "That Hideous Strength" does no damage to this evaluation of the formula.
Jan
22
comment Why do I get one extra wrong solution?
Quite generally, when you get a wrong solution as a result of a computation like this, you can usually pinpoint the source of the error by just substituting the wrong value, in this case $x=1$, into each step of the computation. Substituting into the original equation produces a false result ($2-1=-1$); substituting into the final step gives a true result ($1=1$ or $1=4$). Plug $x=1$ into every step and see where the transition from false to true occurs.
Jan
21
awarded  Enlightened
Jan
18
comment Ordered sets $\langle \mathbb{N} \times \mathbb{Q}, \le_{lex} \rangle$ and $\langle \mathbb{Q} \times \mathbb{N}, \le_{lex} \rangle$ not isomorphic
The notation for the ordered pairs seems reversed. For example, $\phi(0,0)$ is in $\mathbb N\times\mathbb Q$, so its first component should be in $\mathbb N$ and the second in $\mathbb Q$. Your notation $(q,n)$ suggests (though it doesn't strictly imply) the opposite; later, you use that there's nothing between $(q,n-1)$ and $(q,n)$, which does imply that you meant for $n$ to be in $\mathbb N$, not $\mathbb Q$.
Jan
18
comment Prove matrix $X'XA$ equals $X'XB$ iff $XA = XB$
(I see Robert Israel posted the same answer while I was typing mine.) Another way to proceed, once you have $Y'Y=0$ is to add up the diagonal entries of $Y'Y$. Of course the result is $0$, but it's also the sum of the squares of all the entries of $Y$. So $Y=0$ follows without having to introduce the vectors $V$.
Jan
18
answered Prove matrix $X'XA$ equals $X'XB$ iff $XA = XB$
Jan
17
comment Proving that a set is not consistent
Aha! It seems you're asking only about propositional logic, not first-order logic. In that case, yes, substitute "valuation" for "model". ("Model" would be the appropriate concept in first-order logic, where the same result holds.)
Jan
17
comment Proving that a set is not consistent
If you don't know what models are, then what do you mean by "$\Sigma\models\psi$ and what do you mean by "satisfiable"? The usual definitions of those concepts involve models.
Jan
17
answered Proving that a set is not consistent
Jan
16
comment How's it possible for a bounded subset of the reals not to contain it's $\inf$ or $\sup$?
If $A$ is the set of numbers $x$ such that $0<x<1$, then the supremum of $A$ is 1, which is not an element of $A$.
Jan
16
comment Do we gain anything interesting if the stabilizer subgroup of a point is normal?
The action of $G$ (in the transitive case) need not be regular, not need the cardinalities of $G$ and $S$ be equal. But the action of $G$ induces an action of $G/G_s$, and that is regular.
Jan
3
answered What are these diagrams called? And, what are some good *free* books/notes where I can learn about them?
Jan
3
comment How to prove the expressiveness of first-order logic formulas with equality over the empty signature?
You presumably mean not only these three sorts of sentences but also their (finitary) propositional combinations, such as "there are at most $n$ elements" or "the number of elements is 3 or 17 or 2016."
Jan
1
comment Show the representation is an integral multiple of the regular representation
I conjecture that Joey Zou was hinting at the following argument. The trivial representation has multiplicity 1 in the regular representation, so it has multiplicity $n/|G|$ in your representation with character $\chi$. But the multiplicity of any irreducible representation in any (actual) representation is an integer.
Dec
13
comment Existence of Hamel basis, choice and regularity
I'm not aware of any result removing the need for regularity here.
Dec
12
awarded  Nice Answer
Dec
10
comment On the meaning of the complex measure $\int_{\mathbb{C}} d z d \bar{z}$
@PhilosophiæNaturalis The product $dz\,d\bar z$ is a purely imaginary multiple of an area element. I see no reason to use the bsolute value of the Jacobian in this complex context. Nor do I see a reasonable meaning for "orientation" since that refers to the signs of real numbers.