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1d
comment If $\phi$ holds for all standard models of ZF and ZF proves this, then does ZF prove $\phi$?
I thought I was writing my first comment shortly before falling asleep, but apparently it was actually shortly after.
1d
comment If $\phi$ holds for all standard models of ZF and ZF proves this, then does ZF prove $\phi$?
You're right. My mistake is that the alleged equivalence between "Con(ZF) holds in all standard models" and Con(ZF) tacitly assumed that the "all standard models" part isn't vacuous. I'll delete my incorrect comment (since this one contains enough information to reconstruct it if anyone wants to).
2d
comment Cardinality of a set of natural sequences
The proof shows not only that it's uncountable but that it has the cardinality of the continuum.
Jul
25
comment What is ordinal expression of $\infty$?
I'm not aware of any notion of "cardinal expression" that would apply to $\infty$.
Jul
25
comment On Kolmogorov complexity of first and last half of a string
You don't need to code for a Turing machine; you need to code for a universal machine. Notice first that there is a non-universal (in fact very far from universal) machine that just prints its input. For this ridiculous machine, the length of a program that prints z is just the length of z. Therefore, a universal machine must have a program that prints z and has length bounded by the length of z plus a constant.
Jul
25
comment On Kolmogorov complexity of first and last half of a string
I think you want to arrange for the input to encode both the program for computing $y$ and the string $z$. Unfortunately, you can't just concatenate them; you need to tell the machine where the one ends and the other starts. That can be done with a logarithmic number of additional bits, to tell how long the first part of the concatenation is. I don't immediately see a way to avoid that additional $\log n$.
Jul
24
comment On Kolmogorov complexity of first and last half of a string
Try the algorithm that first computes $y$ and prints it, and then prints $z$. Since $z$ has length $n/2$, the instruction "then print $z$" has length only $(n/2)+O(1)$.
Jul
24
comment $\mathbb{P}_{\kappa}$ forces $\text{non}(\mathcal{M})\leq \kappa$ and $\text{cov}(\mathcal{M})\leq \kappa$
For Question 1, instead of using the Cohen reals obtained by reducing the Hechler reals, you could also use the Cohen reals that are added at limit stages of the iteration because you use finite supports.
Jul
24
answered $\mathbb{P}_{\kappa}$ forces $\text{non}(\mathcal{M})\leq \kappa$ and $\text{cov}(\mathcal{M})\leq \kappa$
Jul
24
answered Question regarding Cramer's rule proof
Jul
24
comment Implication in linear logic
@StevenTaschuk Thanks.
Jul
24
comment Implication in linear logic
Since the OP asked about $p\to p$, it would be worthwhile to add that the linear implication from $p$ to itself is valid in linear logic. Also, when I've seen $\to$ used in linear logic, it's always meant either linear implication (when folks like me can't draw a lollipop) or intuitionistic implication (linear implication from $!a$ to $b$), and either of these meanings will make $p\to p$ valid.
Jul
23
comment Logical equivalence - Russell's Paradox
What do you mean by "the set R is equivalent to the logical statement ..."? How can a set be equivalent to a statement? They're two entirely different sorts of things.
Jul
22
comment Abstract enunciation of the Good Set Principle in measure theory
@Kolmin The error in your statement is that you need to assume that $\mathcal G$ is a $\sigma$-algebra. That does not follow from the things you assumed. If you add the assumption that $\mathcal G$ is a $\sigma$-algebra, then your induction principle becomes correct.
Jul
21
answered Drawing circumference issue
Jul
20
comment Euler's Identity in Degrees
I upvoted this to support the assertion that $180^\circ=\pi$, i.e., that "degree" as an angle measure is the pure number $\pi/180$. It seems to need some support because two other answers deny it.
Jul
19
comment Why does Gödel's (First) Incompleteness Theorem apply to ZFC?
I haven't read Smullyan's book, so I don't know what formalization he uses. The incompleteness proof for theories that "contain" Robinson's Q is available in many places, for instance Peter Hinman's book "Fundamentals of Mathematical Logic". Shoenfield's book "Mathematical Logic" uses a very similar theory that he calls N in place of Q, but it's the same idea. The original source for Q is, I believe, a book "Undecidable Theories" by Tarski, Mostowski, and Robinson.
Jul
19
comment Why does Gödel's (First) Incompleteness Theorem apply to ZFC?
Here "contains Robinson arithmetic Q" needs to be taken in the sense "admits an interpretation of Q", because theories like ZF, having different primitive notions from those of Q, do not literally contain Q.
Jul
19
comment Coproducts in $\mathsf{Grp}$
@Exterior The coproduct in the category of abelian groups also fails to match the coproduct of sets (which is just disjoint union).
Jul
18
comment How is this derivative paradox solved?
In the original form of the question, which didn't say "$(\in\mathbb Z)$", the error is that "till $x$ times" makes no sense. After the edit, inserting "$(\in\mathbb Z)$" (along with other corrections), the error is that it makes no sense to differentiate a function that is defined only on $\mathbb Z$.