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Oct
20
comment Set of ordinals less than a given ordinal
@AsafKaragila Oh, right. But equivalence classes are so big and clumsy :(
Oct
20
comment Set of ordinals less than a given ordinal
Isnt't the set of ordinals less than $\alpha$ simply $\alpha$ itself?
Oct
20
answered A Vector Space is a Set - Axiom or Derivation?
Oct
20
comment Argument that two given finite groups are not isomorphic.
You could use the relations to explicitly write down a group multiplication table, taking $x^iy^j$, $0\le i\le 3, 0\le j\le 1$ as group elements.
Oct
19
comment $n$ persons who make telephone calls
@Alkibiades Actually the upper bound is $2n-3$: Let $a_1, \ldots, a_{n-1}$ call $a_n$, after which $a_n$ and $a_{n-1}$ know everything. Then let $a_1, \ldots,a_{n-2}$ call $a_n$.
Oct
19
comment Logically speaking, why can variables be substituted?
@user101939 It is called Leibniz law
Oct
19
comment Winning strategy writing a binary number
@JoyentanujDas Lemma 1 still holds. In order to adapt lemma 2, it is sufficient if there are at least four rounds after the given bit pattern. "Divisible by $3$, but not by $9$" would be sufficient. The result is divsible by $3$ if B picks the right number of $1$s in the last four rounds (and if zero/four $1$s would be okay, so would be three/one); swap consecutive $0$ and $1$ moves to deals with divisibility by $9$.
Oct
19
revised Winning strategy writing a binary number
added 260 characters in body
Oct
19
answered Winning strategy writing a binary number
Oct
19
comment Showing unique prime factorization in first-order logic?
@BabyDragon That does not work for $4=2\cdot 2$.
Oct
19
comment Induction proof of $a^r \ge 1$
What's in a name? A natural by any other name than $n$ would smell as sweet
Oct
19
comment Showing unique prime factorization in first-order logic?
Can you even express $a=b\cdot c$?
Oct
19
answered If A, B, C, D are non-invertible $n \times n$ matrices, is it true that their $2n \times 2n$ block matrix is non-invertible?
Oct
19
comment Can we construct a sequence of functions such that…
Can $\int_E\max\{f_n,0\}$ or $\int_E\min\{f_n,0\}$ be infinite? What can be said about $\int_E\min\{f_n,0\}$ as $n\to \infty$?
Oct
19
answered Finding the limit of a sequence $n^{\sin\frac{\pi}{n}}$
Oct
19
comment Boolean algebra generated by value sets of polynomials over $\mathbb{N}$
Consider the polynomials $X+2$ and $X+1$.
Oct
19
comment Is there always exist a polynomial with a unique zero point?
This is in principle the same answer as by @user33433, but without Galois theory.
Oct
19
answered Is there always exist a polynomial with a unique zero point?
Oct
19
comment Is there always exist a polynomial with a unique zero point?
Note that with $K=\mathbb R$ and $\alpha=i$ this construction indeed gives us the "obvious" example $x_1^2+\ldots +x_m^2$. - Looking closely at the product definig $f$ one sees: Another way to obtain $f(X,Y)$ is to simply take $f(X,Y)=Y^d p(\frac XY)$ where $p$ is irreducible (and without root) of degree $d$
Oct
19
comment Is there always exist a polynomial with a unique zero point?
For $K=\mathbb R$ we could take $x_1^2+\ldots +x_m^2$, but how do we get this polynomial from merely knowing that $X^2+1$ has no root? And what if for general $K$ we just have an arbitrary $f$ without roots?