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Aug
4
comment Time evolution of the worldlines of 2 particles
Since it is not a scalar field, whether the components are constant or not depends on the coordinate system. Remember, we are in a curved spacetime.
Aug
4
comment Question on Empty set and Power set
"A set may have another set as a member." — actually in ZFC, a set may only have other sets as members.
Aug
4
comment 7 Drinks - 7 Flavors - Infinite variety?
@JasonPolak: I would argue that "flavour" only exists in our perception (there exists a mix of chemicals in the drink, but how that translates into flavour depends on out taste buds and the way their signal is processed in our brain). So it is reasonable to say that if nobody is able to taste the difference in flavour, the difference in flavour doesn't exist.
Aug
4
comment Time evolution of the worldlines of 2 particles
"I think all the components of $F_{ab}$ are constant" — in which coordinate system?
Aug
4
comment Diameter of a triangle
Isn't there an edge for each pair of vertices in a simplex? Then the largest distance between two vertices is the longest edge also in $\mathbb R^n$.
Aug
4
comment Can every set be a group?
"a cardinal number is a special kind of ordinal" — I thought this was only true for the $\aleph$ cardinals, which with AC are all cardinalities, but not without AC.
Aug
4
comment Can every set be a group?
"(Under the axiom of choice,) every set has a cardinality" Why does this statement need the axiom of choice? I mean, every set has a bijection to itself (the identity), and thus, every set should have a cardinality (because as I understand it the cardinality is defined by the sets it has a bijection to, or is that wrong?).
Aug
4
comment Can every set be a group?
The empty set admits no group because there has to be a neutral element.
Aug
2
accepted The “depth” of a set
Aug
2
comment The “depth” of a set
@ZhenLin: Thank you.
Aug
2
asked The “depth” of a set
Aug
2
comment trace of $(A^4-A^3)$
For $2\times 2$ matrices you have: $2\det A = (\operatorname{tr} A)^2 - \operatorname{tr} (A^2)$.
Aug
2
revised Whether $(0,1)$ and $(0,1]$ are homeomorphic
edited body
Aug
2
comment How can I find the example of $f(x)$ such that $\,\lim_{x\to\infty}f(x) \neq 0$?
About your question: Why do you think the problem is wrong?
Aug
2
comment How can I find the example of $f(x)$ such that $\,\lim_{x\to\infty}f(x) \neq 0$?
Of course. There's nothing wrong with your notation. I just wanted to make you aware of a shorter way to write it. The "I don't know if this notation is common" refers to the shorter notation I gave.
Aug
2
comment How can I find the example of $f(x)$ such that $\,\lim_{x\to\infty}f(x) \neq 0$?
Just a remark: I don't know if this notation is common, but I know the much shorter notation $\mathbb R_0^+$ for the set $\mathbb R^+\cup \{0\}$.
Aug
1
comment Is $0$ a natural number?
The cardinality of sets of sets can certainly be $0$: All members of the empty set are sets. Indeed, in ZF all sets are sets of sets.
Aug
1
comment Is $0$ a natural number?
@Kaveh: Not including $0$ simplifies other things. For example, you can then define rational numbers as (equivalence classes of) pairs of an integer and a natural number; no explicit exception for $0$ needed. And also the equivalence relation can then be easily stated by $(a,b) \equiv (ac,bc)$ for any $c\in\mathbb N$ (again, no exception needed).
Aug
1
comment Is $0$ a natural number?
Indeed, Wikipedia confirms that "whole number" is even more ambiguous than "natural number" because it may not only take both meanings of "natural number", but in addition also the meaning of "integer".
Aug
1
comment Is $0$ a natural number?
Now if you say "one" is the name of the initial natural number, then "take $0$ to be one" would be interpreted as "take $0$ to be the initial number", that is, "start the natural numbers with $0$".