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Jul
2
awarded  Curious
Jun
4
comment Looking for a field isomorphic to $\Bbb{Q}$
Let $\alpha: \mathbb{Q} \to \mathbb{N}$ be a bijection and define operations $\times_\alpha, +_\alpha$ on $\mathbb{N}$ so that $\alpha$ is an isomorphism of fields. Your question should be more specific and well defined, otherwise I would not be surprised if your question is closed.
May
18
comment Conjecture on OEIS A167055
How high have you not disproved the conjecture?
May
14
answered Functional equation $f(x)=f(\sqrt{x})$
Apr
14
awarded  Yearling
Mar
18
awarded  Disciplined
Mar
16
accepted How many normal subgroups of the free group on $n$ generators?
Mar
15
asked How many normal subgroups of the free group on $n$ generators?
Mar
13
comment Can $\mathbb R$ be enumerated after all?
In bijection with some ordinal(that is uncountable), not the natural numbers.
Mar
9
comment Let $A$ be a subset of $C$ and $B$ a proper subgroup of $C$
@DonAntonio Okay, I was thinking you were going to do some something with reverse inclusion ($A \leq B$ iff $B \subseteq A$) and see if, with the help of Zorn's lemma get a much smaller set than $C \setminus B$ (which would be maximal with the subset relation). As for the hint $Bc$ and $c^{-1}B$ are disjoint from $B$. The idea would be to pick a $ b \in B $ and an element from $Bc$ and one from $ c^{-1}B $ that multiply to get $b$.
Mar
9
comment When should I start learning Set Theory?
This question will be useful and may answer it for you. I think most of the really important stuff showed up in the answers to your question (some basics on cardinals, ordinals, and well ordering principle/axiom of choice) and I think that can be picked up as you go along. Although, if you want to study set theory or foundations of mathematics then you probably could start now.
Mar
9
answered An example of an ordered, uncountable set in $\Bbb R$?
Mar
9
comment Let $A$ be a subset of $C$ and $B$ a proper subgroup of $C$
@DonAntonio I am curious as to what you were planning (finding a minimal $A$?)
Mar
9
comment An example of an ordered, uncountable set in $\Bbb R$?
In the context of real analysis, particularly studying continuity of the the real numbers, I don't think these longer sequences add anything due to some nice topological properties of the reals. In more general topological situations you do need a more general concenpt than sequences though. You may want to look into nets and filters.
Mar
9
comment An example of an ordered, uncountable set in $\Bbb R$?
I would bet that is what the professor means, but one can have "sequences" that are longer than the natural numbers, or even uncountable (I put quotes since sequences are typically assumed to be countable). I guess technically your prof's definition includes these longer sequences and as others have pointed out, within ZFC, you can order the whole real line, or any subset, so there is always a next thing in the list.
Mar
9
comment An example of an ordered, uncountable set in $\Bbb R$?
If you are familiar with functions a definition I like for a sequence of reals is a function $f: \mathbb{N} \to \mathbb{R}$, and the $n^{th}$ term in the sequence would be $f(n)$. If taken as the definition implies all sequences are countable in length. (Sorry if my first comment was a tad confusing, my eyes skimmed over the word sequence in the post)
Mar
9
comment An example of an ordered, uncountable set in $\Bbb R$?
A sequence can be seen as an ordered list and is typically considered countable but there are generalizations to include uncountable lists.
Mar
9
comment An example of an ordered, uncountable set in $\Bbb R$?
If you are sticking with the typical order on the reals there are no uncountable ordered subsets of the reals. If there are no restrictions on ordering then every subset can be ordered in ZFC.
Mar
9
comment Let $A$ be a subset of $C$ and $B$ a proper subgroup of $C$
Certainly $\langle C \setminus B \rangle \cup B = C$.
Mar
9
comment Let $A$ be a subset of $C$ and $B$ a proper subgroup of $C$
I would show that any group is not the union of two proper subgroups.