bof
Reputation
1,989
Next privilege 2,000 Rep.
 1d awarded Yearling Nov 10 comment Replacing Axiom of Choice with Axiom of Countable Choice @NoahSchweber In defining what it means for a well-orderable regular limit cardinal $\aleph_\alpha$ to be strongly inaccessible, do you require $\kappa\lt\aleph_\alpha\implies2^\kappa\lt\aleph_\alpha$ or merely $\kappa\lt\aleph_\alpha\implies2^\kappa\not\ge\aleph_\alpha$? Nov 10 comment Replacing Axiom of Choice with Axiom of Countable Choice @AndrésCaicedo How are limit beth cardinals defined in ZF? How is "strongly inaccessible defined"? If strongly inaccessible cardinals exist, is there necessarily a smallest one? Nov 10 comment Replacing Axiom of Choice with Axiom of Countable Choice Any model of ZF + AC + CH is also a model of ZF + ACC + CH. Likewise, any model of ZF + AC + not-CH is also a model of ZF + ACC + not-CH. Nov 10 comment Replacing Axiom of Choice with Axiom of Countable Choice Surely, if CH could be proved or disproved with ACC, then it could be proved or disproved in set theory with the full AC? So why is 2) a question? Nov 10 comment Replacing Axiom of Choice with Axiom of Countable Choice I thought it was automatic that, if you weaken an axiomatic theory, then anything that was undecidable before is still undecidable? Oct 30 comment Characterizing (up to permutations) finite sequences of real numbers By "unordered sequence" do you mean a multiset? Oct 26 awarded Necromancer Oct 24 comment What are some reasonable-sounding statements that are independent of ZFC? If I remember right, the funny statement is that there is a uncountable famility of entire functions which assumes only countably many different values at each point in the complex plane. Oct 24 comment Does this version of Hadwiger's conjecture hold for graphs with infinite chromatic number? Can't you simplify the argument slightly? If $G$ has isolated vertices, remove them all, and obtain a subgraph $M$ with no isolated vertices. If $G$ has no isolated vertices, remove all edges incident with some fixed vertex $v,$ and obtain a subgraph $M$ with one isolated vertex. Oct 18 comment Examples of common false beliefs in mathematics @Qiaochu: I thought the correct generalization of$$\operatorname{lcm}(a,b)=\frac{a\cdot b}{\operatorname{gcd}(a,b)}$$was $$\operatorname{lcm}(a,b,c)=\frac{a\cdot b\cdot c\cdot \operatorname{gcd}(a,b,c)}{\operatorname{gcd}(a,b)\cdot \operatorname{gcd}(a,c)\cdot \operatorname{gcd}(b,c)}$$Another false belief I guess. Maybe not a common one. Oct 16 comment The Turán problem for graphs with given chromatic number To have a complete answer to your question one would have to know for which values of $r,t,n$ there exists a $K_{r+1}$-free graph on $n$ vertices with chromatic number $t,$ right? I'm certainly no expert on graph coloring, but I would have thought that was an open problem even for $r=2.$ What is the minimum number of vertices for a $t$-chromatic triangle-free graph? Aug 25 comment Injective subset function You are right that it still works for infinite $X$ if $F(x)$ is always finite. But of course it's trivially false as stated, with no finiteness assumption. I wonder if the OP knows that and just forgot to mention finiteness, or if he really wanted it to be true with no restrictions? Aug 17 comment Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed? This must be a dumb question, but can you prove (without regularity) that a proper class must have an infinite subset? If $C$ were a proper class with no infinite subsets, then the class of all (finite) subsets of $C$ would be well-founded under reverse inclusion, but would have no "minimal" element. Aug 15 comment A Banach-Tarski game Don't you want to say that $bigcup Z_{0,i}$ and $\bigcup Z_{1,i}$ are disjoint unions? Aug 6 comment Statements reliant on conjectures Why didn't the axiom of constructibility make your list? Aug 6 comment Statements reliant on conjectures Does the simple negation of the continuum hypothesis really have a lot of interesting consequences? Aug 3 comment Is there a structure theorem or group law for finite groups generated by two elements? According to Theorem 2.1 of F. Levin, Factor Groups of the Modular Group, J. London Math. Soc 43 (1968), 195-203, every countable group is embeddable in a $2$-generator group with generators of prescribed orders $p\ge3$ and $q\ge2.$ Jun 17 awarded Enlightened Jun 14 comment Collection of graduate research projects in Real Analysis Picking a research problem from the list of answers you're going to get seems kind of risky. How many other graduate students are going to look at this list and pick the same problem you did?