François G. Dorais
Reputation
32,578
119/100 score
 2d comment explicit big linearly independent sets @Todd I'm not sure what you mean by "known" but observe that a linear combination of $T_r$ is a series $\sum a_n/n!$ where the $a_n$ come from a finite set of rational numbers (a sum of some of the coefficients). Consider $C_k$ to be the set of all numbers of the form $\sum a_n/n!$ where the $a_n$ come from $\{q_0,q_1,\ldots,q_k\}$. Observe that $C_k$ is a closed nowhere dense set. Baire category shows that $\bigcup_k C_k \neq \mathbb{R}$. This uses DC but I think it's possible to make this argument effective and prove this in ZF alone. Apr 26 revised Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space links to references Apr 19 comment A Model where Dedekind Reals and Cauchy Reals are Different @PaulTaylor: Ah! I didn't understand what $\leq$ you were talking about. You're right, there are better constructive definitions of Dedekind cuts. I used decidable cuts especially to bring out the comparability issue. Apr 19 comment Intersections of families of open sets ordered by well-inside relation in Euclidean space @MadHatter: The key is that $\operatorname{Cl}(\bigcup_{n=m}^\infty V_n) \subseteq \operatorname{Cl} U_m$. So $\operatorname{Cl} A \subseteq \operatorname{Cl} V_0 \cup \cdots \cup \operatorname{Cl} V_{4n} \cup \operatorname{Cl} U_{4n+4}$ and take the limit as $n \to \infty$. Apr 18 comment A Model where Dedekind Reals and Cauchy Reals are Different @PaulTaylor: I'm not sure I understand. $\mathsf{LLPO}$ is definitely about non-observable properties. In other words, it says either you can't observe one or you can't observe the other. If it were about observable properties, it would be constructively true! Apr 12 comment Must $L_\alpha$ be correct about well-foundedness? The second paragraph seems false. The subsets of $\omega$ in $L_{\omega_1^{CK}}$ are precisely the hyperarithmetic sets. Famously, Harrison showed that there are pseudowellorderings: recursive orderings of $\omega$ that are not wellfounded but have no hyperarithmetic descending sequences. Apr 11 awarded Good Answer Apr 10 awarded Nice Question Apr 9 revised Can one divide by the cardinal of an amorphous set? added 2 characters in body Apr 9 asked Can one divide by the cardinal of an amorphous set? Apr 8 comment The Continuum Hypothesis and Countable Unions Ah! That makes sense and 4 is even more interesting now. Apr 8 comment The Continuum Hypothesis and Countable Unions Interesting. Does Truss use an inaccessible? Item 2 implies that $\omega_1$ is inaccessible to reals, so I suppose he must use one somewhere. Or am I missing something? Apr 8 comment Cofinality of countable ordinals in ZF, and in toposes @AsafKaragila: Yes, fixed. Thanks! Apr 8 revised Cofinality of countable ordinals in ZF, and in toposes correction Apr 7 revised Cofinality of countable ordinals in ZF, and in toposes addendum Apr 7 revised Cofinality of countable ordinals in ZF, and in toposes added references Apr 7 revised Cofinality of countable ordinals in ZF, and in toposes added references Apr 7 answered Cofinality of countable ordinals in ZF, and in toposes Apr 2 awarded Nice Answer Mar 29 awarded Great Answer