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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