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Yes, for starters there is the arithmetical hierarchy, where enumerable = $\Sigma^0_1$ and it continues $\Pi^0_1$, $\Delta^0_2$, $\Sigma^0_2$ etc. See also the Computability Menagerie.

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I …

I'm wondering where the relative probabilistic distance or Jaccard distance was first studied: $$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$ where $\overline A$ is the complement of $A …

$A'$ is the discrete partition of $A$. That is, we think of it as a partition of $A$ induced by the finest equivalence relation, the identity relation.

Here is a counterexample of size 23. Let $m=6$ and let $$P=\{0,a_1,\dots,a_m,1\}\cup\{b_{ij}: 1\le i<j\le m\}$$ where $0<a_i<b_{jk}<1$ whenever $i$ is distinct from $j$ and $k$. The cardinality of $P$ …

Question 1: Inspired by the ones you found we can see that there are infinitely many solutions as follows: $$(x,y,z;n) = (k-1,\quad k(k-1),\quad k-1;\quad 2^k)$$ for any $k\ge 0$. Edit re: Question 2 …

Yes, this should follow from the elementary bound. The point is that having a Kalmar elementary time bound is "closed under" searches through exponentially large collections. Suppose $N=W(r,k)$ is lea …

Yes, there is now Pavlovic's characterization of Turing computability in terms of the monoidal computer, based on monoidal categories. http://arxiv.org/abs/1208.5205

If you restrict attention to TMs that always halt, then: One measure of complexity of a Turing machine is its running time, the maximum number of steps taken before it halts on inputs of length $n$, …

This is in Downey and Hirschfeldt: Algorithmic randomness and complexity, Theorem 6.1.3, which cites Chaitin, G. Information-theoretical characterizations of recursive infinite strings, Theoretical C …

$\mathscr F$-indistinguishability. In analogy with Topological indistinguishability.

A year after you posted the question, Fritz showed the common theory of all such lattices is undecidable: https://arxiv.org/abs/1607.05870 In reponse to @MattF's query I'll post an example of how i …

There is Burris and Sankappanavar's free book A Course in Universal Algebra.

Yes, this has been studied and is indeed known as ordered geometry or the study of betweenness spaces: https://en.m.wikipedia.org/wiki/Ordered_geometry

Here is the computability-theoretic version: Suppose $G=G_0\oplus G_1$ is 1-generic and suppose $X$ is computable from both: so $X=\Phi_1^{G_0}=\Phi_2^{G_1}$. Then also $X=\Phi^{G_0}=\Phi^{G_1}$ where …