Yes.
The following is due to M. Spector (Ultrapowers without the axiom of choice. J. Symbolic Logic 53 (1988), no. 4, 1208–1219; DOI: 10.1017/S0022481200028024, JSTOR, projecteuclid.)
The ultrapower embedding $j\colon M\to M^I/U$ is elementary if and only if for every family of non-empty set...3
An ideal $\mathcal{I}$ on the positive integers $\mathbf{N}$ is a P-ideal if for every sequence $(A_n)$ of sets in $\mathcal{I}$ there exists $A \in \mathcal{I}$ such that $A_n\setminus A$ is finite for all $n$.
Moreover, an ideal $\mathcal{I}$ is said to be analytic if (equipping $\mathcal{P}(...
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You wrote:
Suppose our intuition for the phrase "subset of $X$" comes from the idea of having an effective total function $X \rightarrow \{0,1\}$ that returns an answer in a finite amount of time. In this case, the subsets of $X$ ought to form a Boolean algebra.
Unfortunately, this is not a...16
While reading a paper (pdf) about the history of modern logic, I learned that some opinions (about deductive/axiomatic mathematics) typically attributed to David Hilbert can be traced back to Moritz Pasch. After googling for Moritz Pasch, I was surprised to learn that he had found important impli...
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For the history of first-order logic I strongly recommend "The Road to Modern Logic-An Interpretation", José Ferreirós, Bull. Symbolic Logic v.7 n.4, 2001, 441-484. The abstract at least is freely available.
Apart from first order logic and higher order logic there are several less well known l...
