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Once you have set-theoretic definitions of a few basic notions, like the ordered pairs, ordered triples, and functions that you mentioned in your question, plus natural numbers, you can define other c …

At Sergei Akbarov's suggestion, I'm copying my comment into this answer. In the presence of the other usual axioms of set theory, including especially the axiom of regularity, the axiom of global cho …

The result you asked about follows instantly from Fagin's proof of the zero-one law for finite graphs. He shows that all of Gaifman's extension axioms have asymptotic probability 1, and "diameter $\l …

This abelian group, which can also be described as the quotient of the direct product $\prod_{\mathbb N}\mathbb Z$ by the direct sum $\sum_{\mathbb N}\mathbb Z$, is isomorphic to the direct sum of the …

The notion of classification seems to depend rather strongly on the particular field of mathematics, and in the (very few) cases I'm acquainted with, it seems easier to give a criterion for unclassifi …

I assume your "average change in uncertainty" $S(p)$ in the case of a binary variable is meant to be the usual entropy, $-p\log p-(1-p)\log(1-p)$. In that case, your two formulas for the change in un …

Do you have a reason to prefer NBG over MK (Morse-Kelley theory of sets and classes)? If not, you might look at the development of MK in the Appendix (if I remember correctly) of Kelley's book "Genera …

I'm not sure what is meant by "illustrating" a basis, but the axiom of choice is needed even to prove the existence of a basis for $\mathbb Q_p$ over $\mathbb Q$.

Here's a variant of the argument that gives a slightly stronger result, by making explicit the use of the Baire category theorem that is "hidden" in some of the previous proofs. Suppose, toward a c …

ZFC plus $\text{MA}_{\aleph_1}$ is consistent relative to ZFC, while analytic determinacy has a little bit of large cardinal strength, namely the existence of sharps of reals. So ZFC+MA cannot prove …

Being away from home, I can't easily check references, but here's an outline of the proofs for what you quoted from googling: Finite linear orders satisfy the statements There is a first element and …

Here's a counterexample for (A). Let $V=\{0,1,2\}$, let $E$ consist of $\{0,1\}$ and $\{1,2\}$, each with weight 1. The shortest-path metric would be unchanged if we add a third edge $\{0,2\}$ with …

I think "selector" usually refers to choosing just one element from each set in a family. For the concept you described here, I've seen names like "blocker" or "blocking set", but I haven't seen them …

(1) A holomorphic manifold is also (or "can also be viewed as") a smooth manifold, and that lets you define integration. To put it another way, you do have partitions of unity, just not holomorphic o …

Assuming that by "interval" you mean something of the form [x,y] (as in the earlier question you linked to), you could just represent intervals by ordered pairs $(x,y)\in P\times P$ with $x\leq y$. T …