\(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable  family  of denumerable subsets of \({\Bbb R}\) is denumerable.

Lebesgue measure is countably additive.

The Krein-Milman Theorem: Let \(K\) be a compact convex set in a locally convex topological vector space \(X\). Then \(K\) has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in  \(K\).) Rubin, H./Rubin, J. [1985] p. 177.

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.

Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family  of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\).  Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S.

\({\cal P}(\Bbb R)\) is well orderable.

Every non-well-orderable set has an infinite, Dedekind finite subset.

\(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\).

There is a subset of \({\Bbb R}\) which is not Borel.

There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}.

\(\Bbb Z\) (the set of integers under addition) is amenable.  (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

\(MC(\infty,\infty,\hbox{ even})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 2\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is even.

In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.

There are exactly \(2^{\aleph_0}\) Borel sets in \(\Bbb R\). G. Moore [1982], p 325.

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

The set of all finite subsets of a Dedekind finite set is Dedekind finite. Jech [1973b] p 161 prob 11.5.

Aczel's Realization Principle: On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points.

Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact.

Every non-well-orderable set has an infinite, Dedekind finite subset.

Every non-well-orderable set has an infinite subset with a Dedekind finite power set.

For every set \(S\), if the only linearly orderable subsets of \(S\) are the finite subsets of \(S\), then either \(S\) is finite or \(S\) has an amorphous subset.

Every infinite tree has either an infinite chain or an infinite antichain.

Every Lindelöf metric space is second countable.

If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\).

\(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.