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

Lebesgue measure is countably additive.

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

\(DC(\aleph _{1})\):  Given a relation \(R\) such that for every  subset \(Y\) of a set \(X\) with \(|Y| < \aleph_{1}\) there is an \(x \in  X\)  with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow  X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).

\(SPI\): Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.
Jech [1973b], p 172 prob 8.5.

\(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\).

\(PW\):  The power set of a well ordered set can be well ordered.

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

If \(X\) is a \(T_2\) space with at least two points and \(X^{Y}\) is hereditarily metacompact then \(Y\) is  countable. (A space is metacompact if every open cover has an open point finite refinement. If \(B\) and \(B'\) are covers of a space \(X\), then \(B'\) is a refinement of \(B\) if \((\forall x\in B')(\exists y\in B)(x\subseteq y)\). \(B\) is point finite if \((\forall t\in X)\) there are only finitely many \(x\in B\) such that \(t\in x\).) van Douwen [1980]

\(SVC\): 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}.

The Banach-Tarski Paradox: There are three finite partitions \(\{P_1,\ldots\), \(P_n\}\), \(\{Q_1,\ldots,Q_r\}\) and \(\{S_1,\ldots,S_n, T_1,\ldots,T_r\}\) of \(B^3 = \{x\in {\Bbb R}^3 : |x| \le 1\}\) such that \(P_i\) is congruent to \(S_i\) for \(1\le i\le n\) and \(Q_i\) is congruent to \(T_i\) for \(1\le i\le r\).

\(\Bbb Z\) (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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

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.

The set of all denumerable subsets of \(\Bbb R\) has power \(2^{\aleph_0}\).

If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\).

\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every  set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 81(\(n\)).  

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

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

  \(C(\infty ,2)\):  Every family of pairs has a choice function.

\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.

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

\(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\).

\(EFP\ Ab\): Every Abelian group is a homomorphic image of a free projective Abelian group.

\(KW(WO,<\aleph_0)\),  The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\)  then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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 \(E_i\).