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

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

Lebesgue measure is countably additive.

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

\(C(WO,{\Bbb R})\):  Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.

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

There is an aleph whose cofinality is greater than \(\aleph_{0}\).

\(EI\ Ab\): For every Abelian group \(A\) there is an injective Abelian group \(G\) and a one to one homomorphism from \(A\) into \(G\).

There is a non-trivial injective Abelian group.

\(SVC\): 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(\(n\)).  

Every compact \(T_2\) space is weakly  Loeb. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.

\(MC_\omega(\aleph_0,\infty)\): For every denumerable family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

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

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

A product of non-empty, compact \(T_2\) topological spaces is non-empty.