Cardinal successors 2:  For every cardinal \(m\) there is a cardinal \(n\) such that \(m < n\) and \((\neg  \exists  p)(m < p < n)\).

If \(X\) is finite then \(X^\omega\) is compact.

Closed subspaces of limited amorphous spaces arel imited amorphous.  (A space is limited amorphous if each amorphous subset is compact.)

Products of limited amorphous spaces are limited amorphous. (A space is limited amorphous if each amorphous subset is compact.)

The box product of a well ordered family of limited amorphous spaces is limited amorphous.  (A space is limited amorphous if each amorphous subset is compact.)

Cancellation  for surjective cardinal equivalence:
For every \(k\in\omega -\{ 0\}\), \((\forall x\forall y)(kx =^* ky\) implies \(x =^* y)\)
(The expression \(x =^* y\) means that there is a function \(f\) from \(x\) onto \(y\) and a function \(g\) from \(y\) onto \(x\).)

Cancellation for injective cardinal inequality:
For every \(k\in\omega - \{ 0\}\), \((\forall x \forall y)(kx \le ky\)implies \(x\le y)\)
(\(\le \) is the usual (injective) cardinal ordering.

F. Riesz Theorem:  A Hilbert space is finite dimensional iff its closed unit ball is compact.

The following are equivalent in a Hilbert  space:

1. The closed unit  ball  is  sequentially compact.
2. Each unconditionally convergent series converges absolutely.
3. Each ortho-normal system is Dedekind finite.

Zermelo's Fixed Point Theorem:  For every partially ordered set \((X,\le)\), if every well ordered subset has a least upper bound then every \(f: X\rightarrow X\) satisfying \(\forall t \in X\), \(t\le f(t)\) has a fixed point.

For all partially ordered sets \((X,\le)\), if there is a \(\sup\) function \(\sigma\) on the well ordered subsets of \(X\) then every \(f: X\rightarrow X\) satisfying \(\forall t \in X\), \(t\le f(t)\) has a fixed point.

Caristi's Fixed Point Theorem:  If \((X,\rho)\) is a complete metric space and \(\phi : X\rightarrow{\Bbb R}\) is bounded above and upper semi-continuous then in the Br\ondsted  ordering(\(x\le y\) iff \(\rho(x,y)\le\phi(y) - \phi(x))\) every \(f: X\rightarrow X\)satisfying \(\forall t\in X\), \(t\le f(t)\) has a fixed point.

A \(T_1\) topological space is Dedekind finite if and only if every subspace has a dense Dedekind finite subset.

If \(X\) and \(Y\) are topological spaces with dense Dedekind finite subsets so is \(X\times Y\).

A Lindelöf space with a dense, Dedekind finite subset is compact.

A Lindelöf space with a Dedekind finite basis is compact.

For all cardinal  numbers \(m\)  and \(n\) and Dedekind finite cardinals \(p\), if \(p + m = p + n\) then \(m = n\).

Every infinite Dedekind finite field is the direct limit of a strictly increasing \(\omega \)-sequence of finite subfields.

If \((P,\le)\) is a quasi-order such that \(P\) is non-empty, well orderable and every antichain is finite then there is a \(P\)-generic filter.

If a set \(S\) is hereditarily countable (that is, every \(y\in TC(S)\) is countable) then the rank of \(S\) is \(<\omega_2\).

For any class of similar (universal) algebras \(K\),\(HSP\ K = Mod\ Eq\ K\). (The class of homomorphic images of subalgebras of products of non-empty families of algebras in \(K\) is the same as the class of algebras which satisfy the set of equations holding in every algebra in \(K\).) (The proofs of forms [0 U] and [0 V] in ZF depend on the axiom of regularity.)

For every class \(K\) of similar algebras \(SP\ K \supseteq PS\ K\). (For any class \(K\) of similar algebras, \(S\ K\) is the class of all subalgebras of elements of \(K\) (not closed under isomorphism) and\(P\ K\) is the class of algebras isomorphic to  direct  products  of algebras in \(K\)). (The proofs of forms [0 U] and [0 V] in ZF depend on the axiom of regularity.)

For every class \(K\) of similar algebras \(SP^{r}K \supseteq P^{r}S\ K\). (For any class \(K\) of similar algebras, \(S\ K\) ist he class of all subalgebras of elements of \(K\) (not closed under isomorphism) and \(P^{r}K\) is the class of all algebras isomorphic to reduced direct products of algebras in \(K\).)

\(DC(\boldsymbol\Sigma^1_2)\): If \(P \subseteq{}^{\omega}\omega\times{}^{\omega}\omega\) and \(P\) has domain \({}^{\omega}\omega\),\(P \in\boldsymbol\Sigma^1_2\), then for any \(y\in {}^{\omega}\omega\)there is a sequence \(\langle x_{k}: k\in\omega\rangle\) of elements of\({}^{\omega }\omega\) with \(x_{0}=y\) and \(\langle x_k,x_{k+1}\rangle \in P\) for all \(k \in\omega \). Kanovei [1979].

Suppose \(X\) and \(M\) are sets, \(F\) is a function from\({\cal P}(X) - \{\emptyset\}\) into \(M\) and \(G\) is a function on \(M\) such that \(\forall z\in {\cal P}(X) - \{\emptyset\}\), \(G(F(z))\in {\cal P}(z)- \{\emptyset\}\).  Then there are unique \(W\) and \(R\) such that \(W \subseteq M\), \(R\subseteq W\times W\) and \(R\) is an irreflexive well ordering of \(W\),the range of \(G|W\) is a partition of \(X\), and \(\forall w\in W\), \(w =F(X -(\bigcup\{G(\nu): \nu\in W\) and \(\nu\mathrel R w\}))\).

\(CT_{1}\):  Restricted Compactness Theorem for Propositional Logic I:  If \(\Sigma\) is a set of formulas in a propositional language such that every finite subset of \(\Sigma\)is satisfiable and if each propositional variable occurs in at most one formula in \(\Sigma\), then \(\Sigma\) is satisfiable.

If \(X\subseteq \Bbb R\) is perfect, then \(|X|= 2^{\aleph_0}\).

There are \(2^{\aleph_0}\) perfect subsets of\({\Bbb R}\).

For all ordinals \(\beta\), well ordered cardinals\(\gamma \ge 1\) and all \(n\in\omega\), there is a well ordered cardinal\(\kappa\) such that \(\kappa\to (\beta)^{n}_{\gamma}\).

The number of well ordered subsets of a set:

The number of well orderings of a set:  

For any \(n\in\omega\), \(n\neq 0\), \(|s(x)|\not\le |x^n|\)and \(|x^n|< |w(x)|\) if \(|x|\) exceeds some small finite value depending on\(n\).  (\(s(x) \) is the set of well orderable subsets of \(x\) and \(w(x)\) is the set of well orderings of subsets of \(x\).)

No infinite well orderable set is amorphous.

Turinici's Fixed Point Theorem for Hausdorff Spaces: If \((X,\le)\) is a directed, partially ordered set and \(\tau\) is aHausdorff topology on \(X\) such that

1. \((X,\le)\) is upper semi-continuous with respect to \(\tau\),
2. Every well ordered subset of \((X,\le)\) has a unique limit as a net,

Every separable metric space is second countable.

For every set \(A\), there is an \(\aleph\) which cannotbe covered by fewer than \(|A|\) sets each of cardinality \(< |A|\).

Assume \(V\) is a separable vector space, \(D\) is a countable dense subset of \(V\), and \(p :V \to \Bbb R\) satisfies\(p(x+y) \le p(x) + p(y)\), and\((\forall t\ge 0)(\forall x\in V)(p(tx) = tp(x))\). Also assume that \(p\) satisfies the following condition:

No maximal open free filter in a \(T_2\) topologicalspace has a countable filter base.

Countably compact pseudometric spaces are Baire.

Using the discrete topology on 2, \(2^m\) is compact for every well ordered cardinal number \(m\). \ac{Keremedis}\cite{1999b}.

If \(E\) is a separable normed topological vector space, then for every continuous sublinear functional \(p\) on \(E\) there is a linear functional \(f\) on \(E\) such that \(f\leq p\).

For all cardinals \(m\) and \(n\), if \(2m=2n\),then \(m=n\). (Brought to our attention by W. Felscher.)

The Modified Ascoli Theorem: For any set \(F\) of continuous functions from \(\Bbb R\) to \(\Bbb R\), the following conditions are equivalent:

1. Each sequence in \(F\) has a subsequence that converges continuously to some continuous function (not necessarily in \(F\)).
2. 1. For each countable subset \(G\) of \(F\) and each\(x\in {\Bbb R}\), the set \(G(x) = \{ g(x) : g\in G\}\) is bounded, and
   2. Each countable subset of \(F\) is equicontinuous.
   \par\ac{Rhineghost} \cite{2000}

Every complete lattice is constructively \(\cal F\)-complete.  

Every complete lattice is constructively \(\cal W\)-complete.

Every super second countable subspace of \(\mathbb R\)is separable.

\(\aleph_{\alpha+1}\) is not an inaccessible cardinal for any ordinal \(\alpha\).