FORM 3: 2ra = ra 17

F O R M 3. 2ra = m: For all infinite cardinals ra, 2m — m. Tarski [1954a].

[3 A] For all cardinals m and n, n m and m infinite — • m + n — m.

Halpern/Howard [1970].

[3 B] Ho - m = m: For all infinite cardinals ra, Ho • m — m. Halpern/Howard

[1970].

[3 C] Definability of cardinal addition as the least upper bound: (For all car-

dinals x, y and z)(x + y — z iff (x z and y z and (Vw)(x, y u — • z ^)).

Haussler [1983] and Tarski [1949a].

[3 D] Every pair of cardinals has a least upper bound in the usual cardinal

ordering and for all cardinals n and 7, if nn 7 for all n £ a;, then Hoft 7.

Hickman [1979b].

[3 E] E(IV, V): For all infinite cardinals m, m + 1 = m implies 2m = m.

Howard/Yorke [1989] and note 94.

[3 F] E(III, V): For every set X, \iV{X) is Dedekind infinite then 2\X\ = \X\.

Howard/Yorke [1989] and note 94.

[3 G] E(II, V): For every set X, if there is a non-empty family of subsets of X

which is linearly ordered by C and has no maximal element then 2\X\ — \X\.

Howard/Yorke [1989] and note 94.

[3 H] E(Ia, V)\ For every set X, if 2\X\ \X\ then X is amorphous. Howard/

Yorke [1989], notes 85(H) and 94.

F O R M 4. Every infinite set is the union of some disjoint family of denumerable

subsets. (Denumerable means = H0.) Whitehead [1902].

F O R M 5. C(Ho, Ko5 ^ ) : Every denumerable set of non-empty denumerable subsets

of M has a choice function. Sageev [1975].

F O R M 6. UT(H0,

HO, HO,M):

The union of a denumerable family of denumerable

subsets of R is denumerable. G. Moore [1982] and note 3.

[6 A] Every uncountable subset of 1 R contains a condensation point. G. Moore

[1982] p 205 and 324, Sierpihski [1918].

[6 B] Every uncountable subset of 1 R has two condensation points. G. Moore

[1982] p 205 and 324, Sierpihski [1918].

F O R M 7. There is no infinite decreasing sequence of cardinals. Schoenflies [1908].

F O R M 8. C(Ko,oo): Every denumerable family of non-empty sets has a choice

function. Sierpihski [1918].

[8 A] PC(oo,oo,oo): Every infinite family of non-empty sets has an infinite

subfamily with a choice function. Brunner [1983b].

[8 B] PC(Ko,oo, 00): Every denumerable family of non-empty sets has an

infinite subfamily with a choice function. Monro [1975].

[8 C] Every cr-compact space is Lindelof. Brunner [1982b] and note 43.

[8 D] Every cr-compact, locally compact T2 space is weakly Lindelof. Brunner

[1982b] and note 43.

[8 E] In a metric space every sequentially continuous real valued function is

continuous. Brunner [1982c].