Listing notes

[94 O] \(\rightarrow\) Form 74.

Form 43 implies Form 113

[9 A] is equivalent to Form 9

For the relationships between Form 71

Form 9 (Dedekind finite = finite) is equivalent to [9 C] (The image of a Dedekind finite set is Dedekind finite.)

Form 9 (Dedekind finite = finite) and [9 D] (The union of a Dedekind finite family of finite sets is Dedekind finite) are equivalent.

Form 39 does not imply Form 43

Form 40 does not imply Form 44

This note contains the definitions necessary to understand the forms in Brunner [1984d], that is, Form 126.

Definitions for forms [14 R] through [14 U], [14 BH] through [14 BL], and [86 B]. See Banaschewski [1981], Banaschewski [1983], Banaschewski [1997], Paseka [1989], and Vickers [1989].

This note contains the proof that forms 14 (The Boolean Prime Ideal Theorem), [14 BM], [14 BN], and [14 BO] are equivalent.

Definitions for forms [14 W], [70 A], [52 H] through [52 L], Form 372 and [372 A] through [372 D] .These are modifications of definitions from Schechter [1996a] and Schechter [1996b].

Form 99 (Rado's selection lemma) and Form 62 (\(C(\infty ,<\aleph _{0})\)) imply the Boolean prime ideal theorem (Form 14)

A proof that [14 AA] implies [14 Z]

Form 10 is equivalent to [10 F]

A proof that Form 284 impiies Form 61

Definitions for form [14 AB]

Form 233 is true in \(\cal N1\)

These are definitions from Brunner [1982b] and results similar to the equivalence of [8 C] and [8 D] to Form 8 and [10 H] to Form 10. We also include some results from Brunner [1987b].

Form 62 (\(C(\infty,< \aleph_0)\)) + Form 57 implies Form 9 is not provable in \(ZF\).

All the properties of Form 17 (Ramsey's Theorem I) proved by Blass also hold of Form 325

Jech [1982] shows that [34 A] implies Form 34

Results and consequences of von Rimscha [1982]

Ramsey's does and does not imply...

Results of Gitik

If \(\Gamma\) and \(\Delta\) are Dedekind finite cardinals the following are equivalent:

1. lub\(\{\Gamma,\Delta\}\) exists,
2. \(\Gamma\) and \(\Delta\) are comparable and
3. glb \(\{\Gamma ,\Delta \}\) exists.

References of class forms of \(AC\)

Equivalences in Feferman's model, \(\cal M2\) shown in Truss [1978]

Form 322 (\(KW(WO,\infty)\)) is true in \(\cal N1\)

Form 203 implies Form 94

Form 3 implies Form 53

[40 B] implies Form 208

[62 E] implies Form 62

In this note we give definitions from Morillon [1988] for forms [14 L], [14 BP] through [14 BX],[14 CC] through [14 CH], [118 I] through [118 T], Form 331, Form 332, Form 343, Form 344 ,and [345 C] through [345 E].

Form 207\((\alpha)\) implies Form 209

A general method for adding Dependent Choice (Form 43) to models of \(ZF\) is described

It is shown that \(C(\infty ,\lt \aleph _{\alpha }) \rightarrow  C(\infty ,\aleph _{\alpha })\)is not a theorem of \(ZF\) for regular \(\aleph _{\alpha }\)...

Several weak forms of \(AC\) are known to be equivalent to \(AC\) in \(ZF\)...

Form 126 (\(MC(\aleph_0,\infty)\)) implies Form 82 and Form 185

In Rav [1977], it is shown that under the assumption of \(RL\) (Form 99) and certain conditions on a set \(X\) and filter \(\cal F\) in \({\cal P}(X)\), \({\cal F}\) can be extended to an ultrafilter.

In Rav [1977] forms [14 AL], [14 AM], [14 AO] and [14 AP] are shown to follow from Form 14 (BPI).

Form 30 implies Form 83

Form 14 (BPI) implies [14 AR]

Form 31 (\(UT(\aleph^0,\aleph^0,\aleph^0)\)) does not imply Form 9 (Dedekind finite = finite).

Form 269 is  false in the first Fraenkel model \(\cal N1\) of \(ZF^{0}\)

Some relationships between forms 271(\(n\)) and 45(\(n\)) are given

We give proofs due to \ac{A.~Rubin} that\(MC(\infty,\infty,\hbox{ even})\) (334) is true in \(\cal N2\) and\(MC(\infty,\infty,\hbox{ odd})\) (333) is true in \(\cal N2^*(3)\).\proclaim{Theorem 1} \(MC(\infty,\infty,\hbox{ even})\) is true in \(\cal N2\).

Form 65 (Krein-Milman Theorem) does not imply Form 14 (BPI) in \(ZF^{0}\)

If \(N=N(A,\cal G,S)\)  is a permutationmodel, then the \(2m=m\) principle (form 3) is false in \(N\) if thefollowing holds in \(N\)...

Relationships between the forms Form 336(\(n\)) and Form 342(\(n\))

A proof that Form 304 (There does not exist a \(T_{2}\) topological space \(X\) such that every infinite subset of \(X\) contains an infinite compact subset.) is true in \(\cal N1\).

Definitions from Cowen [1990] for forms [14 BD] through [14 BG].

In this note we list some relationshipsbetween forms which are provable in \(ZF^0\) and are not readily available from table 1

We shall prove that Form 10(\(C(\aleph_0,<\aleph_0)\)) + Form 163 (Every non-well orderable set has an infinite Dedekind finite subset.) implies Form 231 (\(UT(WO,WO,WO)\)) and also that Form 133 (Every set is either well orderable or has an infinite amorphous set.) implies Form 231.

A proof that Form 314 implies Form 119

In this note we prove that form [30 G] (Every \(\subseteq\)-chain in \(\cal P(X)\) is contained in a maximal \(\subseteq\)-chain) is equivalent to Form 30

The following definitions and results for forms [8 M], [8 N], [43 Q], [126 B] through [126 F] are from Keremedis [2000a].

Prove forms [1 F] and [1 CC] are equivalent to Form 1

Form 375 (Tietze-Urysohn Extension Theorem) implies Form 119 \(C(\aleph_0\),uniformly orderable with order type of the integers).

Form 67 (\(MC\), Multiple Choice Axiom) implies Form 375 (Tietze-Urysohn Extension Theorem).

Form 214 implies Form 152

Form 234 implies Form 282

Form 385 implies Form 70

A proof that Form 409 implies Form 62.

Form 144 is true in \(\cal N14\),\(\cal N15\), \(\cal N17\), \(\cal N18\), \(\cal N36(\beta)\), \(\cal N37\), and \(\cal N41\).

Form 133 implies Form 340

Form 341 implies Form 10

Definitions for forms [0 AV], [8 AP] through [8 AS], [94 X], and Form 424 from Gutierres [2004].

[218 B] implies Form 218