1999 Changes and additions for the Consequences of the Axiom of Choice Project

1. Changes June 1999
   • Add the following items to the bibliography:
     

     Kolany, A.

     [1999] Rad\`o selection lemma and other combinatorial statements uniformly proved, preprint.

     Herrlich, H./Keremedis, K.

     [1998] Products, the Baire category theorem, and the axiom of dependent choice, preprint October 12.

   • Add Goldblatt [1985] as a reference for form [345 A].
   • In form [43 D] add the word "filter": "... there is a ${\Cal D}$ generic filter $G$ with $p\in G$."
   • Forms [14 CM] and [43 W] through [43 AC] have been added. Also, the first definition in note 28 has been revised. For these additions and revisions choose one of the two below:
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     Put [14 CM] in CHOICE FORMS, V. Conditional Choice and put [43 W] through [43 AC] in TOPOLOGICAL FORMS I. Baire Category Type Theorems and in II. Product Theorems.
   • In the bibliography, change Banaschewski [1998] to:
     

     [1998] Choice functions and compactness conditions , Math. Logic Quart. 44, 427-430.

     
     change Felgner, U./Truss, J. K. [1999] to:
     

     [1999] The independence of the prime ideal theorem from the order extension theorem , Fund. Math. 64, 199-215.

     
     and change Keremedis, K./Tachtsis, E. [1999] to:
     

     Howard, P./Keremedis, K./Rubin, J./Stanley, A./Tachtsis E.,

     [1999] Non-constructive properties of the real numbers , preprint.

   • Add the following items to the bibliography:

     Keremedis, K./Tachtsis, E.

     [1999a] On Loeb and weakly Loeb Hausdorf spaces preprint.

     Keremedis, K./Tachtsis, E.

     [1999b] The countable axiom of choice for finite sets does not imply compact metric spaces are separable preprint.

   • Add the two forms:
     

     [5 A] Partial Choice for Countable Families of Countable Sets of Reals: Every countable family of non-empty countable sets of real numbers has an infinite subset with a choice function. (See the proof of the equivalence of 94 and [94 N].)

     FORM 338. $UT(\aleph_0,\aleph_0,WO)$: The union of a denumerable number of denumerable sets is well orderable. Note 4.

     Add [5 A] to CHOICE FORMS, Part IV, Partial Choice and add form 338 to CARDINAL NUMBER FORMS, Part III Cardinality of Unions.

   • Form [8 F] should be a new form, form 339. This leaves [8 F] blank.
     

     FORM 339. Martin's Axiom $(\aleph_{0})$: Whenever $(P\le)$ is a non-empty, ccc quasi-order (ccc means every anti-chain is countable) and ${\Cal D}$ is a family of $\le\aleph_0$ dense subsets of $P$, then there is a ${\Cal D}$ generic filter $G$ in $P$. Kunen [1980], Shannon [1990], and note 47.

     Remove [8 F] and add form 339 to ORDERING RELATIONS, Part II, Versions of Martin's Axiom.

   • It was shown in Howard/Keremedis/Rubin/Stanley [1999] that forms 6 and 36 are equivalent. Thus, omit form 36 and add form [6 C]:

     [6 C] If $A\subseteq{\Bbb R}^n$ and $A\bigcap B$ is countable for every bounded $B$ then $A$ is countable. G. Moore [1982] p 36, Keremedis/Howard/Rubin/Stanley/Tachtsis [1999].

   • Add a new form 36:

     FORM 36. Compact T$_2$ spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.) Keremedis/Tachtsis [1999a].

2. Changes July 1999

   • Add a new note 4. In the Table of Contents for Part IV replace 4 by: A proof that 32 does not imply 338. For the new note 4, choose one of the two below.
     

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   • We have added a new model N55. For a description of this new model choose one of the two below.
     

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   • Several additions to rfb1.tex (PART V) have been made. (After down loading the most recent version of rfb1.tex, run the program mk_book.c to get the updated versions of book1 and book2.)
3. Additions and Changes August 1999

   • For additions to the bibliography click on one of the following.
     

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   • For additions to the Numerical List of Forms click on one of the following.

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   • For additions to the Topical List of Forms click on one of the following.
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   • For additions to the models click on one of the following.
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   • For additions to the notes click on one of the following.

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4. Additions and Changes October 1999

   • For additions to the bibliography click on one of the following.
     

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   • For additions to the notes click on one of the following.
     

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   • For additions to the Numerical List of Forms click on one of the following.

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