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Second point is adding to the results of §2,§3 which say that (in §3 with no large cardinals) we can force {\aleph_1} < \mathfrak{b} = \mathfrak{d} < \mathfrak{a}. We like to have {\aleph_1} < \mathfrak{s} \le \mathfrak{b} = \mathfrak{d} < \mathfrak{a}. For this we allow in §2,§3 the sets K_t to be uncountable; this requires non-essential changes. In particular, we replace usually {\aleph_0}, {\aleph_1} by \sigma , \partial. Naturally we can deal with \mathfrak{i} and similar invariants.

Third we proofread the work again. To get \mathfrak{s} we could have retained the countability of the member of the I_t-s but the parameters would change with A \in I_t, well for a cofinal set of them; but the present seems simpler.

We intend to continue in [Sh:F2009].

Part A: A revised version of the guide in [Sh:g], with corrections and expanded to include later works.

Part B: Corrections to [Sh:g].

Part C: Contains some revised proof and improved theorems.

Part D: Contains a list of relevant references.

Recent (July 2022) additions

• §14 = 12a.2 on: no choice

• On inner models 12.29, see [Sh:805].

• On Black Boxes and abelian groups 14.44 - 14.53, see [Sh:750] and [Sh:898]

• On somewhat free scales 2.16, see [Sh:1008],

• On n-dimensional Black Boxes, quite free abelian groups such that Hom(G,\mathbb{Z}) = \{0\}, 14.56, see [Sh:1028],

• Survey on the existence of universal models; in particular, abelian groups 14.32, see [Sh:1151].

(1) If ZF is consistent then the following theory is consistent “ZF + DC(\omega_{1}) + Every set of reals has Baire property” and

(2) If ZF is consistent then the following theory is consistent “ZFC + ‘every projective set of reals has Baire property’ + ‘any union of \omega_{1} meager sets is meager’ ”.

To 666: problem on equalities of x’s; deal with co-\kappa-Souslin deal with the k-notation and the \alpha-notation.

2012.8.16 In Poland - July- has written a new try , in order to get a Borel relation with aleph_alpha rectangle but no more, for any countable ordinal alpha. What was written was only the forcing. Yesterday, proofread it, expand - still has to write kappa-Delta pair proof, and thoughts about more than \aleph_y w_1 But the rank in [522, §4] seem not OK, will try to revise

(a) I_{\kappa,\lambda}^+\mathop{\longrightarrow}\limits^\kappa (I_{\kappa,\lambda}^+,\omega+1)^2,

(b) I_{\kappa,\lambda}^+\mathop{\longrightarrow}\limits^\kappa [I_{\kappa,\lambda}^+]_{\kappa^+}^2 if \kappa is a limit cardinal, and

(c) I_{\kappa,\lambda}^+ \mathop{\longrightarrow}\limits^\kappa (I_{\kappa,\lambda}^+)^2 if \kappa is weakly compact.

Since we think the introduction may be helpful to some readers (as opposed to the referee, who found the diagrams "mystifying") we preserve it in this form. The overview is supposed to complement the paper, and not to be read independently. The emphasis of this note is on giving the reader some vague understanding, at the expense of correctness of the claims.

Is it consistent that for some set X, {\rm cov}({\rm NULL}\restriction X)=\lambda is a weakly inaccessible cardinal (so X not null of course) while {\rm cov}(\rm meagre) is small, say it is \aleph_1.

Assuming CH , there is a non-P-point ultra-filter on the natural numbers which is preserved by Sacks forcing and many more.

This answer a question communicated to me by Mark Poor

This a work in preparation by Baldwin, J. T., Laskowski, M. C. and Shelah, S.

We deal with linear orders which are countable unions of scattered ones with unary predicates; it is self contained.

The strong (\mathbf{W},\mathbf{V})-covering lemma for (\lambda,\kappa) states that \big([\lambda]^{< \kappa} \big)^{\!\mathbf{W}} is a stationary subset of [\lambda]^{< \kappa}, which means that for every model M \in \mathbf{V} with universe \lambda and vocabulary of cardinality < \kappa, there is N \prec M with universe \in \big([\lambda]^{< \kappa} \big)^{\!\mathbf{W}}.

We give sufficient conditions for the strong (\mathbf{W},\mathbf{V})-covering lemma to hold, which are satisfied in the classical cases where the original lemma holds (i.e. covering, squares, and reals). In fact, we place stronger conditions on M. The proof does not use fine structure theory, but only some well-known combinatorial consequences thereof.

We use this to solve problems about the aspects of adding a real to a universe \mathbf{V}.

Earlier versions appeared as [Sh:b][XIII,§1-4] in the author’s book Proper Forcing (Springer-Verlag 940, 1982), and later versions as Chapter VII of Cardinal Arithmetic (Oxford University Press, Clerendon Press, Vol. 24).

For PC_{\aleph_0}-representable AECs we investigate the conclusion of having not too many non-isomorphic models in \aleph_1 and \aleph_2, but we have to assume 2^{\aleph_0} < 2^{\aleph_1} and even 2^{\aleph_1} < 2^{\aleph_2}.

NOTE: if U is countable, the gap in cardinals disappear even for the weaker (i.e. finer) equivalence every such family is equivalent to a family of equivalence relations (if you want just to follow a proof of this then read):
0.6: definition
1.5 page 5: how to represent equivalence relations, hence later we can deal with a family with one isomorphism type
2.2 page 7: finish the analysis for interpreting cases of monadic quantifier
3.5 page 15: finish the analysis of interpreting one to one functions, concerning uniformity see the remark at the bottom of page 15
4.5 page 16: definition of \lambda_2 (R)
4.17 page 23: reduction on a relation of cardinality \le \lambda_2 (R)
Def 5.1 page 26: defining \lambda_3
Definition 5.11A(?) defining K^{word} page 3(?)
Fact 5.2 page 26(?) \lambda_2 \in \{\lambda_3,\;lambda_2^+\}
5.11,5.12, 5.14 pages 31,32,33: analyses the case \lambda_2\neq\lambda_3,
rest on the case of equality page 35 line -16; defining \chi page 35 line -2

6.11 page 40: finishing this case and from now on \chi\ge cardinality(R) when U has cardinality \aleph_0
Explanation of the from of the paper:
The original aim of this article was to prove that for every K (a family of relations on U on a fixed arity) its quantifier is equivalent to one for KU, a family of equivalence relations (all such classes are assumed to be closed under isomorphism). Sections 1-6 were written for this and realize it to large extent. Clearly it suffice to deal with I with one isomorphism type as long as the interpretations are uniform. But two essential difficulties arise (1) the quantifier \exists^{word}_{\lambda} (a well ordering of length \lambda), provably is not biinterpretable with \exists_K for any family K of equivalence classes. This is not so serious: just add another case. (2) It is consistent that there are cardinals \chi,\lambda satisfying \chi\le\lambda \le 2^\chi such that R is e.g. a family of \chi sunsets of \lambda, again we cannot reduce this to equivalence relations in general (see section 8)
Only under the assumptions V=L and considering more liberal notion of bi-interpretability (\equiv_{exp} rather than \equiv_{int} the desired result is gotten. However when the gap degenerates for any reason we get the original hope (note \exists^{word}_\omega is second order quantification).

We prove (in ZFC) the existence of such B of density character \lambda and cardinality \lambda^{\aleph_0} whenever \lambda>\aleph_0. We can conclude answers to some questions from Monk’s list. We use a combinatorial method from [Sh:45],[Sh:172], that is represented in Section 1.

For a model, being resplendent is a strengthening of being \kappa-saturated. Restricting ourselves to the case \kappa > |T| for transparency, to say a model M is \kappa-resplendent means:

when we expand M by < \kappa individual constants \langle c_i : i < \alpha \rangle, if (M, c_i)_{ < \alpha } has an elementary extension expandable to be a model of T' where {\rm Th}((M, c_i)_{i < \alpha} ) \subseteq T', |T'| < \kappa then already (M, c_i)_{i < \alpha} can be expanded to a model of T' .

Trivially, any saturated model of cardinality \lambda is \lambda-resplendent. We ask: how many \kappa-resplendent models of a (first order complete) theory T of cardinality \lambda are there? We restrict ourselves to cardinals \lambda = \lambda^\kappa + 2^{|T|} and ignore the case \lambda = \lambda^{<\kappa} + |T| < \lambda^\kappa. Then we get a complete and satisfying answer: this depends only on T being stable or unstable. In this case proving that for stable T we get few, is not hard; in fact, every resplendent model of T is saturated hence it is determined by its cardinality up to isomorphism. The inverse is more problematic because naturally we have to use Skolem functions with any \alpha < \kappa places. Normally we use relevant partition theorems (Ramsey theorem or Erdős-Rado theorem), but in our case the relevant partitions theorems fail so we have to be careful.

2. copy abstract- how was it a5Xiv-ed without abstract

In the third section we show stronger results concerning linear orders. If for each linear order I of cardinality \lambda > \aleph_0 we can attach a model M_I \in K_\lambda in which the linear order can be embedded such that for enough cuts of I, their being omitted is reflected in M_I, then there are 2^\lambda non-isomorphic cases. We also do the work for some applications.