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We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing., Comment: 24 pages. Commentary concerning this article can be made at http://jdh.hamkins.org/strongly-uplifting-cardinals-and-boldface-resurrection

Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_\delta$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_\delta$ are uniformly definable in their set-forcing extensions by posets admitting a gap at $\delta$, using a ground model parameter. In particular, this means that models of ${\rm ZF}+{\rm DC}_\delta$ are uniformly definable in their forcing extensions by posets of size less than $\delta$. We also show that it is consistent for ground model definability to fail for models of ${\rm ZFC}^-$ of the form $H_{\kappa^+}$. Using forcing, we produce a ${\rm ZFC}$ universe in which there is a cardinal $\kappa>\!>\omega$ such that $H_{\kappa^+}$ is not definable in its Cohen forcing extension. As a corollary, we show that there is always a countable transitive model of ${\rm ZFC}^-$ violating ground model definability. These results turn out to have a bearing on ground model definability for models of ${\rm ZFC}$. It follows from our proof methods that the hereditary size of the parameter that Woodin used to define a ${\rm ZFC}$ model in its set-forcing extension is best possible., Comment: 17 pages. Discussion and commentary concerning this article can be made at http://boolesrings.org/victoriagitman/2013/06/25/on-ground-model-definability/

We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of an uplifting cardinal., Comment: 25 pages. To be published in the Archive for Mathematical Logic. Commentary concerning this article can be made at http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC- in which $\omega_1$ is singular, in which every set of reals is countable, yet $\omega_1$ exists, in which there are sets of reals of every size $\aleph_n$, but none of size $\aleph_\omega$, and therefore, in which the collection axiom sceme fails; there are models of ZFC- for which the Los theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC- for which the Gaifman theorem fails, in that there is an embedding $j:M\to N$ of ZFC- models that is $\Sigma_1$-elementary and cofinal, but not elementary; there are elementary embeddings $j:M\to N$ of ZFC- models whose cofinal restriction $j:M\to \bigcup j``M$ is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC- to a $\Sigma_1$-formula or a $\Pi_1$-formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory $ZFC^-$, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach., Comment: 22 pages; commentary concerning this article can be made at http://jdh.hamkins.org/what-is-the-theory-zfc-without-power-set

This case study of a school-based parent center in a low-income Latino community, based on action research, investigated ways in which low-income parents became involved with their child's school, and identified barriers and supports to the implementation of various parent involvement activities. Data were collected through participant observation, document analysis, parent surveys, and parent and staff interviews. Findings indicated that parent surveys provided an important needs assessment tool that was useful in developing programs. The school's "open-door" policy was identified by staff as the strongest school practice linking the school and families. Both parents and staff identified the principal as the key to establishing parent involvement as a school priority. There was a high level of enthusiasm, support, and energy generated for parent participation during the first year, but father participation in parent center programs was very limited. Collaboration with community agencies provided a high quality parent center, and a great deal of effort was expended in developing collaborative partnerships. A wide range of parent services and programs were offered; parents perceived this diversity as important. English as a Second Language classes were parents' number one priority and were consistently well attended. Over the 4 years of the study, program priorities shifted from families' basic needs toward increased involvement in learning activities at home, decision-making, governance, and advocacy. (Appendices include sample excerpts from the principal log and the parent survey. Contains 67 references.) (KDFB)