410 research outputs found
Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few
results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her
P.h.d. thesis. We also observe that if is a model of ZFC, then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -distributive and is -complete.
Further we observe that if is a model of ZF + , then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -strategically closed and is
-complete.Comment: Revised versio
Compactness of powers of \omega
We characterize exactly the compactness properties of the product of \kappa\
copies of the space \omega\ with the discrete topology. The characterization
involves uniform ultrafilters, infinitary languages, and the existence of
nonstandard elements in elementary estensions. We also have results involving
products of possibly uncountable regular cardinals.Comment: v2 slightly improve
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