DEPENDENT CHOICE, PROPERNESS, AND GENERIC ABSOLUTENESS
| Autor: | David Asperó, Asaf Karagila |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Classical mathematics
Forcing (recursion theory) Logic Absoluteness 010102 general mathematics Mathematics - Logic 0102 computer and information sciences Base (topology) 01 natural sciences Primary 03E25 Secondary 03E55 03E35 03E57 03A05 Mathematics::Logic Philosophy Mathematics (miscellaneous) 010201 computation theory & mathematics FOS: Mathematics Independence (mathematical logic) Proper forcing axiom Axiom of choice Set theory 0101 mathematics Logic (math.LO) Mathematical economics Mathematics |
| Zdroj: | The Review of Symbolic Logic. 14:225-249 |
| ISSN: | 1755-0211 1755-0203 |
| DOI: | 10.1017/s1755020320000143 |
| Popis: | We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to DC-preserving symmetric submodels of forcing extensions. Hence, ZF+DC not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in ZF, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in ZF+DC and ZFC. Our results confirm ZF+DC as a natural foundation for a significant portion of "classical mathematics" and provide support to the idea of this theory being also a natural foundation for a large part of set theory. Comment: 23 pages; final version |
| Databáze: | OpenAIRE |
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