614,137 research outputs found
On properties of theories which preclude the existence of universal models
We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality ? when certain cardinal arithmetic assumptions about ? implying the failure of GCH (and close to the failure of SCH) hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to non-universality, as was raised by the earlier work of Shelah. One of our examples is the theory View the MathML source for which non-universality results similar to the ones we obtain are already known; hence we may view our results as an abstraction of the known results from a concrete theory to a class of theories. We show that no theory with the oak property is simple
Quotients of Strongly Proper Forcings and Guessing Models
We prove that a wide class of strongly proper forcing posets have quotients
with strong properties. Specifically, we prove that quotients of forcing posets
which have simple universal strongly generic conditions on a stationary set of
models by certain nice regular suborders satisfy the -approximation
property. We prove that the existence of stationarily many -guessing
models in , for sufficiently large cardinals ,
is consistent with the continuum being arbitrarily large, solving a problem of
Viale and Weiss
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