SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard
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16 MENACHEM MAGIDORcan be a good guidance for sharpening our concepts and getting a betterand better axiom systems for Set Theory. In wh<strong>at</strong> sense the imaginedset is specified? As first approxim<strong>at</strong>ion let us say th<strong>at</strong> a set is simplyspecified as any s<strong>at</strong>isfying a certain property. So we should rephraseour slogan asIf a set s<strong>at</strong>isfying a given property is possible and thereis no clear obstruction to the existence of such set thensuch set exists.Wh<strong>at</strong> do we mean by ”possible”? I think th<strong>at</strong> a good approxim<strong>at</strong>ionis ”can be forced to exists” So let us tryIf one can force the existence of a set s<strong>at</strong>isfying a givenproperty and there is no clear obstruction to its existencethen such a set exists.Still this principle is problem<strong>at</strong>ic. One can introduce by forcing a setwhich is an enumer<strong>at</strong>ion of all the reals of order type ω 1 but also onecan introduce by forcing a list of ω 2 different reals. Of course it isinconsistent to have sets s<strong>at</strong>isfying both properties. The st<strong>at</strong>ement iseven more problem<strong>at</strong>ic if in the property one allow parameters say agiven set A. Because suppose th<strong>at</strong> our parameter A is uncountable .By using Levy’s collapse one can make A countable, so introduce a 1-1mapping between A and ω. Obviously such a mapping does not existsin the our universe.I consider forcing axioms as an <strong>at</strong>tempt to try and get a consistentapproxim<strong>at</strong>ion to the above intuitive principle by restricting theproperties we talk about and the the forcing extensions we use. Therestriction of the forcing notions is usually following the intuition of allowingonly forcing notions th<strong>at</strong> do not make a very dram<strong>at</strong>ic change inthe universe , like making an uncountable set countable.This is somewh<strong>at</strong>similar to restricting in the interpret<strong>at</strong>ions of the modalities ”itis possible th<strong>at</strong>...” the set of possible universes to universes which arenot too different from the current universe.So we restrict the forcingwhich we consider to ”mild” forcing extensions.When we extend the universe by using forcing, wh<strong>at</strong> we add to theuniverse is the generic object with respect to the forcing notion. Sinceany other set introduced by the forcing notion is defined (over theground model) from the generic filter, our typical forcing axiom is ofthe form:For a given class of ”mild” forcing notions P and forevery forcing notion P ∈ P there is a rich family offilters in P which are generic enough