The DDG and Very Large Cardinals Daniel Marini Universit` a degli Studi di Torino 26/03/2019
Set-Theoretic Geology It concerns a switch in perspective of the forcing method. One asks himself if the universe 푉 might have arisen by generic extension 푊[ 퐺 ] for some class 푊 and some 푊 -generic filter 퐺 ⊆ 퐏 ∈ 푊 . We shall assume some knowledge of set theory and basic notions of forcing. Some notations and preliminary results We will work in ZFC set theory, consisting of the axioms of Extensionality (Ext), Foundation (Fnd), Pairing (Prn), Union (Unn), Power set (Pwr), Infinity (Inf), Separation (Spr), Collection (Clt), and Choice (AC). Sometimes, we need to refer to some weakened form of this theory. Mostly, ZFC − Pwr , ZFC 훿 , and ZF . 1 Universit ` a degli Studi di Torino Daniel Marini
A transitive class 푊 is an inner model of ZF Theorem if and only if it contains all ordinals, is almost universal, and closed under the G¨ odel operations. In particular, being an inner model of ZFC is first-order expressible. Let ̇ 퐐 be a 퐏 -name for a forcing in 푉 . Let Theorem 퐺 ⊆ 퐏 ∈ 푉 be 푉 -generic and 퐻 ⊆ val( ̇ 퐐, 퐺 ) be 푉 [ 퐺 ] - generic. There exists a filter 퐺 ∗ 퐻 contained in the forcing 퐏 ∗ ̇ 퐐 ∈ 푉 such that 푉 [ 퐺 ][ 퐻 ] = 푉 [ 퐺 ∗ 퐻 ] . 2 Universit ` a degli Studi di Torino Daniel Marini
Sometimes it is useful to think about forcing with complete Boolean algebras. In such cases, we consider the Boolean completion of our forcing. Theorem Let 퐺 ⊆ 퐁 be generic. If 푊 is an inner model of ZFC and 푉 ⊆ 푊 ⊆ 푉 [ 퐺 ] , then there is a complete sub-algebra 퐃 of 퐁 such that 푊 = 푉 [ 퐃 ∩ 퐺 ] . Moreover, 푉 [ 퐺 ] is a generic extension of 푊 , since 퐁 is forcing equivalent to the iteration 퐃 ∗ ̌ 퐁 ∕ ̇ 퐺 0 , where 퐺 0 = 퐃 ∩ 퐺 and 퐁 ∕ ̇ 퐺 0 = { ̌ 푝 ∈ ̇ ̌ 푏 ∈ ̌ 퐵 ∶ ∀ ̌ 퐺 0 ( ̌ 횤 ( ̌ 푝 ) ∥ ̌ 푞 )} . 3 Universit ` a degli Studi di Torino Daniel Marini
Definition Let 푊 ⊆ 푉 be a transitive model of ZFC and 훿 ∈ 푉 be a cardinal. We say that 푊 exhibits the • 훿 - cover property for 푉 if for every 퐴 ∈ 푉 with 퐴 ⊆ 푊 and |퐴| 푉 < 훿 , there is 퐵 ∈ 푊 such that 퐴 ⊆ 퐵 and | 퐵 | 푊 < 훿 ; • 훿 - approximation property for 푉 if for all 퐴 ∈ 푉 such that – 퐴 ⊆ 푊 ; – for all 퐵 ∈ 푊 with | 퐵 | 푊 < 훿 , 퐴 ∩ 퐵 ∈ 푊 ; then 퐴 ∈ 푊 . Lemma Let 푊[ 퐺 ] ⊇ 푊 be a generic extension by a non-trivial forcing notion 퐏 ∈ 푊 with 퐺 a ⟨푊, 퐏⟩ -generic filter. If 훿 = |퐏| + , then 푊 satisfies the 훿 -approximation and 훿 -cover properties for 푊 [ 퐺 ] . 4 Universit ` a degli Studi di Torino Daniel Marini
Definability of Grounds A ground 푊 of 푉 is an inner model of ZFC such that 푉 can be obtained by set-forcing extension over 푊 . We will also write GRD( 푊, 푉 ) . Theorem [The Ground-Model Definability Theorem] There exists a specific first-order formula Φ GRD ( 푦, 푥 ) such that 푉 = 푊 [ 퐺 ] is a generic extension of a ground 푊 by ⟨푊, 퐏⟩ -generic filter 퐺 with 퐏 ∈ 푊 if and only if there is 푟 ∈ 푊 such that 푊 = { 푥 ∈ 푉 ∶ Φ 푉 GRD ( 푟, 푥 )} . We will write Φ GRD ( 푟, 푉 ) in place of 푊 . 5 Universit ` a degli Studi di Torino Daniel Marini
The formula Φ GRD Let Φ ′ GRD ( 훿, 푧, 퐏, 퐺 ) be the formula: (I) 훿 is a regular cardinal; (II) 퐏 ∈ 푧 is a forcing of size less than 훿 ; (III) 퐺 is a ⟨푧, 퐏⟩ -generic; (IV) for every ℶ -fixed point 훾 > 훿 of cofinality greater than 훿 , there exists a transitive structure 푀 of height 훾 such that (i) 푀 is a model of ZFC 훿 ; (ii) 푧 = (2 <훿 ) 푀 ; (iii) 푀[ 퐺 ] = 푉 훾 ; (iv) 푀 훿 -cover 훿 -approximation has the and properties for 푉 . Φ GRD ( 푟, 푥 ) is ∃ 훾 ( Φ ′ ) , where 훾 and GRD ( 훿, 푟, 퐏, 퐺 ) ∧ 푥 ∈ 푊 훾 푊 훾 are as in (IV), 훿 = ( |퐏| + ) 푉 , and 푟 = (2 <훿 ) 푊 . 6 Universit ` a degli Studi di Torino Daniel Marini
Hence, we are able to define the class of all grounds of 푉 as {Φ GRD ( 푟, 푉 ) ∶ 푟 ∈ 푉 } . How are grounds seen in different universes? 7 Universit ` a degli Studi di Torino Daniel Marini
Hence, we are able to define the class of all grounds of 푉 as {Φ GRD ( 푟, 푉 ) ∶ 푟 ∈ 푉 } . How are grounds seen in different universes? If 푈 is an inner model of ZFC satisfying Proposition Φ GRD ( 푟, 푉 ) ⊆ 푈 ⊆ 푉 , then Φ GRD ( 푟, 푈 ) = Φ GRD ( 푟, 푉 ) . On the other hand, for every generic extension 푉 [ 퐺 ] and every 푟 ∈ 푉 there exists 푠 ∈ 푉 such that Φ GRD ( 푟, 푉 ) = Φ GRD ( 푠, 푉 ) = Φ GRD ( 푠, 푉 [ 퐺 ]) . 7 Universit ` a degli Studi di Torino Daniel Marini
Hence, we are able to define the class of all grounds of 푉 as {Φ GRD ( 푟, 푉 ) ∶ 푟 ∈ 푉 } . How are grounds seen in different universes? If 푈 is an inner model of ZFC satisfying Proposition Φ GRD ( 푟, 푉 ) ⊆ 푈 ⊆ 푉 , then Φ GRD ( 푟, 푈 ) = Φ GRD ( 푟, 푉 ) . On the other hand, for every generic extension 푉 [ 퐺 ] and every 푟 ∈ 푉 there exists 푠 ∈ 푉 such that Φ GRD ( 푟, 푉 ) = Φ GRD ( 푠, 푉 ) = Φ GRD ( 푠, 푉 [ 퐺 ]) . Proof: The parameter 푟 in Φ GRD ( 푟, 푉 ) can be 2 <훿 relativized in the ground itself, with any regular cardinal 훿 ≥ ( |퐏| + ) 푉 . Let us assume 훿 > ( | 퐑퐎 ( 퐏 ) | ) 푉 . The class 푈 is a generic extension by a complete subalgebra of 퐑퐎 ( 퐏 ) , so the same parameter 푟 suffices. 7 Universit ` a degli Studi di Torino Daniel Marini
Hence, we are able to define the class of all grounds of 푉 as {Φ GRD ( 푟, 푉 ) ∶ 푟 ∈ 푉 } . How are grounds seen in different universes? If 푈 is an inner model of ZFC satisfying Proposition Φ GRD ( 푟, 푉 ) ⊆ 푈 ⊆ 푉 , then Φ GRD ( 푟, 푈 ) = Φ GRD ( 푟, 푉 ) . On the other hand, for every generic extension 푉 [ 퐺 ] and every 푟 ∈ 푉 there exists 푠 ∈ 푉 such that Φ GRD ( 푟, 푉 ) = Φ GRD ( 푠, 푉 ) = Φ GRD ( 푠, 푉 [ 퐺 ]) . Proof: For the remaining implication, Φ GRD ( 푟, 푉 ) is a ground of 푉 [ 퐺 ] , hence there is a parameter 푠 ∈ 푉 [ 퐺 ] which represents it. We can conclude applying the previous result to Φ GRD ( 푠, 푉 [ 퐺 ]) ⊆ 푉 ⊆ 푉 [ 퐺 ] . □ 7 Universit ` a degli Studi di Torino Daniel Marini
The DDG Definition • The Downward Directed Ground hypothesis , DDG , is the formula: ∀ 푟, 푠 ∈ 푉 ∃ 푡 ∈ 푉 ( Φ GRD ( 푡, 푉 ) ⊆ Φ GRD ( 푟, 푉 ) ∩ Φ GRD ( 푠, 푉 ) ). • The strong DDG is the formula: ⋂ ∀ 푋 ∈ 푉 ∃ 푡 ∈ 푉 ( Φ GRD ( 푡, 푉 ) ⊆ Φ GRD ( 푟, 푉 ) ). 푟 ∈ 푋 Definition The mantle ℳ 푊 of a model of set theory 푊 is the intersection of all of its grounds. Formally, ℳ 푊 = ⋂ Φ GRD ( 푟, 푊 ) . 푟 ∈ 푊 We will simply write ℳ instead of ℳ 푉 . 8 Universit ` a degli Studi di Torino Daniel Marini
Remark: The mantle is a parameter-free first-order definable class that is transitive and contains all ordinals. Consequences of the DDG Theorem The following holds: (i) If DDG holds, then for any ground 푊 of 푉 we have ℳ 푊 = ℳ 푉 . (ii) If for every ground 푊 of 푉 one has ℳ 푊 = ℳ 푉 , then the mantle is an inner model of ZF . (iii) Furthermore, if the strong DDG holds, then the mantle is an inner model of ZFC . 9 Universit ` a degli Studi di Torino Daniel Marini
Theorem The following holds: (i) If DDG holds, then for any ground 푊 of 푉 we have ℳ 푊 = ℳ 푉 . Proof: Any ground of 푊 is a ground of 푉 , hence ℳ 푉 ⊆ ℳ 푊 . Vice versa, if 푎 ∉ ℳ 푉 , there is a ground 푊 ′ such that 푎 ∉ 푊 ′ and so 푎 ∉ 푊 ∩ 푊 ′ . By directedness, there exists a ground 푊 ⊆ 푊 ∩ 푊 ′ , which is a ground of 푊 . As before, this means that ℳ 푊 ⊆ ℳ 푊 . But 푎 ∉ 푊 implies 푎 ∉ ℳ 푊 , as requested. 9 Universit ` a degli Studi di Torino Daniel Marini
Theorem The following holds: (ii) If for every ground 푊 of 푉 one has ℳ 푊 = ℳ 푉 , then the mantle is an inner model of ZF . Proof: Since every ground is a transitive class which contains all ordinals and is closed under the G¨ odel operations, then by definition so is ℳ (remember that G¨ odel operations are absolute for transitive models). It suffices to show that ℳ is almost universal. Let 퐴 ⊆ ℳ and 훼 = rank( 퐴 ) . In 푉 , for any ground 푊 we have 푉 훼 ∩ ℳ = 푉 훼 ∩ ( 푊 ∩ ℳ ) = ( 푉 훼 ) 푊 ∩ ℳ 푊 ∈ 푊. By the arbitrary choice of 푊 , we deduce that 퐴 ⊆ 푉 훼 ∩ ℳ ∈ 푊 , that is, ℳ is almost universal. 9 Universit ` a degli Studi di Torino Daniel Marini
Recommend
More recommend
