A microscopic approach to Souslin trees constructions Forcing and - PowerPoint PPT Presentation

Special and specializable λ + -trees Definition A λ + -tree is special if it is the union of λ many antichains. Definition A λ + -tree is specializable if it is special in some extended universe of ZFC with the same cardinal structure. Theorem (Baumgartner-Mailtz-Reinhardt, 1970) An ℵ 1 -tree is Aronszajn iff it is specializable. 8 / 32

Special and specializable λ + -trees Definition A λ + -tree is special if it is the union of λ many antichains. Definition A λ + -tree is specializable if it is special in some extended universe of ZFC with the same cardinal structure. Theorem (Baumgartner-Mailtz-Reinhardt, 1970) An ℵ 1 -tree is Aronszajn iff it is specializable. Theorem (implicit in David, 1990) If V = L, then for every regular λ , the canonical λ -complete λ + -Souslin tree constructed using fine structure, is specializable. 8 / 32

Non-specializable λ + -Souslin trees Theorem (Baumgartner, 1970’s, building on Laver) � ℵ 1 entails a non-specializable ℵ 2 -Aronszajn tree. 9 / 32

Non-specializable λ + -Souslin trees Theorem (Baumgartner, 1980’s, improving Devlin) GCH + � ℵ 1 entails a non-specializable ℵ 2 -Souslin tree. 9 / 32

Non-specializable λ + -Souslin trees Theorem (Baumgartner, 1980’s, improving Devlin) GCH + � ℵ 1 entails a non-specializable ℵ 2 -Souslin tree. Theorem (Cummings, 1997) ℵ 1 ≤ λ <λ = λ + ♦ λ entails a non-specializable λ -complete λ + -Souslin tree. 9 / 32

Non-specializable λ + -Souslin trees Theorem (Baumgartner, 1980’s, improving Devlin) GCH + � ℵ 1 entails a non-specializable ℵ 2 -Souslin tree. Theorem (Cummings, 1997) ℵ 1 ≤ λ <λ = λ + ♦ λ entails a non-specializable λ -complete λ + -Souslin tree. Theorem (Cummings, 1997) If λ is a singular cardinal of countable cofinality, � λ + CH λ and µ ℵ 1 < λ for all µ < λ , then there exists a non-specializable λ + -Souslin tree. 9 / 32

Non-specializable λ + -Souslin trees Theorem (Baumgartner, 1980’s, improving Devlin) GCH + � ℵ 1 entails a non-specializable ℵ 2 -Souslin tree. Theorem (Cummings, 1997) ℵ 1 ≤ λ <λ = λ + ♦ λ entails a non-specializable λ -complete λ + -Souslin tree. Theorem (Cummings, 1997) If λ is a singular cardinal of countable cofinality, � λ + CH λ and µ ℵ 1 < λ for all µ < λ , then there exists a non-specializable λ + -Souslin tree. Theorem (Cummings, 1997) If λ is a singular cardinal of uncountable cofinality, � λ + CH λ and µ ℵ 0 < λ for all µ < λ , then there exists a non-specializable λ + -Souslin tree. 9 / 32

To sum up The construction of λ + -Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular. 10 / 32

To sum up The construction of λ + -Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular. Some of them also depend on whether or not λ is of countable cofinality. 10 / 32

To sum up The construction of λ + -Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular. Some of them also depend on whether or not λ is of countable cofinality. Question Do one really have to come up with such a long list of variations each time that a fundamental construction is discovered? 10 / 32

To sum up The construction of λ + -Souslin trees often makes an explicit distinction between the case that λ is a regular cardinal and the case that λ is singular. Some of them also depend on whether or not λ is of countable cofinality. Question Do one really have to come up with such a long list of variations each time that a fundamental construction is discovered? Isn’t there any automatic translation between the different cardinals? 10 / 32

An idea Find a proxy! 1. Introduce a family of combinatorial principles from which the constructions can be carried out uniformly; 2. Prove that this operational principle is a consequence of the “usual” hypotheses. 11 / 32

An idea Find a proxy! 1. Introduce a family of combinatorial principles from which the constructions can be carried out uniformly; 2. Prove that this operational principle is a consequence of the “usual” hypotheses. This part is done only once, and then will be utilized each time that a new construction is discovered. 11 / 32

The proxy principle Goal The proxy principle will allow to translate constructions from one cardinal to another, to calibrate the hypotheses needed to carry a construction, and will capture all known ♦ -based constructions. 12 / 32

The proxy principle Goal The proxy principle will allow to translate constructions from one cardinal to another, to calibrate the hypotheses needed to carry a construction, and will capture all known ♦ -based constructions. Definition P ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts that ♦ ( κ ) holds, and so is the corresponding P − ( κ, µ, R , θ, S , ν, σ, ̟ ). 12 / 32

The proxy principle Definition P ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts that ♦ ( κ ) holds, and so is the corresponding P − ( κ, µ, R , θ, S , ν, σ, ̟ ). 12 / 32

The proxy principle Definition P ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts that ♦ ( κ ) holds, and so is the corresponding P − ( κ, µ, R , θ, S , ν, σ, ̟ ). Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ R is a binary relation over P ( κ ); 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ R is a binary relation over P ( κ ); 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ R is a binary relation over P ( κ ); ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; 12 / 32

The proxy principle Example of a binary relation R ⊑ , where D ⊑ C iff ∃ β such that D = C ∩ β . Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; 12 / 32

The proxy principle Example of a binary relation R ⊑ χ , where D ⊑ χ C iff D ⊑ C or otp( C ) < χ . Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; 12 / 32

The proxy principle Example of a binary relation R ⊑ ∗ , where D ⊑ ∗ C iff ∃ α < sup( D ) with D \ α ⊑ C \ α . Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; 12 / 32

The proxy principle Example of a binary relation R χ ⊑ ∗ , where D χ ⊑ ∗ C iff cf(sup( D )) < χ or D ⊑ ∗ C . Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every sequence � A i | i < θ � of cofinal subsets of κ 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every sequence � A i | i < θ � of cofinal subsets of κ , and every S ∈ S , 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every sequence � A i | i < θ � of cofinal subsets of κ , and every S ∈ S , there exists δ ∈ S 12 / 32

The proxy principle Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every sequence � A i | i < θ � of cofinal subsets of κ , and every S ∈ S , there exists δ ∈ S with |C δ | < ν such that: 12 / 32

The proxy principle Recall succ σ ( C ) = { α ∈ C | otp( C ∩ α ) = j + 1 for some j < σ } . Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every sequence � A i | i < θ � of cofinal subsets of κ , and every S ∈ S , there exists δ ∈ S with |C δ | < ν such that: ◮ ∀ i < min { δ, θ }∀ C ∈ C δ sup { β ∈ C | succ σ ( C \ β ) ⊆ A i } = δ ; 12 / 32

The proxy principle Recall succ ̟ ( C ) = { α ∈ C | otp( C ∩ α ) = j + 1 for some j < ̟ } . Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subsets of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every sequence � A i | i < θ � of cofinal subsets of κ , and every S ∈ S , there exists δ ∈ S with |C δ | < ν such that: ◮ ∀ i < min { δ, θ }∀ C ∈ C δ sup { β ∈ C | succ σ ( C \ β ) ⊆ A i } = δ ; ◮ ∀ i < min { δ, θ } sup � C ∈C δ { β ∈ C | succ ̟ ( C \ β ) ⊆ A i } = δ , unless ̟ = 0. 12 / 32

Default values Don’t worry, we have some default values! Whenever omitted, let θ = 1 , S = { κ } , ν = 2 , σ = 1 , ̟ = 0. 13 / 32

Default values Don’t worry, we have some default values! Whenever omitted, let θ = 1 , S = { κ } , ν = 2 , σ = 1 , ̟ = 0. Definition P − ( κ, µ, R , θ, S , ν, σ, ̟ ) asserts the existence of �C δ | δ < κ � s.t.: ◮ for every limit δ < κ , C δ is a collection of club subset of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every sequence � A i | i < θ � of cofinal subsets of κ , and every S ∈ S , there exists δ ∈ S with |C δ | < ν such that: ◮ ∀ i < min { δ, θ }∀ C ∈ C δ sup { β ∈ C | succ σ ( C \ β ) ⊆ A i } = δ ; ◮ ∀ i < min { δ, θ } sup � C ∈C δ { β ∈ C | succ ̟ ( C \ β ) ⊆ A i } = δ , unless ̟ = 0. 13 / 32

Default values Don’t worry, we have some default values! Whenever omitted, let θ = 1 , S = { κ } , ν = 2 , σ = 1 , ̟ = 0. Special case P − ( κ, µ, R ) asserts the existence of �C δ | δ < κ � such that: ◮ for every limit δ < κ , C δ is a collection of club subset of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; 13 / 32

Default values Don’t worry, we have some default values! Whenever omitted, let θ = 1 , S = { κ } , ν = 2 , σ = 1 , ̟ = 0. Special case P − ( κ, µ, R ) asserts the existence of �C δ | δ < κ � such that: ◮ for every limit δ < κ , C δ is a collection of club subset of δ ; ◮ 0 < |C δ | < µ for all δ < κ ; ◮ if C ∈ C δ and β ∈ acc( C ), then ∃ D ∈ C β with ( D , C ) ∈ R ; ◮ for every cofinal A ⊆ κ , there exists δ < κ with C δ = { C δ } , such that sup(nacc( C δ ) ∩ A ) = δ . 13 / 32

A Souslin tree from the weakest principle Let κ denote a regular uncountable cardinal. Proposition P ( κ, κ, ⊑ ∗ , 1 , { κ } , κ ) entails a κ -Souslin tree. 14 / 32

A Souslin tree from the weakest principle Let κ denote a regular uncountable cardinal. Proposition P ( κ, κ, ⊑ ∗ , 1 , { κ } , κ ) entails a κ -Souslin tree. Proposition P ( κ, κ, ⊑ ∗ , 1 , { E κ ≥ χ } , κ ) entails a χ -complete κ -Souslin tree, provided | α | <χ < κ for all α < κ . 14 / 32

A Souslin tree from the weakest principle Let κ denote a regular uncountable cardinal. Proposition P ( κ, κ, ⊑ ∗ , 1 , { κ } , κ ) entails a κ -Souslin tree. Proposition P ( κ, κ, χ ⊑ ∗ , 1 , { E κ ≥ χ } , κ ) entails a χ -complete κ -Souslin tree, provided | α | <χ < κ for all α < κ . 14 / 32

Sanity check #1 Let λ denote an uncountable cardinal. Theorem (Jensen, 1972) If λ <λ = λ and ♦ ( E λ + λ ) holds, then there exists a λ -complete λ + -Souslin tree. 15 / 32

Sanity check #1 Let λ denote an uncountable cardinal. Theorem (Jensen, 1972) If λ <λ = λ and ♦ ( E λ + λ ) holds, then there exists a λ -complete λ + -Souslin tree. Theorem ♦ ( E λ + λ ) entails P ( λ + , 2 , λ ⊑ , { E λ + λ } ) . 15 / 32

Sanity check #1 Let λ denote an uncountable cardinal. Theorem (Jensen, 1972) If λ <λ = λ and ♦ ( E λ + λ ) holds, then there exists a λ -complete λ + -Souslin tree. Theorem ♦ ( E λ + λ ) entails P ( λ + , 2 , λ ⊑ , { E λ + λ } ) . Corollary If λ <λ = λ and ♦ ( E λ + λ ) holds, then there exists a λ -complete λ + -Souslin tree. 15 / 32

Sanity check #2 Let λ denote an uncountable cardinal. Theorem (Jensen, 1972) If there exists S ⊆ λ + for which � λ ( S ) + ♦ ( S ) holds, then there exists a λ + -Souslin tree. 16 / 32

Sanity check #2 Let λ denote an uncountable cardinal. Theorem (Jensen, 1972) If there exists S ⊆ λ + for which � λ ( S ) + ♦ ( S ) holds, then there exists a λ + -Souslin tree. Theorem � λ + CH λ entails P ( λ + , 2 , ⊑ , { E λ + ≥ θ | θ < λ } ) . 16 / 32

Sanity check #2 Let λ denote an uncountable cardinal. Theorem (Jensen, 1972) If there exists S ⊆ λ + for which � λ ( S ) + ♦ ( S ) holds, then there exists a λ + -Souslin tree. Theorem � λ + CH λ entails P ( λ + , 2 , ⊑ , { E λ + ≥ θ | θ < λ } ) . Corollary If � λ + CH λ holds, then for every χ < λ with λ <χ = λ , there exists a χ -complete λ + -Souslin tree. 16 / 32

Sanity check #3 Let λ denote an uncountable cardinal. Theorem (Gregory, 1976) If λ <λ = λ, 2 λ = λ + and exists a nonreflecting stationary subset of E λ + <λ , then there exists a λ + -Souslin tree. 17 / 32

Sanity check #3 Let λ denote an uncountable cardinal. Theorem (Gregory, 1976) If λ <λ = λ, 2 λ = λ + and exists a nonreflecting stationary subset of E λ + <λ , then there exists a λ + -Souslin tree. Theorem If 2 λ = λ + and there exists a nonreflecting stationary subset of E λ + <λ , then P ( λ + , 2 , λ ⊑ ∗ , { E λ + λ } ) holds. 17 / 32

Sanity check #3 Let λ denote an uncountable cardinal. Theorem (Gregory, 1976) If λ <λ = λ, 2 λ = λ + and exists a nonreflecting stationary subset of E λ + <λ , then there exists a λ + -Souslin tree. Theorem If 2 λ = λ + and there exists a nonreflecting stationary subset of E λ + <λ , then P ( λ + , 2 , λ ⊑ ∗ , { E λ + λ } ) holds. Corollary (Kojman-Shelah, 1993) If λ <λ = λ, 2 λ = λ + and there exists a nonreflecting stationary subset of E λ + <λ , then there exists a λ -complete λ + -Souslin tree. 17 / 32

Sanity check #4 Let λ denote an uncountable cardinal. Theorem (Shelah, 1984) If 2 ℵ 0 = ℵ 1 , NS ℵ 1 is saturated, then there exists an ℵ 2 -Souslin tree. 18 / 32

Sanity check #4 Let λ denote an uncountable cardinal. Theorem (Shelah, 1984) If 2 ℵ 0 = ℵ 1 , NS ℵ 1 is saturated, then there exists an ℵ 2 -Souslin tree. Theorem If 2 ℵ 1 = ℵ 2 , NS ℵ 1 is saturated, then P ( ℵ 2 , 2 , ℵ 1 ⊑ ∗ , { E ℵ 2 ℵ 1 } ) holds. 18 / 32

Sanity check #4 Let λ denote an uncountable cardinal. Theorem (Shelah, 1984) If 2 ℵ 0 = ℵ 1 , NS ℵ 1 is saturated, then there exists an ℵ 2 -Souslin tree. Theorem If 2 ℵ 1 = ℵ 2 , NS ℵ 1 is saturated, then P ( ℵ 2 , 2 , ℵ 1 ⊑ ∗ , { E ℵ 2 ℵ 1 } ) holds. Corollary If 2 ℵ 0 = ℵ 1 , NS ℵ 1 is saturated, then there exists an ℵ 1 -complete ℵ 2 -Souslin tree. 18 / 32

And so on.. 19 / 32

And so on.. Okay, so you seem to found a way to redirect all ♦ -based constructions of Souslin trees through a single construction. You haven’t yet shown me anything new! 19 / 32

κ -trees which cohere modulo finite Definition A subtree T of <κ κ is said to be coherent if for all δ < κ : ◮ if x , y ∈ T δ , then { α < δ | x ( α ) � = y ( α ) } is finite; ◮ if x , y ∈ δ κ , and { α < δ | x ( α ) � = y ( α ) } is finite, then x ∈ T δ iff y ∈ T δ . 20 / 32

κ -trees which cohere modulo finite Definition A subtree T of <κ κ is said to be coherent if for all δ < κ : ◮ if x , y ∈ T δ , then { α < δ | x ( α ) � = y ( α ) } is finite; ◮ if x , y ∈ δ κ , and { α < δ | x ( α ) � = y ( α ) } is finite, then x ∈ T δ iff y ∈ T δ . Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a coherent ℵ 1 -Souslin tree. 20 / 32

κ -trees which cohere modulo finite Definition A subtree T of <κ κ is said to be coherent if for all δ < κ : ◮ if x , y ∈ T δ , then { α < δ | x ( α ) � = y ( α ) } is finite; ◮ if x , y ∈ δ κ , and { α < δ | x ( α ) � = y ( α ) } is finite, then x ∈ T δ iff y ∈ T δ . Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a coherent ℵ 1 -Souslin tree. Theorem (Veliˇ ckovi´ c, 1986) ♦ ℵ 1 entails a coherent ℵ 2 -Souslin tree. 20 / 32

κ -trees which cohere modulo finite In my talk at “Young Set Theory 2011” workshop, I asked about the consistency of a coherent λ + -Souslin tree for λ singular. Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a coherent ℵ 1 -Souslin tree. Theorem (Veliˇ ckovi´ c, 1986) ♦ ℵ 1 entails a coherent ℵ 2 -Souslin tree. 20 / 32

κ -trees which cohere modulo finite In my talk at “Young Set Theory 2011” workshop, I asked about the consistency of a coherent λ + -Souslin tree for λ singular. Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a coherent ℵ 1 -Souslin tree. Theorem (Veliˇ ckovi´ c, 1986) ♦ ℵ 1 entails a coherent ℵ 2 -Souslin tree. Theorem P ( κ, 2 , ⊑ , κ ) entails a coherent κ -Souslin tree. 20 / 32

κ -trees which cohere modulo finite Theorem ◮ ♦ ( ℵ 1 ) entails P ( ℵ 1 , 2 , ⊑ , ℵ 1 ) Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a coherent ℵ 1 -Souslin tree. Theorem (Veliˇ ckovi´ c, 1986) ♦ ℵ 1 entails a coherent ℵ 2 -Souslin tree. Theorem P ( κ, 2 , ⊑ , κ ) entails a coherent κ -Souslin tree. 20 / 32

κ -trees which cohere modulo finite Theorem ◮ ♦ ( ℵ 1 ) entails P ( ℵ 1 , 2 , ⊑ , ℵ 1 ) ◮ ♦ λ entails P ( λ + , 2 , ⊑ , λ + ) Theorem (Veliˇ ckovi´ c, 1986) ♦ ℵ 1 entails a coherent ℵ 2 -Souslin tree. Theorem P ( κ, 2 , ⊑ , κ ) entails a coherent κ -Souslin tree. 20 / 32

κ -trees which cohere modulo finite Theorem ◮ ♦ ( ℵ 1 ) entails P ( ℵ 1 , 2 , ⊑ , ℵ 1 ) ◮ ♦ λ entails P ( λ + , 2 , ⊑ , λ + ) ◮ � λ + CH λ entails P ( λ + , 2 , ⊑ , λ + ) for λ singular Theorem P ( κ, 2 , ⊑ , κ ) entails a coherent κ -Souslin tree. 20 / 32

κ -trees which cohere modulo finite Theorem ◮ ♦ ( ℵ 1 ) entails P ( ℵ 1 , 2 , ⊑ , ℵ 1 ) ◮ ♦ λ entails P ( λ + , 2 , ⊑ , λ + ) ◮ � λ + CH λ entails P ( λ + , 2 , ⊑ , λ + ) for λ singular Corollary If � λ + CH λ holds for λ singular, then there exists a coherent λ + -Souslin tree. Theorem P ( κ, 2 , ⊑ , κ ) entails a coherent κ -Souslin tree. 20 / 32

κ -trees which cohere modulo finite Theorem ◮ ♦ ( ℵ 1 ) entails P ( ℵ 1 , 2 , ⊑ , ℵ 1 ) ◮ ♦ λ entails P ( λ + , 2 , ⊑ , λ + ) ◮ � λ + CH λ entails P ( λ + , 2 , ⊑ , λ + ) for λ singular ◮ V = L entails P ( κ, 2 , ⊑ , κ ) for all regular uncountable κ which is not weakly compact Corollary If � λ + CH λ holds for λ singular, then there exists a coherent λ + -Souslin tree. Theorem P ( κ, 2 , ⊑ , κ ) entails a coherent κ -Souslin tree. 20 / 32

κ -trees which cohere modulo finite Theorem ◮ ♦ ( ℵ 1 ) entails P ( ℵ 1 , 2 , ⊑ , ℵ 1 ) ◮ ♦ λ entails P ( λ + , 2 , ⊑ , λ + ) ◮ � λ + CH λ entails P ( λ + , 2 , ⊑ , λ + ) for λ singular ◮ V = L entails P ( κ, 2 , ⊑ , κ ) for all regular uncountable κ which is not weakly compact Corollary If � λ + CH λ holds for λ singular, then there exists a coherent λ + -Souslin tree. Corollary If V = L, then any regular uncountable κ is not weakly compact iff there exists a coherent κ -Souslin tree. 20 / 32

A concept of “being productive” for Souslin trees Definition A κ -Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes � t i | i < n � from a fixed level δ < κ , i < n t i ↑ is again κ -Souslin. the product tree of the upper cones � 21 / 32

A concept of “being productive” for Souslin trees Definition A κ -Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes � t i | i < n � from a fixed level δ < κ , i < n t i ↑ is again κ -Souslin. the product tree of the upper cones � Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a free ℵ 1 -Souslin tree. 21 / 32

A concept of “being productive” for Souslin trees Definition A κ -Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes � t i | i < n � from a fixed level δ < κ , i < n t i ↑ is again κ -Souslin. the product tree of the upper cones � Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a free ℵ 1 -Souslin tree. Jensen construct the levels of the tree by recursion, where the nodes of limit level α are obtained by forcing with finite conditions over some countable elementary submodel that knows about the diamond sequence and the tree constructed so far. 21 / 32

A concept of “being productive” for Souslin trees Definition A κ -Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes � t i | i < n � from a fixed level δ < κ , i < n t i ↑ is again κ -Souslin. the product tree of the upper cones � Theorem (Jensen, 1970’s) ♦ ( ℵ 1 ) entails a free ℵ 1 -Souslin tree. Jensen construct the levels of the tree by recursion, where the nodes of limit level α are obtained by forcing with finite conditions over some countable elementary submodel that knows about the diamond sequence and the tree constructed so far. Genericity entails freeness. 21 / 32

A concept of “being productive” for Souslin trees Definition A κ -Souslin tree T is free, if for every nonzero n < ω and any sequence of distinct nodes � t i | i < n � from a fixed level δ < κ , i < n t i ↑ is again κ -Souslin. the product tree of the upper cones � Observation CH + ♦ ( E ℵ 2 ℵ 1 ) entails a free ℵ 2 -Souslin tree. 21 / 32