‎First Fix the following notation:‎ ‎

$‎‎\forall ‎\kappa‎\in Card~~~Tp(‎\kappa‎):="‎\kappa‎~has~tree~property"$‎ ‎‎‎‎

The large cardinals as "monsters of heaven" live everywhere in the land of mathematics. Most of them appear in topology, measure theory, infinitary logic, category, model theory, etc. But some of them live in a strange misty place, "the forest of tall trees"! One of the most important problems of set theory is to discover that which one of these monsters have a forestial nature? The forestial nature of a monster gives us very important information about his possible behavior at many combinatorial situations in set theory and model theory. Now define the following concepts:‎ ‎

Definition (1): ‎Let ‎‎$‎A‎$‎, ‎$‎‎B‎$ be ‎large ‎cardinal ‎axioms, $I‎$‎, ‎$‎J‎$‎ ‎be ‎sub ‎classes ‎of ‎‎$‎Ord‎$ , ‎‎$‎‎\lbrace ‎‎\alpha‎‎_{i}‎‎\rbrace‎_{i\in I}, ‎\lbrace ‎‎‎‎‎‎\beta‎‎_{j}‎‎\rbrace‎_{j\in J}\subseteq Ord$ , $\lbrace ‎‎‎‎‎‎\alpha‎‎_{i}‎‎\rbrace‎_{i\in I}\cap ‎‎\lbrace ‎‎‎‎‎‎\beta‎‎_{j}‎‎\rbrace‎_{j\in J}=‎\emptyset‎‎‎$

‎then respectively every ‎statement ‎in ‎the ‎form ‎‎$(‎‎‎\star)‎$ and ‎$‎‎(‎\star‎‎\star‎)$‎ ‎called a‎ ‎"weak" and "strong" tree property "equation".‎ ‎ ‎‎ $(‎\star‎)~~Con(ZFC+A+\bigwedge_{i\in I}Tp(\aleph_{‎\alpha‎_{i}}))\Longleftrightarrow‎ Con(ZFC+B+\bigwedge_{j\in J}Tp(\aleph_{‎\beta‎_{j}}))$ ‎‎ ‎

$(‎\star\star‎)~~(A+\bigwedge_{i\in I}Tp(\aleph_{‎\alpha‎_{i}}))\Longleftrightarrow ‎(‎B+\bigwedge_{j\in J}Tp(\aleph_{‎\beta‎_{j}}))$ ‎ ‎

Example (1): Note to the following ‎well ‎known ‎results:‎ ‎

‎ A‎ ‎wea‎k tree property equation: ‎ ‎‎ ‎ $Con(ZFC+Tp(\aleph_{2}))\Longleftrightarrow ‎Con(ZFC+‎\exists~a~weakly~compact~cardinal‎‎‎)‎‎‎$‎‎ ‎

‎‎ A strong tree property equation:
‎ ‎ $(‎\kappa‎‎~is~strongly~inaccessible~+~Tp(‎\kappa‎))‎\Longleftrightarrow (‎‎‎‎\kappa‎‎~is~weakly~compact)‎‎‎$‎‎

‎‎‎ ‎ Remark (1): ‎It ‎seems ‎there are ‎many weak ‎tree ‎property "‎unequalities" like the following results, ‎but strangely the weak and strong "equalities" are too rare. These equalities uncover some fundamental relations between large cardinal axioms and combinatorial properties of other cardinals and so could be very useful and valuable.

‎‎‎ A‎ ‎weak ‎tree ‎property ‎unequality ‎discoverd by Menachem ‎Magidor:‎ ‎ ‎‎ $‎Con(ZFC+ ‎‎‎\exists‎~\mathtt{0}^{\sharp}‎)‎\Longleftarrow‎ ‎‎‎Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))‎‎$‎ ‎

‎‎ ‎Another ‎weak ‎tree ‎property ‎unequality ‎discoverd ‎by ‎Uri ‎Abraham:‎ ‎ ‎‎‎ ‎$‎‎‎Con(ZFC+‎\exists‎~a~supercompact~cardinal~with~a~weakly~compact~cardinal~above~it‎) ‎‎\Longrightarrow‎ ‎Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))‎$‎ ‎

‎ ‎ Question (1): ‎Is ‎there ‎any ‎known large ‎cardinal ‎axiom stronger than "$‎‎‎\mathtt{0}^{\sharp}‎$ exists" ‎and ‎weaker ‎than "there is a supercompact cardinal with a weakly compact cardinal above it " ‎‎like ‎$‎‎A‎$ ‎in ‎which ‎we have an ‎"weak tree property equality" ‎in ‎Magidor ‎and ‎Abraham's ‎results? ‎What ‎about ‎"strong tree property equality"? ‎‎In ‎the ‎other ‎words‎: is there any large cardinal axioms ‎$‎A‎$, ‎$B$, ‎‎$C$‎ ‎which ‎the ‎foll‎owing statements be true?‎ ‎

‎ $(1)~Con(ZFC+A)\Longleftrightarrow ‎Con(ZFC+Tp(\aleph_{2})+Tp(\aleph_{3}))‎‎‎‎$‎ ‎ ‎‎ ‎$‎‎(2)~B \Longleftrightarrow (C+Tp(\aleph_{2})+Tp(\aleph_{3}))$‎ ‎ ‎‎

Question (2): Are there any other known weak or strong tree property equality different from statements in example ‎$‎‎(1)$?‎

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    $\begingroup$ Please avoid manually adding spaces in equations, as a rule, they don't display well on devices other than your own. Also, what was the point of redefining all logical notations? The usage of + is standard for axiomatic systems but your definitions are incorrect: ZFC is not a sentence and your $\mathcal{L}$ is undefined. $\endgroup$ – François G. Dorais Aug 27 '13 at 13:25
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    $\begingroup$ Dear Francois, your critic remarks are true. About changing the logical symbols I want to add an algebraic form to the statements just based on a personal intuition from model theory. But based on your comment I think this is rather unclear and I will change them. $\endgroup$ – user36136 Aug 27 '13 at 13:43

For a strong special tree property equation, here is Theorem 9.4 in Todorcevic´s "Coherent Sequences":

$\kappa$ is strongly inaccessible + $sTp(\kappa) \Longleftrightarrow \kappa$ is Mahlo.

Where $sTp(\kappa)$ would mean "there are no special $\kappa$-Aronszajn trees".


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