When cardinals determine the power set: inner models and Härtig quantifier logic

We show that the predicate “x is the power set of y” is Σ1(Card)$\Sigma _1(\operatorname{Card})$ ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here Card$\operatorname{Card}$ is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to VI$V_I$ , the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has a cardinal strong up to one of its ℵ‐fixed points, and ℓI$\ell _{I}$ , the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) ℓI$\ell _I$ is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.


MATHEMATICAL LOGIC QUARTERLY
Because () ∩ L ⊆ L || + , it is clear that  holds if V = L.It needs a little more to see that  holds also if V = L  .The purpose of this paper is to show that  holds if V = L[E] for a sequence E of extenders and there is no inner model with a Woodin cardinal.This establishes the consistency of  with large cardinals below the first Woodin cardinal, relative to the existence of such cardinals.It remains an open question how far this result can be extended.In particular, the question whether  is consistent with a supercompact cardinal relative to the consistency of a supercompact cardinal, posed in [21], remains open.
Note that the axiom  by no means limits the size of the continuum.For example,  is consistent with the negation of the Continuum Hypothesis, relative to the consistency of  ([21, Theorem 7]).The axiom fails if we add a Cohen real ( [18,Theorem 1.7]).For a stronger result, let  ′ be the set of sets of hereditary cardinality less than 2  .It is consistent relative to the consistency of  that  ′ ≺ Σ 1 (Card) V, i.e., that  ′ reflects all Σ 1 (Card)-predicates, and then certainly  fails ( [16,Theorem 26]).
Let   denote the extension of first order logic by the quantifier .Early results on   indicated that it is quite a strong logic and the question arose whether it is as strong as second order logic.This cannot be literally true for in a finite unary vocabulary the logic   is decidable while second order logic is certainly not.However, there is a deeper sense in which the answer depends on .To see what this means we need to introduce some notation.Suppose  * is an abstract logic.We are mainly interested in the cases that  * is   or second order logic.A class  of models of a fixed vocabulary  is  * -definable if there is an  * -sentence  such that  is the class of models of .A class  of models, again of a fixed vocabulary , is Σ( * ) (-definable) if there is an  * -sentence , with a possibly larger vocabulary, such that  is the class of relativized reducts of models of .Finally, a class  is said to be Δ( * ) (-definable) if both  and its complement in the class of all models of the vocabulary  are Σ( * ).It was shown in [9] that we can regard Δ( * ) as an abstract logic, as it is closed under finite unions and intersections as well as complementation, and (in a sense which is made precise in [9]) also under quantification.It is called the Δ-extension of  * .Intuitively, Δ( * ) is the closure of  * under "recursive" operations.
Two logics are said to be equivalent if they have the same definable model classes.For first order logic and   1  , the logic and its Δ-extension are equivalent-a consequence of the Craig Interpolation Theorem [3].A logic is called Δ-closed if it is equivalent to its own Δ-closure.Of course, the Δ-extension of any logic is itself Δ-closed.The logics are called Δ-equivalent if their Δ-extensions are equivalent.
Historically the first example of Δ-equivalence was the observation that the logic ( 0 ), with the quantifier and weak second order logic  2  , with second order quantifiers over finite sets, relations and functions, are Δ-equivalent.Moreover, the infinitary logic  HYP with conjunctions and disjunctions over recursive sets of formulas is Δ-equivalent to ( 0 ) and  2   .
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MATHEMATICAL LOGIC QUARTERLY
The interest in the Δ-operation stems from the fact that it preserves many model-theoretic properties.After this short introduction to the Δ-operation we can state the connection between  and the logic   : The following conditions are equivalent: 2.   and second order logic are Δ-equivalent.
Our applications here concern semantic concepts related   in particular to classifying the complexity of the validities set   already mentioned and to the Löwenheim number of   : Definition 1.1.The Löwenheim number of a logic   * is the least cardinal  so that for any  ∈  * , if  has a model then it has a model of cardinality ≤ .
As intimated we apply this principally here to the logic   .In this case we write   for this Löwenheim number.A number of points are established concerning properties of   .Firstly, that   is of cofinality , and is moreover a fixed point of the ℵ function, thus   = ℵ   .
Secondly,   is not to be confused with the Löwenheim-Skolem-Tarski number, LST( * ), which involves the notion of structures suitable for a logic  * to have elementary substructures of size less than the cardinal.This LST number may not exist (large cardinals are required for example to show that it exists for second order logic 2 , or for   ).We also have that LST(  ) is larger than   , as can be easily seen by building a model  from the countable number of witnesses of   being what it is defined to be, and noting that  cannot have an elementary submodel with respect to   of cardinality <   .
For the rest of this paper we let  be the least weakly inaccessible cardinal, if it exists.If we write something such as " < " this is taken to assert that  exists and that the inequality holds.The following problem was open for several decades.Problem 1.3 (cf.[20]).If there are weakly inaccessible cardinals, can we have   <  ?As   < LST(  ) (cf. footnote 2) we thus have by the last theorem the consistency of   <  relative to that of a supercompact.That LST(  ) exists, and is equal to  is of necessity a large cardinal notion: as is further shown in [8,Theorem 10] for  ∈ Card,  > LST(  ) we have the failure of □  .The failure everywhere of this combinatorial principle is known to imply large cardinals in inner models, and, as only an example, projective determinacy.
This begs the question of a lower bound to the consistency strength of the simpler assertion   < .In [20, Proposition 2.8] it is shown there is no generic extension of L, or L  in which   < .We note here that essentially what was shown there is that   <  implies that  # exists, and, if L  exists, that also  † exists.One aim of the paper is to give a modest improvement to this (cf.Theorem 3.2 & Corollary 3.4).

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MATHEMATICAL LOGIC QUARTERLY Definition 1.4 (The   validities).We denote by   the set of sentences of the logic   true in all structures.
Notice that this notion of validity is over all such structures, not just those of a fixed signature.It is easy to see that the usual first order structure of arithmetic can be captured by an   sentence, and from this it follows that the validities in   are not arithmetically definable.In [19] it is shown that   is neither Σ 1 2 nor Π 1 2 definable.By an Inner Model we mean a transitive proper class model of the  axioms, which thus contains -the class of all ordinals.Inner models which are of the form L[E] (E a coherent sequence of measures or extenders) are built under various assumtions concerning the size of inner models in the universe: as usual, following on its inventor Solovay [15],  # denotes that there is no "sharp for L " the constructible universe.Further "¬ † " abbreviates the assumption that there is no sharp for any inner model with a single measurable cardinal; similarly let "¬ ¶ " ("not-Oh-pistol") abbreviate the assumption that there is no sharp for any inner model with a strong cardinal.We shall also use "¬ k " ("not-Oh-kukri", not "not-Oh-kay") for the assertion that there is no sharp for any inner model with a proper class of measurable cardinals (cf.[23]).Finally we shall mention "¬ † ", (or "¬ sword "-"not-Oh-sword") for Jensen's assertion that there is no mouse  with a measure that concentrates on smaller measurables, i.e, has a measurable  with () > 0 (cf.[24]).
Here  is intended to be a coherent sequence of extenders; this can be in the sense of Jensen [5], or as exposited in [24]; this is constructed according to such a sequence and is a very particular such model K = L[E K ].The reader may also take this as the model built in [17], although nothing in this article turns on the kind of indexing chosen for the extender sequence.In fact here we shall work with extenders suitable for building the core model K below (or at) a Woodin cardinal for which our official reference will be [6].
The core models K built under any of these anti-large cardinal assumptions, may be very different.However for each of them (under the appropriate hypothesis) we have the Weak Covering Lemma (due in various models to Jensen and Mitchell, and below a Woodin cardinal, by Mitchell, Schimmerling and Steel assuming countable closure [12], and Mitchell and Schimmerling [11] without): Theorem 1.5 (Weak Covering Lemma, WCL(K); [11,12]).Suppose there is no inner model of a Woodin cardinal.If  ≥  2 then cf ( +K ) ≥ ||.In particular if  is a singular cardinal, then  + =  +K .Further, in [10] Mitchell showed that assuming there is no inner model with a measurable cardinal  with () =  ++ then for cardinals , if we had that  = df cf () <  whilst  was regular in K, and that  was -closed (in particular   < ), then  was measurable in K if  = , and moreover () K ≥  if  > .
Vickers showed countable closure was unnecessary in a smaller universe, and did not require that  be a V-cardinal: Theorem 1.6 (Vickers [22]).Assume ¬ † .If  >  2 , and Cox proves this then in much larger models without such restrictions: In particular for singular cardinals ,  is either singular or measurable in K.

We show:
Theorem 1.8.Assume that there is no inner model of a Woodin cardinal, and that there is a measurable cardinal.Then   is neither Σ 1  3 nor Π 1 3 . Moreover: Corollary 1.9.Assume V = L[E] but there is no inner model of a Woodin cardinal.Then   is neither Σ   nor Π   .
In § 3.2 we work under a further restricted notion of 'smallness' of our models: we assume that no inner model (and so in particular no L[E] model) has a measurable cardinal  which is strong up to some larger fixed point of the ℵ-function.We write this as requiring that there is no inner model of "() = ℵ  =  > ".We then can show that K must have measurable cardinals: Corollary 1.10.Assume   < .Suppose there is no inner model of "() = ℵ  =  > ".Then the order type of the measurable cardinals of K below the Löwenheim number   is   .
We further have (Corollary 3.5) a new example of Weak Covering.Theorem 1.11.Assume   < .Suppose there is no inner model of "() = ℵ  =  > ".Then the WCL(K) holds at the weakly inaccessible  itself, that is:  + =  +K .By  − we mean the Zermelo-Fraenkel axioms, taken with the axiom scheme of Collection, but with the power set axiom removed.By the mouse ordering we mean the relation  ≤ *  which holds between set or proper classed-sized "mice" so that in the standard method of coiteration to comparable structures ( ∞ ,  ∞ ) that  ∞ is a (not necessarily proper) initial segment of the  ∞ hierarchy.(Cf.[24, § 5.4].)We assume some slight familiarity with these notions and the construction there of L[E] hierarchies with E both a coherent sequence of measures, and also of extenders.

SOME RESULTS ON CORE MODELS AND CARDINALS
We first observe that any inner model  which has the same cardinals as V will have the same core model as V, if those core models are thin: Consequently: Corollary 2.2.Assume ¬  and that  is an inner model with Card = Card  , then Once   exists then for such an  as above, we may not have that K  = K but we shall nevertheless have that K  is universal.(Recall that a weasel  is a class-sized mouse, and it is said to be universal if in the comparison with any other mouse or weasel  then  absorbs .That is if the comparison iteration of (, ) is to the models ( ∞ ,  ∞ ) then  ∞ is an initial segment (not necessarily proper) of  ∞ .(Cf. [24, § 6.3].)A universal weasel is "as good as" being the full K in many respects.Below a Woodin cardinal an inner model  whose K  is universal will for example have the same reals as the true K V .If we strengthen the assumption to that of ¬ ¶ in Lemma 2.3, then the universality of K  implies that it is a simple iterate of the true K.This is a result of Jensen and Mitchell [17].

MATHEMATICAL LOGIC QUARTERLY
Thus on the K side some critical point is moved repeatedly by ultrapowers out through the ordinals.However these ultrapower maps are continuous at the successor of such critical points.We thus also have that for some stage of the iteration  0 ∈  before which all truncations and all reductions in fine structural degree of the embeddings   , on the branch  (if any) have occurred, that for later  ∈ ∖ 0 : (iv)    0 , ( Thus there is some fixed  ∈ Reg V so that cf V ( +   ) =  for  0 ≤  ∈ .
Let  >  0 be some larger limit cardinal of V, with   =  ∈ .
Claim 1.  is inaccessible in K  (and all subsequent   for  ∈ ).
Proof of Claim 2. The first equality holds by using (iii) that   0, " ⊆ , after an argument by induction on  <  that   0, () = ; this uses the inaccessibility of  in the relevant models, and the usual argument as for measures, that if  ′ is the -successor of , as   , ′ ∶   ⟶ Ult(  ,    ) =   ′ is a map from an extender of length < , then   , ′ fixes .This, using (iii), clearly holds into direct limits for -limit  ′ ≤  .
For the second equality again by induction on  <   show that   0, " + 0 is cofinal in  +  .Again this is the same argument as for measures and holds into direct limits for -limit  ≤  .The last equation is just by elementarity.□ Now suppose  ∈  had been chosen with cf V () ≠  (where  comes from (v)).By Claim 2 cf V ( + 0 ) = cf V ( +  ) and by the comparison process  +  <  + (because  +  =  +  and the latter must be less than  + ).Thus cofinality  +  has cofinality  in V. However by assumption on Card  we have  + =  + whilst  +K  < + .As the WCL(K) holds in  we cannot have that  is singular in  and so, by WCL(K) again, (cf ( +K  ) = ))  .Putting these facts together cf V ( +K  ) is now not equal to .Contradiction.□ Corollary 2.4.Suppose there is no inner model with a Woodin cardinal.Let  be any inner model so that: is stationary for two different values of , then K  is universal.
Proof.If the conclusion failed then choose a value of  which makes the given class stationary, but for a  different from the  of the last proof, which was the cofinality of the successor of the critical point used on the  side, along the cub class  contained in the main branches  ∩ .□ For our application later we remark that the comparison argument in the proof of Lemma 2.3 works for sufficiently large .Corollary 2.5.Suppose there is no inner model with a Woodin cardinal.Let  be any transitive model of this statement together with a sufficiently large number of  axioms, and with Card  = Card ∩, then K  is weakly universal: that is for any mouse  with   <   then  < * K  .
Proof.We may assume   is a strong limit cardinal and that sufficiently many axioms are true in  to define K  and prove the WCL(K  ).Then taking  as some -singular cardinal below   , but with   <  for a given  as in the statement of the Corollary, we may assume the Weak Covering Lemma holds in  of K  , and in particular that + (=  + ) =  +K  .Set  0 =  and  0 = df K  |  + , and suppose for a contradiction that  0 ≤ *  0 .Then the comparison to models (  + ,   + ) requires  + steps, as indicated, with   + an initial segment of   + , and there being no truncations of models or in the degree of ultrapowers taken on the main branch [0,  + ]  0 on the  0 -side.But now we obtain a contradiction, as  + must be a limit of critical points, and so inaccessibles in   + , but cannot be so in   + .□

The complexity of the Härtig logic validities
The following is modelled on a proof that   is neither Σ 1 2 nor Π 1 2 using Shoenfield Absoluteness of Σ 1 2 sentences of L (cf. [19]).Theorem 3.1.Assume that there is no inner model of a Woodin cardinal, and that there is a measurable cardinal.Then   is neither Σ 1  3 nor Π 1 3 .
Proof.Let Υ be an   -sentence so that Υ  holds iff  is (isomorphic to) a transitive model of   (for some ,   an unspecified large number of axioms) with Card  = Card ∩ On  .Take such a transitive  in which Υ  holds.Let  = () = On  .By the above Corollary 2.5 K  is weakly universal, in particular, here for countable mice (meaning it absorbs any mouse  ∈ ).As any real of K is in some countable mouse, comparison of that mouse with K  shows that the real is itself in K  .Hence in particular ℝ K = ℝ K  .Our assumption on the existence of a measurable cardinal implies that K is Σ 1 3 -correct in V (cf.[17, Theorem 7.9]-recall here that the second measurable cardinal Ω mentioned in this reference is only there to enable the construction of K; since the Jensen-Steel result of [6] this upper measurable cardinal is redundant and a single measurable cardinal suffices).We thus have that Σ 1  3 -correctness in V holds for K  and so  too.Let Φ() define , a complete Σ 1  3 set-which is perforce not Π 1 3 -definable.Then: For, if Φ() holds it will hold in  of any model  with Υ  by the Σ 1 3 -correctness we have just outlined.Hence the quoted formula on the right hand side is   -valid, i.e., it is in   .Conversely, if ¬Φ(), then for a sufficiently large  ∈ Card we have: (Υ ∧ ¬Φ()) V  , and thus the right hand side fails.Hence  is reducible to   making the latter not Π 1  3 .By taking complements the same argument shows that   is not Σ 1  3 .□ We shall improve this later in the case of V being an L[E] model at Corollary 3.10.

The Löwenheim number for the Härtig logic, 𝓁 𝐼
In this subsection we show under a slightly more restrictive smallness assumption that the core model has measurable cardinals unbounded below the Löwenheim number   for the Härtig logic.
Proof.Suppose not, and the measurables of K below   are bounded by  0 <   .We obtain a contradiction.Let Ψ be a sentence of   that only has models of size at least  0 .There is then a transitive model  which is correct about cardinals, and in which there is an ordinal   where  ⊧ "  is the least weakly inaccessible".Further require that  contains a model  of the sentence Ψ which we may take as having cardinality in  less than   .We may require  to be a model of   a sufficiently large fragment of  that is sufficient for the inductive construction of K and to prove the Weak Covering Lemma for it.We may also require that the height of , say θ, is such that there is   <  < θ with  a strong limit cardinal in  with cf  () =  1 .
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MATHEMATICAL LOGIC QUARTERLY
Lastly require that in K  the measurables are bounded by the size of the model , just as they are bounded by  0 in K.Such a model can easily be found in V.By the definition of   we can then assume there is such an  with  = df On  ∈ Card, with  0 <  <   .
Proof.Suppose for a contradiction that  0 ≤ *  0 , and let ( 0 ,  0 ) ⟶ (  ,   ) be their coiteration, with  0 iterating past  0 , and thus with   an initial segment of   .We let the iteration maps be respectively   , and   , for  ≤  ≤ .By this supposition there can be no truncation of any model   on the -side of the coiteration.
Proof of Claim.Otherwise we should have for some  <  + that there is some   so that   , + (  ) =  + .By the usual arguments there is in fact a c.u.b.  ⊆  + with  <  ∈  implying that   , (  ) =   , and with  =   the critical point of the map   , .However | 0 | <  + so likewise there is a c.u.b.  ⊆  unbounded in  + with   , (  ) =   for  <  ∈ .By our assumption (ii) on  0 , the comparison to  + is the same as that for some   and   |  (where   is the critical point of   ,+1 and   <  + is some ordinal with ( +  )   ≤   ).But these two mice are of cardinality <  + , and this coiteration is completed in less than  + stages.A contradiction and so the Claim holds.□ However then   + ⊧ " + is a successor cardinal", whilst   + believes that  + is a limit of critical points   , and so inaccessible.But this contradicts the comparison process.□ Corollary 3.4.Suppose there is no inner model of "∃(() = ℵ  =  > )".Then K  = * K   .
Proof.Clearly * ≥ holds by our smallness assumption as K  is universal for all mice of cardinality < .So suppose K  * > K   .Our smallness assumption implies that for no  <  is there some mouse  with (() ≥ )  .However then the supposition ensures there must be some  0 a proper initial segment of K   , with both | 0 | <  and K   ≤ *  0 .Let  > | 0 | be any singular cardinal with   <  < .The latter implies that  is singular in , and thence, by WCL in , that  + =  + =  +K  .
But now notice that it cannot be the case that for every such  satisfying the above that in K  there is some measurable   <  with K  ⊧ "(  ) ≥  + ", since a regressive function argument in  (using that (cf () > )  ) would show then that for some  0 <  we would have K  ⊧ "( 0 ) ≥ ", thereby contradicting our smallness supposition.
However then for some such  0 we have for  0 = K  ⊧ " + 0 =  +K 0 ", to which we can apply the last lemma and then deduce that K   + 0 is universal for mice of size less than  + 0 .Hence  0 < * K   + 0 , a contradiction.□ Setting Ñ0 = df K  with P0 = df K  , we consider the coiteration of ( Ñ0 , P0 ) to, say ( Ñ , P ).Note that by stage   in the models ( Ñ  , P  ) we have lined up all the total measures to agreement below   .Claim.At stage   Ñ  must be truncated to some  *   to form an ultrapower to Ñ  +1 .
Proof of Claim.Note at stage   both models Ñ  , P  are of height the cardinal  <   : no truncation has occurred.By the WCL in V over K inherent in Theorem 1.7,   , being a singular cardinal, is either singular or measurable in K.The latter fails by our assumption that such measurables below   actually are all below  0 , and our construction that enforces  0 ≤ || <   .Nevertheless   is inaccessible in  and thus so in K  , and in the intermediate models   for  ≤   .However the iteration of K  to Ñ  preserves the singularity of   in the models Ñ (by induction on the stages  ≤   , using that  Ñ ,  ''  ⊆   ).Thus a truncation must be taken of Ñ  to remove the subset of   which is a witness to the singularity of   , if we are to have agreement between the final models.
However the result of this is that K  ≤ *    +1 , whilst the last corollary establishes that K  , being = * equivalent to K  , was universal for mice of cardinality < .Contradiction.This establishes the theorem.□ A further variant of the method of Theorem 3.2 yields: Corollary 3.5.Assume   < .Suppose there is no inner model of "∃(() = ℵ  =  > )".Then: (i) the WCL(K) holds at , i.e.  + =  +K ; (ii) there is  ≤  <  ++ which is measurable in K.
Proof.Assume for a contradiction that one of these two conclusions fail.Run the above argument but now using an   sentence Ψ 1 stating: "∃ δ( δ is the least weakly inaccessible with either (a) δ+ > δ+K or (b) no  with δ ≤  < δ++ measurable in K)".
Then the model  in the last argument, if it additionally is a model of Ψ 1 , has cardinality some θ less than   with either, call it case (a),  + = (  ) + > (  ) +K  , or otherwise, call it case (b), no  in [  , (  ) ++ ) measurable in K  .Corollary 3.4 still holds as before.
Suppose case (a) could hold, then by the Weak Covering of Theorem 1.7 in , we have: Either   is singular in K θ = df Ñ0 and so also in Ñ  and then we should have to do a truncation of Ñ  to remove the singularising sequence, as   inaccessible in K  implies that it is so in P  .
However a truncation on the Ñ side of the coiteration violates the fact that K   is universal for set sized mice, i.e, mice  with   < .So case (a) cannot hold.Now if case (b) were to hold, then as   is singular in V, again by Theorem 1.7 in V,   is singular or measurable in K.If the former we have that Ñ  ⊧ "  is singular" whilst P  ⊧ "  is inaccessible".Then we have to truncate Ñ  and iterate away a singularizing sequence, and leave behind   as regular; this leads to a contradiction as before.If the latter, that is if   is measurable in K, then we should have that Ñ  has   as a measurable K-cardinal (as   is not moved in the iteration of Ñ0 to Ñ  ).However then the mouse Ñ  ||  + = ⟨ Ñ  ,   + ⟩ with   + the order zero measure with critical point   , can neither be iterated out beyond K   , (as above, K   is universal for mice of cardinality less than ), nor, we shall argue, can it be absorbed as a measure on the K  -side in some P somewhere above  ++ , since the cofinality of the successor of its critical point   is  + .The reason being in more detail that, if we set  =  ++ =  ++ , then at the 'th stage of the coiteration we should have that  is measurable in Ñ .If we had agreement between the power sets of  in the two models Ñ and P we then should have that  + Ñ =  + P <  + .Moreover the map  P 0, ↾ +K  is cofinal into  + P .We thus have that  +K  <  + , but, moreover by Weak Covering in , that  ⊧ "cf ( +K  ) = ".Hence cf ( + P ) =  whilst cf ( + Ñ ) =  + <  ++ = .Hence () must be different in the two models.This clash can again only be resolved by a truncation at stage  on the Ñ side.But this leads to a contradiction as before.
Hence if the measure   + is to be absorbed and this clash avoided, then it must happen at a stage before  ++ , and so as a measure with critical point on the P side, below  ++ .By elementarity then, there is a measure in P0 = K  below  ++ .So Case (b) fails. □ From these conclusions one might expect that it is indeed  itself which is measurable in K, but the proof falls just short of that.We should then have that  is a measurable limit of measurables of K. Still the result here is somewhat unusual, when not even strange, and perhaps reflects the fact that we are far from an optimal lower bound.Moreover it seems we can push this argument further: Theorem 3.6.Assume that   < .Suppose there is no inner model of "() = ℵ  =  > ".Then for every weakly inaccessible cardinal  there is  ≤  <  ++ , which is measurable in K, and moreover  + =  +K .
as being a premouse is a Δ 1 notion, being defined by just first order properties over the structure, and the rest considered as Σ 1 (Card).□ Corollary 3.10.Assume V = L[E] but there is no inner model of a Woodin cardinal.Then   is neither Σ   nor Π   .
Proof.Now let Υ 1 be an   -sentence so that Υ  1 holds iff  is (isomorphic to) a transitive model of   (for some  large), with Card  = Card ∩ On  , and is also a sound premouse which thinks all its initial segments are iterable mice.
Take such a transitive  in which Υ  holds.
Then such an  is correct about power sets, indeed has domain V  = L  [E], in particular is obviously correct about finitely iterated power sets of ℕ by the last Lemma.Then use the template of Theorem 1.8.□ Obviously this can be extended to show the undefinability of   over much higher types.

OPEN QUESTIONS
Question 4.1.Can we improve the lower bound in Theorem 3.2?The consistency of   less than the first weakly inaccessible is obtained from a supercompact cardinal.There is thus a wide gap here.
Question 4.2.Is it consistent relative to large cardinals that there be a proper class of weakly inaccessible cardinals with   less than the least such?
In the next question  ec denotes the equicofinality quantifier of Shelah [14].Let (,  ec ) extend first order  by adding both sorts of formulae.
Question 4.4.How much of the above works for the logic (,  ec )? (Cf.[8]).We may be able to reflect down below the first weakly Mahlo, but can we get further measurables in K as a result?

Lemma 2 . 3 .
Suppose there is no inner model with a Woodin cardinal.Let  be any inner model with Card  = Card, then K  is universal.Proof.Suppose this failed.We give here a standard application of the Comparison Lemma (cf.[13,Theorem 7.1]).Our supposition implies, with the proof of the Comparison Lemma, that there is a cub class  of points  < On on the main branches  = [0, ∞]  and  = [0, ∞]  of the iteration trees  ,  resulting from the coiteration of  0 = K and  0 = K  to ( ∞ ,  ∞ ) with the following properties, for  <  ∈ , and   , (  , respectively) the iteration maps: (i)  ⊆ , ; (ii)   , (  ) =   = ; (iii)   , "  ⊆   .M L Q

Question 4 . 5 .Question 4 . 6 .
How large can a cardinal be and still be consistent with the statement " = () is Σ 1 (Card)"?Is it consistent relative to the existence of a supercompact cardinal that there is no proper inner model  with Card  = Card V ?A C K N O W L E D G M E N T SThe project has received funding from the Academy of Finland (grant No 322795) and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 101020762).
To see what this means, suppose  * and  + are Δ-equivalent.Then 1.  * satisfies the Compactness Theorem if and only if  + does.This generalizes to weaker form of compactness, such as -compactness.2.  * satisfies the Downward Löwenheim-Skolem Theorem 1 if and only if  + does.This generalizes to modifications of the Downward Löwenheim-Skolem Theorem, such as the Downward Löwenheim-Skolem Theorem down to , meaning that if a sentence has a model it has a model of cardinality ≤ . 3.  * is effectively axiomatizable 2 if and only if  + is.More generally, the decision problems of  * and  + are recursively isomorphic.