Model theoretic characterizations of large cardinals

Abstract

We consider compactness characterizations of large cardinals. Based on results of Benda [Ben78], we study compactness for omitting types in various logics. In Lκ,κ, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopěnka's Principle.Fact 1.2. κ is measurable iff every theory T ⊂ L κ,κ that can be written as a union of an increasing κ-sequence of satisfiable theories is itself satisfiable.Magidor [Mag71, Theorem 4] showed that extendible cardinals are the compactness cardinals of second-order logic, and Makowsky [Mak85] gives an over-arching result that Vopěnka's Principle is equivalent to the existence of a compactness cardinal for every logic (see Fact 3.12 below). This seems to situate Vopěnka's Principle as an upper bound to the strength of cardinals that can be reached by compactness characterizations.However, this is not the case. Instead, a new style of compactness is needed, which we call compactness for omitting types (Definition 3.4). Recall that a type p(x) in a logic L is a collection of L-formulas in free variable x. A model M realizes p if there is a ∈ M realizing every formula This is based on the author's personal impressions. Although the statement seems forgotten, the proof is standard: if T = ∪α<κTα and Mα Tα, then set Mκ := Mα/U for any κ-complete, nonprincipal ultrafilter U on κ. Loś' Theorem for Lκ,κ implies Mκ T .12 Here meant as 'sublogics of L∞,∞.' 13 The notion of "elementary substructure" does appear here, but always in reference to its first-order version.

ON C(n)-EXTENDIBLE CARDINALS

“…Note that j( ) is inaccessible. Moreover, note that ∈ C (2) in M as well: since being in C (2) is Π 2 -expressible, we have that V j(κ) |= ∈ C (2) ; thus, and since M |= j(κ) ∈ C (4) (by Proposition 3.4 in [1]), it follows that M |= ∈ C (2) .…”

“…But note that W ∈ V +2 ⊆ V h( ) ⊆ N , so W is indeed a normal measure on (i.e., in V ). Now, a standard reflection argument shows that the set {α < κ : V κ |= "α is supercompact"} belongs to U; thus, by elementarity, the set {α < : V |= "α is supercompact"} belongs to W. Moreover, note that ∈ C (2) both in V and in N ; the latter because, by elementarity, = h(κ) is C (2) extendible in N and, thus, it belongs to C (2) (indeed C (4) ) by Proposition 3.4 in [1]. Consequently, and since being supercompact is Π 2 -expressible, for every α < we have that α is supercompact in V if and only if it is supercompact in N if and only if it is supercompact in V .…”

“…The following definition is due to Bagaria. 4 As usual in the context of C (n) -cardinals, this is actually a schema of definitions, one for each meta-theoretic natural number n 1.…”

“…As an indication of the strength of these notions, we mention that if there is a C (2) -extendible cardinal, then there are unboundedly many supercompacts in the universe. To see this, let κ be C (2) -extendible and note that κ ∈ C (4) (by Proposition 3.4 in [1]). Meanwhile, the statement "there are unboundedly many supercompact cardinals" is easily seen to be Π 4 -expressible (given the Π 2 -definability of supercompactness).…”

“…Subsequently, the hierarchies of C (n) -cardinals were further studied and extended by the author in [18] and in [20] (see also [15] for another related work). Among these hierarchies, the notion of C (n) -extendibility is of particular interest since it has found several applications in other mathematical contexts, such as category and homotopy theory (see [2] and [3]), as well as model theory (see the recent [4]).…”