Vaught’s Two-Cardinal Theorem and Notions of Minimality in Continuous Logic
Victoria Noquez · Bulletin of Symbolic Logic · 2019
streamline treatments of the interpretability orders * κ of Shelah, the key new notion being that of pseudosaturation.In Chapter 4, we uniformize many ultrafilter constructions in Keisler's order.As a particular application, we prove that for all 3 ≤ k < k , T k+1,k T k +1,k , where T n,k is the theory of the random k-ary n-clique free hypergraph.This improves the previous result of Malliaris and Shelah that T k+1,k T k +1,k for all k < k -1.Borel complexity is a pre-order on sentences of L 1 measuring the complexity of countable models.In Chapter 5, we describe joint work with Richard Rast and Chris Laskowski on this order.They key idea is the following: suppose Φ is a sentence of L 1 .Define css(Φ)ptl to be the set of all sentences φ ∈ L∞ , such that in some forcing extension V[G], φ becomes the canonical Scott sentence of some model of Φ. Define Φ to be the cardinality of css(Φ)ptl (possibly ∞).We show that if Φ ≤B Ψ then this induces an injection from css(Φ)ptl to css(Ψ)ptl, whence Φ ≤ Ψ .This is a potent new method for proving nonreducibilities in ≤B , and we give several applications, including the first example of a complete first order theory T with non-Borel isomorphism relation, but which is not Borel complete.In Chapter 6, we introduce the notion of thickness.The motivation is as follows: suppose Φ is a sentence of L 1 with Φ = ∞; we wish to still be able to apply counting arguments to css(Φ)ptl.The thickness spectrum (Φ, κ) of Φ accomplishes this; roughly, (Φ, κ) ≈ |CSS(Φ)ptl ∩ V κ + |, although care must be taken to ensure that thickness is a ≤B -reducibility invariant.We present several applications of the notion of thickness; in particular, we show that all the Friedman-Stanley jumps of torsion abelian groups are non-Borel complete.We also show that if Φ has the Schr öder-Bernstein property (that is, whenever two countable models of Φ are biembeddable, then they are isomorphic), then under large cardinals, Φ is not Borel complete.In Chapter 7, we describe joint work with Saharon Shelah on the complexity of countable torsion-free abelian groups.In particular, we show that if certain large cardinals fail, then torsion-free abelian groups are aΔ 1 2 -complete, where ≤ aΔ 1 2 is a well-known coarsening of ≤B .