CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC

@article{Montalbn2018CONSERVATIVITYOU,
  title={CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC},
  author={Antonio Montalb{\'a}n and Richard A. Shore},
  journal={The Journal of Symbolic Logic},
  year={2018},
  volume={83},
  pages={740 - 765}
}
Abstract We extend the usual language of second order arithmetic to one in which we can discuss an ultrafilter over of the sets of a given model. The semantics are based on fixing a subclass of the sets in a structure for the basic language that corresponds to the intended ultrafilter. In this language we state axioms that express the notion that the subclass is an ultrafilter and additional ones that say it is idempotent or Ramsey. The axioms for idempotent ultrafilters prove, for example… 
Reduction games, provability and compactness
TLDR
A certain compactness result is established that shows that if an implication $\Mathsf{Q} \to \mathsf{P}$ between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game.

References

SHOWING 1-10 OF 29 REFERENCES
ON IDEMPOTENT ULTRAFILTERS IN HIGHER-ORDER REVERSE MATHEMATICS
  • A. Kreuzer
  • Mathematics
    The Journal of Symbolic Logic
  • 2015
TLDR
It is shown that overACA_0^\omega, the higher-order extension of ACA0, the statement that an idempotent ultrafilter on ℕ exists implies the iterated Hindman’s theorem (IHT).
FROM BOUNDED ARITHMETIC TO SECOND ORDER ARITHMETIC VIA AUTOMORPHISMS
In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as PA, ACA0 (arithmetical comprehension schema with
Hindman's theorem, ultrafilters, and reverse mathematics
  • J. Hirst
  • Mathematics
    Journal of Symbolic Logic
  • 2004
TLDR
This article proves the equivalence of the existence of certain ultrafilters on countable Boolean algebras and an iterated form of Hindman's Theorem, which is closely related to Milliken's The theorem.
Non-Principal Ultrafilters, Program Extraction and Higher-order Reverse Mathematics
TLDR
A program extraction method is provided and it is shown that from a proof of a strictly statement ∀ f ∃ g Aqf(f, g) in a realizing term in Godel's system T can be extracted.
Minimal idempotent ultrafilters and the Auslander-Ellis theorem
We characterize the existence of minimal idempotent ultrafilters (on N) in the style of reverse mathematics and higher-order reverse mathematics using the Auslander-Ellis theorem and variant thereof.
Algebra in the Stone-Čech Compactification by Neil Hindman and Dona Strauss
This is an excellent book. This review is an attempt to convince the reader that this verdict is not the prejudice of an enthusiast but a sober, sound judgement. The title might suggest that the
Ultrafilters in reverse mathematics
We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA0, ATR0, and ${\bf{\Pi^1_1{-}{\rm CA}_0}}$.
An effective proof that open sets are Ramsey
TLDR
A direct proof of this theorem that is easily formalizable in $ATR_0$ is provided.
Hindman's theorem: an ultrafilter argument in second order arithmetic
  • H. Towsner
  • Mathematics
    The Journal of Symbolic Logic
  • 2011
TLDR
Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters and the methods of this proof can be translated into second order arithmetic.
Ultrafilters and multidimensional Ramsey Theorems
Utilizing ultrafilters on the setN of natural numbers which have certain special properties, we prove some simultaneous generalizations of Ramsey's Theorem and several single dimension Ramsey-type
...
...