Hindman's theorem: an ultrafilter argument in second order arithmetic
نویسنده
چکیده
Hindman’s Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.
منابع مشابه
A weak variant of Hindman's Theorem stronger than Hilbert's Theorem
Hirst investigated a slight variant of Hindman's Finite Sums Theorem called Hilbert's Theorem and proved it equivalent over RCA0 to the In nite Pigeonhole Principle for all colors. This gave the rst example of a natural restriction of Hindman's Theorem provably much weaker than Hindman's Theorem itself. We here introduce another natural variant of Hindman's Theorem which we name the Adjacent Hi...
متن کاملNew Bounds on the Strength of Some Restrictions of Hindman's Theorem
We prove upper and lower bounds on the e ective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors implies ACA0. An emerging leitmotiv is that the known lower bounds for Hindman's Theorem and for its restriction to sums of at most 2 elements are already valid f...
متن کاملCompactness Theorem for Some Generalized Second-Order Language
For the first-order language the compactness theorem was proved by K. Gödel and A. I. Mal’cev in 1936. In 1955, it was proved by J. Łoś (1955) by means of the method of ultraproducts. Unfortunately, for the usual second-order language the compactness theorem does not hold. Moreover, the method of ultraproducts is also inapplicable to second-order models. A possible way out of this situation is ...
متن کاملA note on standard systems and ultrafilters
Let (M,X ) |= ACA0 be such that PX , the collection of all unbounded sets in X , admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in X such that M thinks T is consistent. We prove that there is an end-extension N |= T of M such that the subsets of M coded in N are precisely those in X . As a special case we get that any Scott set with a defina...
متن کاملHyperreals and Their Applications
NSA can be introduced in multiple ways. Instead of choosing one option, these notes include three introductions. Section 1 is best-suited for those who are familiar with logic, or who want to get a flavor of model theory. Section 2 focuses on some common ingredients of various axiomatic approaches to NSA, including the star-map and the Transfer principle. Section 3 explains the ultrapower const...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Symb. Log.
دوره 76 شماره
صفحات -
تاریخ انتشار 2011