The proof theoretic strength of the Steinitz exchange theorem
نویسنده
چکیده
We show that the logical theory QLA proves the Cayley–Hamilton theorem from the Steinitz exchange theorem together with a strengthening of the linear independence principle. Since QLA is a fairly weak theory (in the sense that its quantifier-free fragment, LA, translates into tautologies with TC0-Frege proofs—when restricted to the field Q of the rationals), it follows that the proof complexity of matrix algebra can be distilled to the Steinitz exchange theorem. © 2006 Elsevier B.V. All rights reserved.
منابع مشابه
A new proof for the Banach-Zarecki theorem: A light on integrability and continuity
To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...
متن کاملA cone theoretic Krein-Milman theorem in semitopological cones
In this paper, a Krein-Milman type theorem in $T_0$ semitopological cone is proved, in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.
متن کاملA Beck - Fiala-type Theorem for Euclidean Norms
The inequality in (a) is naturally the best possible. Theorem l(a) is connected with questions such as the Beck-Fiala theorem or the Koml6s conjecture (see [2J, [3J and [7]). Combinatorial motivations are presented exhaustively in [8J (see also [6]). In a slightly different form, Theorem l(a) was used in [IJ in the proof that nuclear Frechet spaces satisfy the Levy-Steinitz theorem on rearrange...
متن کاملProof-Theoretic Investigations on Kruskal's Theorem
In this paper we calibrate the exact proof–theoretic strength of Kruskal’s theorem, thereby giving, in some sense, the most elementary proof of Kruskal’s theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction.
متن کاملProof-Theoretic Strength of the Stable Marriage Theorem and Other Problems
We study the proof theoretic strength of several infinite versions of finite combinatorial theorem with respect to the standard Reverse Mathematics hierarchy of systems of second order arithmetic. In particular, we study three infinite extensions of the stable marriage theorem of Gale and Shapley. Other theorems studied include some results on partially ordered sets due to Dilworth and to Dushn...
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:
- Discrete Applied Mathematics
دوره 155 شماره
صفحات -
تاریخ انتشار 2007