A logical view on Tao s finitizations in analysis

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1 A logical view on Tao s finitizations in analysis Jaime Gaspar 1,2 (joint work with Ulrich Kohlenbach 1 ) 1 Technische Universität Darmstadt 2 Supported by the Fundação para a Ciência e a Tecnologia

2 Part I Tao s finitizations

3 Finitizations In 2007 and 2008, Terence Tao wrote essays about finitization of statements in analysis. Soft analysis Deals with: infinite objects (sequences, σ-algebras,...); qualitative properties (convergence, compactness,...). Hard analysis Deals with: finite objects (finite sets, convergent integrals,...); quantitative properties (cardinality of finite sets, bounds,...). Finitization A finitization of a soft analysis statement is an equivalent hard analysis statement.

4 Example 1 Infinite convergence principle Every monotone bounded sequence of real numbers is convergent. Finite convergence principle Every long enough (length M) bounded monotone sequence has arbitrary high-quality (error tolerance ε) long (length F (N)) amounts of stability: ε > 0 F : N N M N (x n ) n=0,...,m [0, 1] monotone N M m, n [N, N + F (N)] ( x m x n ε).

5 Example 2 Let k := {0,..., k 1}. A sequence (A n ) P fin (N) weakly converges to I P inf (N) if for all k N we have A n k = I k for all large enough n N. A function F : P fin (N) N is asymptotically stable near infinite sets (F ASNIS) if stabilizes over all weakly convergent sequences. Infinite pigeonhole principle IPP Every colouring of N with finitely many colours has a infinite colour class. Tao s finitary infinite pigeonhole principle FIPP Every colouring of a large enough initial segment of N with finitely many colours has a big colour class: n N F ASNIS k N f :k n c <n A=f 1 (c) ( A >F (A)).

6 Example 2 IPP Every colouring of N with finitely many colours has a infinite colour class. Tao s FIPP n N F ASNIS k N f : k n c < n A = f 1 (c) ( A > F (A)) Proof of IPP FIPP Assume IPP and, by contradiction, FIPP. We have n N, F ASNIS and a sequence f k : k n such that: ( ) no A k = (f k ) 1 (c) verifies A k > F (A k ). Extend f k to f k : N n. The f k s are in the sequentially compact n N, so (a subsequence of) f k converges to some f : N n. By IPP, f has an infinite colour class f 1 (c). Unfolding f k f we see that A k = (f k ) 1 (c) weakly converges to f 1 (c). Then F stabilizes over (A k ) but A k. So k N ( A k > F (A k )), contradicting ( ).

7 Summary Tao wants to finitize statements in analysis: soft analysis (infinite, qualitative) finitization hard analysis (finite, quantitative) Tao s two prime examples: infinite convergence principle infinite pigeonhole principle finitization almost finitization finite convergence principle Tao s finitary infinite pigeonhole principle Usually argued by contradiction and sequential compactness.

8 Part II Two logical points

9 Gödel functional interpretation PA ω is a Peano arithmetic that deals with N, N N, (N N ) N, N (NN),... The quantifier-free axiom of choice QF-AC is x ya(x, y) f xa(x, f (x)) with A(x, y) quantifier-free. Gödel functional interpretation Is a function A A G x ya G (x, y) with A G (x, y) quantifier-free, such that: PA ω + QF-AC A PA ω A G (t, y), for suitable terms t extracted from a proof of A; PA ω + QF-AC A A G. A G (t, y) is essentially a finitization of A: A G (t, y) is a hard analysis statement; A A G ; PA ω A G (t, y) PA ω (A G with x witnessed by t). Point 1 Gödel functional interpretation finitizes systematically.

10 Sequential compactness vs Heine-Borel compactness Sequential compactness Every sequence has a convergent subsequence. Heine-Borel compactness Every open cover has a finite subcover. Every continuous function is bounded. Reverse mathematics Seeks to find which axioms are need to prove theorems. The axioms considered are the big five subsystems of second order arithmetic: RCA 0 WKL 0 ACA 0 ATR 0 Π 1 1 -CA 0 1st order part PA with Σ 0 1-induction PA with Σ 0 1-induction PA 2nd order part compactness only computable sets some non-computable sets Heine-Borel compactness all arithmetical sets sequential compactness

11 Sequential compactness vs Heine-Borel compactness RCA 0 WKL 0 ACA 0 ATR 0 Π 1 1 -CA 0 Heine-Borel compactness sequential compactness Assume that we proved that A finitizes into B using sequential compactness. Then, at best, ACA 0 A B. If, for example, ACA 0 A, then ACA 0 A B reduces to ACA 0 B, obscuring the role of A. In this sense our finitization is not meaningful. A finitization of A into B that uses sequential compactness is only meaningful if ACA 0 A, B. A finitization of A into B that uses Heine-Borel compactness is already meaningful if WKL 0 A, B. Point 2 We should prefer Heine-Borel compactness.

12 Summary Gödel functional interpretation finitizes systematically. We should prefer Heine-Borel compactness to sequential compactness.

13 Part III A case study

14 Kohlenbach s finitary infinite pigeonhole principle A function F : P fin (N) N is asymptotically stable (F AS) if stabilizes over all chains A 0 A 1 A 2 in P fin (N). IPP G is essentially Kohlenbach s FIPP: Kohlenbach s finitary infinite pigeonhole principle FIPP Every colouring of a large enough initial segment of N with finitely many colours has a big subset of a colour class: n N F AS k N f :k n c <n A f 1 (c) ( A >F (A)). Tao s finitary infinite pigeonhole principle FIPP Every colouring of a large enough initial segment of N with finitely many colours has a big colour class: n N F ASNIS k N f :k n c <n A=f 1 (c) ( A >F (A)).

15 Kohlenbach s finitary infinite pigeonhole principle IPP Every colouring of N with finitely many colours has a infinite colour class. Kohlenbach s FIPP n N F AS k N f : k n c < n A f 1 (c) ( A > F (A)) Theorem IPP and Kohlenbach s FIPP are equivalent in WKL 0. B(f,i) Proof of IPP FIPP {}}{ Prove f : N n i N c < n A = f 1 (c) i ( A > F (A)): f has a infinite colour class f 1 (c) by IPP; F stabilizes over the chain A i := f 1 (c) i but A i so i ( A i > F (A i )). Heine-Borel compactness: φ(f ) := min i B(f, i) is total and continuous on the compact n N, so it has an upperbound k. f only appears in f 1 (c) i with i k, so f k suffices. Theorem IPP and Tao s FIPP are equivalent in ACA 0.

16 Summary Gödel functional interpretation finitizes IPP into Kohlenbach s FIPP. IPP and Kohlenbach s FIPP are provably equivalent in WKL 0 using Heine-Borel compactness.

17 Global summary Tao s finitizations Soft analysis hard analysis. Examples: - infinite convergence principle finite convergence principle; - infinite pigeonhole principle IPP Tao s finitary infinite pigeonhole principle FIPP. Contradiction and sequential compactness argument. Two logical points Gödel functional interpretation finitizes systematically. Prefer Heine-Borel compactness to sequential compactness. Case study Gödel function interpretation: IPP Kohlenbach s FIPP. Heine-Borel compactness: proof of IPP FIPP.

18 References Terence Tao Soft analysis, hard analysis, and the finite convergence principle Terence Tao The correspondence principle and finitary ergodic theory Jaime Gaspar and Ulrich Kohlenbach On Tao s finitary infinite pigeonhole principle To appear in The Journal of Symbolic Logic kohlenbach

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