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| Titel: |
On the Strength of Weak Compactness |
| Auteur: |
Kreuzer, Alexander P. |
| Verschenen in: |
Computability |
| Paginering: | Jaargang 1 (2013) nr. 2 pagina's 171-179 |
| Jaar: |
2013-01-08 |
| Inhoud: |
We study the logical and computational strength of weak compactness in the separable Hilbert space ℓ2. Let weak-BW be the statement the every bounded sequence in ℓ2 has a weak cluster point. It is known that weak-BW is equivalent to ACA0 over RCA0 and thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstraß principle BW. We show that weak-BW is instance-wise equivalent to $\Pi^0_2$-CA. This means that for each $\Pi^0_2$ sentence A(n) there is a sequence $(x_i)_{i\in{\mathbb{N}}})$ in ℓ2, such that one can define the comprehension function for A(n) recursively in a cluster point of (xi)i. As a consequence we obtain that the degrees d≥T 0″ are exactly the degrees that contain a weak cluster point of any computable, bounded sequence in ℓ2. Since a cluster point of any sequence in the unit interval [0,1] can be computed in a degree low over 0′ (see [10]), this also shows that instances of weak-BW are strictly stronger than instances of BW. We also comment on the strength of weak-BW in the context of abstract Hilbert spaces in the sense of Kohlenbach and show that his construction of a solution for the functional interpretation of weak compactness is optimal, cf. [7]. 2010 Mathematics Subject Classification. Primary 03F60; Secondary 03D80, 03B30. |
| Uitgever: |
IOS Press |
| Bronbestand: |
Elektronische Wetenschappelijke Tijdschriften |
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