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Separation and Weak König's Lemma

Published online by Cambridge University Press:  12 March 2014

A. James Humphreys  and
Stephen G. Simpson
A. James Humphreys
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, State College, PA 16802, E-mail:, jimbo@math.psu.edu
Stephen G. Simpson
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, State College, PA 16802, E-mail:, simpson@math.psu.edu
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Abstract

We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 1 , March 1999 , pp. 268 - 278
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1]Brown, Douglas K., Functional Analysis in Weak Subsystems of Second Order Arithmetic, Ph.D. thesis, The Pennsylvania State University, 1987, vii + 150 pages.Google Scholar
[2]Brown, Douglas K., Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic, [18], 1990, pp. 39–50.CrossRefGoogle Scholar
[3]Brown, Douglas K. and Simpson, Stephen G., Which set existence axioms are needed to prove the separable Hahn–Banach theorem?, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 123–144.CrossRefGoogle Scholar
[4]Brown, Douglas K. and Simpson, Stephen G., The Baire category theorem in weak subsytems of second order arithmetic, this Journal, vol. 58(1993), pp. 557–578.Google Scholar
[5]Cameron, Neil, Introduction to Linear and Convex Programming, Cambridge University Press, 1985, 149 pages.Google Scholar
[6]Conway, John B., A Course in Functional Analysis, 2nd ed., Springer-Verlag, 1990, XVI + 399 pages.Google Scholar
[7]Friedman, Harvey, Unpublished communication to Leo Harrington, 1977.Google Scholar
[8]Giusto, Mariagnese and Simpson, Stephen G., Located sets and reverse mathematics, this Journal (1997), accepted for publication, 37 pages.Google Scholar
[9]Harrington, Leo, Unpublished communication to Harvey Friedman, 1977.Google Scholar
[10]Hatzikiriakou, Kostas, WKL0and Stone's separation theorem for convex sets, Annals of Pure and Applied Logic, vol. 77 (1996), pp. 245–249.CrossRefGoogle Scholar
[11]Humphreys, A. James, On the Necessary Use of Strong Set Existence Axioms in Analysis and Functional Analysis, Ph.D. thesis, The Pennsylvania State University, 1996, viii + 83 pages.Google Scholar
[12]Humphreys, A. James and Simpson, Stephen G., Separable Banach space theory needs strong set existence axioms, Transactions of the American Mathematical Society, vol. 348 (1996), pp. 4231–4255.CrossRefGoogle Scholar
[13]Łoś, J. and Ryll-Nardzewski, C., On the application of Tychonoff's theorem in mathematical proofs, Fundamenta Mathematicae, vol. 38 (1951), pp. 233–237.CrossRefGoogle Scholar
[14]Metakides, George, Nerode, Anil, and Shore, Richard A., On Bishop's Hahn–Banach theorem, [15], 1985, pp. 85–91.Google Scholar
[15]Rosenblatt, M. (editor), Errett Bishop, Reflections on him and his Research, Contemporary Mathematics, American Mathematical Society, 1985, xvii + 91 pages.CrossRefGoogle Scholar
[16]Shioji, Naoki and Tanaka, Kazuyuki, Fixed point theory in weak second-order arithmetic, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 167–188.CrossRefGoogle Scholar
[17]Sieg, Wilfried, Fragments of arithmetic, Annals of Pure and Applied Logic, vol. 28 (1985), pp. 33–71.CrossRefGoogle Scholar
[18]Sieg, Wilfried (editor), Logic and Computation, Contemporary Mathematics, American Mathematical Society, 1990, xiv + 297 pages.CrossRefGoogle Scholar
[19]Simpson, Stephen G., Partial realizations of Hilbert's program, this Journal, vol. 53 (1988), pp. 349–363.Google Scholar
[20]Simpson, Stephen G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, 1998, XIV + 445 pages.Google Scholar