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Measures: Back and Forth Between Point sets and Large sets

Published online by Cambridge University Press:  15 January 2014

Noa Goldring
Noa Goldring*
Affiliation:
Department of Mathematics, Occidental College, Los Angeles, CA 90041, USA.E-mail:[email protected]
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Extract

It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theory. In return, the existence of such large sets turned out critical to settling open conjectures about point sets.

How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?

In this essay I dissect one group of such two-way results. Their linchpin is the notion of measure.

§1. Vitali's impossibility result. Our starting point is a problem in measure theory regarding the notion of “Lebesgue measure.” Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 1 , Issue 2 , June 1995 , pp. 170 - 188
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

[1] Drake, F. R., Set theory, North-Holland, 1974.Google Scholar
[2] Folland, G. B., Real analysis, Wiley-Interscience, 1984.Google Scholar
[3] Gödel, K., The consistency of the axiom of choice and of the generalized continuum hypothesis, Proceedings of the National Academy of Science of the USA, vol. 24 (1938), pp. 556–557, reprinted in [4].Google Scholar
[4] Gödel, K., Collected works, Volume II: Publications 1938–1974, edited by Feferman, Solomon (editor-in- chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M. and van Heijenoort, Jean, Oxford University Press, New York, 1990.Google Scholar
[5] Jech, T., Set theory, Academic Press, 1978.Google Scholar
[6] Lusin, N., Sur la classification de M. Baire, Comptes Rendus de l'Academie des Sciences, Paris, vol. 164 (1917), pp. 91–94.Google Scholar
[7] Lusin, N. and Sierpinski, W., Sur quelques proprietes des ensembles (A), Bulletin International de l'Academie des Sciences de Cracovie (1918), pp. 35–48.Google Scholar
[8] Margin, D. A. and Steel, J. R., Projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–126.Google Scholar
[9] Moschovakis, Y. N., Descriptive set theory, North-Holland, 1980.Google Scholar
[10] Raisonnier, J., The measure problem, Israel Journal of Mathematics, vol. 48 (1984), pp. 48–56.Google Scholar
[11] Royden, H. L., Real analysis, The Macmillian Company, 1963.Google Scholar
[12] Shelah, S., Can we take Solovay's inaccessible away?, Israel Journal of Mathematics, vol. 48 (1984), pp. 1–47.Google Scholar
[13] Shelah, S. and Woodin, H., Large cardinals and Lebesgue measure, Israel Journal of Mathematics (1980), pp. 381–394.Google Scholar
[14] Shoenfield, J. R., The problem of predicativity, Essays on the foundations of mathematics (Bah-Hillel, Y. et al., editors), The Magnes Press, Jerusalem, 1961.Google Scholar
[15] Solovay, R., The cardinality of sets of reals, Symposium papers commemorating the sixtieth birthday of Kurt Gödel, Springer-Verlag, 1969, pp. 59–73.Google Scholar
[16] Solovay, R., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56.Google Scholar
[17] Solovay, R., Real valued measurable cardinals, Axiomatic set theory (Scott, D., editor), Proceedings of the Symposium in Pure Mathematics, no. 13, AMS, 1971, pp. 397–428.Google Scholar
[18] Solovay, R., Gödel 1938: Introductory note to 1938, 1939, 1939a and 1940, in Gödel [5].Google Scholar
[19] Ulam, S., Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematica, vol. 16 (1930), pp. 140–150.Google Scholar
[20] Wagon, S., The Banach-Tarski paradox, Cambridge University Press, 1985.Google Scholar
[21] Woodin, W. H., Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proceedings of the National Academy of Sciences of the USA, vol. 85 (1988), pp. 6587–6591.Google Scholar