Skip to main content Accessibility help
×
Home

Martin's axiom and the continuum

  • Haim Judah (a1) and Andrzej Rosłanowski (a2) (a3)

Extract

Since Georg Cantor discovered set theory the main problem in this area of mathematical research has been to discover what is the size of the continuum. The continuum hypothesis (CH) says that every infinite set of reals either has the same cardinality as the set of all reals or has the cardinality of the set of natural numbers, namely

In 1939 Kurt Gödel discovered the Constructible Universe and proved that CH holds in it. In the early sixties Paul Cohen proved that every universe of set theory can be extended to a bigger universe of set theory where CH fails. Moreover, given any reasonable cardinal κ, it is possible to build a model where the continuum size is κ. The new technique discovered by Cohen is called forcing and is being used successfully in other branches of mathematics (analysis, algebra, graph theory, etc.).

In the light of these two stupendous works the experts (especially the platonists) were forced to conclude that from the point of view of the classical axiomatization of set theory (called ZFC) it is impossible to give any answer to the continuum size problem: everything is possible!

In private communications Gödel suggested that the continuum size from a platonistic point of view should be ω2, the second uncountable cardinal. As this is not provable in ZFC, Gödel suggested that a new axiom should be added to ZFC to decide that the cardinality of the continuum is ω2.

Copyright

References

Hide All
[BFS] Brendle, J., Judah, H., and Shelah, S., Combinatorial properties of Heckler forcing, Annals of Pure and Applied Logic, vol. 58 (1992), pp. 185199.
[DF] Van Douwen, E. K. and Fleissner, W. G., Definable forcing axiom: an alternative to Martin's axiom, Topology and Its Applications, vol. 35 (1990), pp. 277289.
[GJS] Goldstern, M., Judah, H., and Shelah, S., Strong measure zero sets without Cohen reals, this Journal, vol. 58 (1993), pp. 13231341.
[HS] Harrington, L. and Shelah, S., Some exact equiconsistency results in set theory, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 178188.
[Jec] Jech, T., Set theory, Academic Press, New York, 1978.
[Ju] Judah, H., Was Gödel right?, Notes, Mathematical Sciences Research Institute, Berkeley, California, 1989.
[JRS] Judah, H., Rosłanowski, A, and Shelah, S., Examples for Souslin forcing, Fundamenta Mathematicae, vol. 144 (1994), pp. 2342.
[JS] Judah, H. and Shelah, S., Souslin forcing, this Journal, vol. 53 (1988), pp. 11821207.
[JS2] Judah, H. and Shelah, S., Souslin absoluteness, preprint.
[JS3] Judah, H. and Shelah, S., Martin's axioms, measurability and equiconsistency results, this Journal, vol. 54 (1989), pp. 7894.
[MP] Miller, A. and Prikry, K., When the continuum has cofinality ω1, Pacific Journal of Mathematics, vol. 115 (1984), pp. 399407.
[MS] Martin, D. A. and Solovay, R. M., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143178.
[Ra] Raisonnier, J., A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel Journal of Mathematics, vol. 48 (1984), pp. 4856.
[RS1] Roslanowski, A. and Shelah, S., More forcing notions imply diamonds, Archive for Mathematical Logic (to appear).
[RS2] Rosłanowski, A. and Shelah, S., Simple forcing notions and forcing axioms, preprint.
[Sh1] Shelah, S., How special are Cohen and random forcings, Israel Journal of Mathematics, vol. 88 (1994), pp. 153174.
[Sh2] Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.
[Si] Sikorski, R., Boolean algebras, 2nd ed., Springer-Verlag, Berlin, 1964.
[T] Todorčević, S., Remarks on Martin's axiom and the continuum hypothesis, Canadian Journal of Mathematics, vol. 43 (1991), pp. 832851.
[Ve] Veličković, B., Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (1992), pp. 256284.

Martin's axiom and the continuum

  • Haim Judah (a1) and Andrzej Rosłanowski (a2) (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed