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# Martin's axiom and the continuum

## 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.

## References

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[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.
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[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.
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[T] Todorčević, S., Remarks on Martin's axiom and the continuum hypothesis, Canadian Journal of Mathematics, vol. 43 (1991), pp. 832851.
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# Martin's axiom and the continuum

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