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Research articles

ScienceAsia 39 (2013): 90-93 |doi: 10.2306/scienceasia1513-1874.2013.39.090

Some remarks on cardinal arithmetic without choice

Supakun Panasawatwong*, Pimpen Vejjajiva

ABSTRACT:     One important consequence of the Axiom of Choice is the absorption law of cardinal arithmetic. It states that for any cardinals m and n, if mn and n is infinite, then m+n=n and if m≠0, mn=n. In this paper, we investigate some conditions that make this property hold as well as an instance when such a property cannot be proved in the absence of the Axiom of Choice. We further find some conditions that preserve orderings in cardinal arithmetic. These results also lead to the conditions that make the cancellation law of cardinal arithmetic hold in set theory without choice.

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Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330 Thailand

* Corresponding author, E-mail: supakun.p@gmail.com

Received 4 Jun 2012, Accepted 8 Jan 2013