Aug 012011

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While exploring theoretical physics and computer science, I commonly encounter large sets whose cardinalities are of interest. Rather than endlessly recalculate these as needed, I would prefer to have a single reference which consolidates all of the salient results. To my knowledge such a work does not exist, so I decided to create it. Consider it a missing chapter on cardinality from Abramowitz and Steguin.

There are many excellent works on the rigorous development of cardinal theory, the more intricate aspects of the continuum hypothesis, and various axiomatic formulations of set theory. Rather than emphasize these, the present work attempts to summarize practical results in cardinal arithmetic as well as list the cardinalities of many common sets. No attempt at rigor or a systematic development is made. Instead, sufficient background is provided for a reader with a basic knowledge of sets to quickly find results they require. Proof sketches offer the salient aspects of derivations without the distraction of formal rigor. Where I perceive that pitfalls or confusion may arise (or where I encountered them myself), I have attempted clarification.

In addition, I included a discussion of infinite bases and integration from the standpoint of cardinality. These are topics that are of interest to me. Hopefully, others will find their mention useful as well.

If you detect any errors in my exposition, wish to offer suggestions for improvement, or know of any omitted references or proofs, I would be grateful for your comments.

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