Cardinal arithmetic for skeptics

• Published in 1992
• Added on 2012-05-02

When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with "consistency" rather than "truth" may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of $2^{\aleph_0}$, cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic.

Links

• http://www.ams.org/journals/bull/1992-26-02/S0273-0979-1992-00261-6/S0273-0979-1992-00261-6.pdf
• http://arxiv.org/pdf/math/9201251v1.pdf

Other information

BibTeX entry

@article{Shelah1992,
	key = {Shelah1992},
	type = {article},
	title = {Cardinal arithmetic for skeptics},
	author = {Shelah, Saharon},
	abstract = {When modern set theory is applied to conventional mathematical problems, it has a disconcerting tendency to produce independence results rather than theorems in the usual sense. The resulting preoccupation with "consistency" rather than "truth" may be felt to give the subject an air of unreality. Even elementary questions about the basic arithmetical operations of exponentiation in the context of infinite cardinalities, like the value of {\$}2^{\{}\aleph{\_}0{\}}{\$}, cannot be settled on the basis of the usual axioms of set theory (ZFC). Although much can be said in favor of such independence results, rather than undertaking to challenge such prejudices, we have a more modest goal; we wish to point out an area of contemporary set theory in which theorems are abundant, although the conventional wisdom views the subject as dominated by independence results, namely, cardinal arithmetic.},
	comment = {},
	date_added = {2012-05-02},
	date_published = {1992-04-01},
	urls = {http://www.ams.org/journals/bull/1992-26-02/S0273-0979-1992-00261-6/S0273-0979-1992-00261-6.pdf,http://arxiv.org/pdf/math/9201251v1.pdf},
	collections = {Attention-grabbing titles,The act of doing maths,Unusual arithmetic},
	url = {http://www.ams.org/journals/bull/1992-26-02/S0273-0979-1992-00261-6/S0273-0979-1992-00261-6.pdf http://arxiv.org/pdf/math/9201251v1.pdf},
	urldate = {2012-05-02},
	year = 1992,
	doi = {10.1090/S0273-0979-1992-00261-6},
	issn = {0273-0979},
	journal = {Bulletin of the American Mathematical Society},
	month = {apr},
	number = 2,
	pages = {197--211},
	volume = 26
}