J. D. Halpern and H. Läuchli. A partition theorem. Transactions of the American Mathematical Society, vol. 124 (1966), pp. 360–367. - J. D. Halpern and A. Lévy. The Boolean prime ideal theorem does not imply the axiom of choice. Axiomatic set theory, Proceedings of symposia in pure mathematics, vol. 13 part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 83–134.

1974 ◽  
Vol 39 (1) ◽  
pp. 181-182
Author(s):  
David Pincus
Keyword(s):  
Prime Ideal ◽  
Rhode Island ◽  
Set Theory ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Partition Theorem ◽  
Pure Mathematics ◽  
The Axiom Of Choice ◽  
Axiomatic Set Theory ◽  
Prime Ideal Theorem

George Boolos. The iterative conception of set. The journal of philosophy, vol. 68 (1971), pp. 215–231. - Dana Scott. Axiomatizing set theory. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 207–214. - W. N. Reinhardt. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 189–205. - W. N. Reinhardt. Set existence principles of Shoenfield, Ackermann, and Powell. Fundament a mathematicae, vol. 84 (1974), pp. 5–34. - Hao Wang. Large sets. Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 309–333. - Charles Parsons. What is the iterative conception of set?Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 335–367.

1985 ◽  
Vol 50 (2) ◽  
pp. 544-547 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Philosophy Of Science ◽  
Set Theory ◽  
American Mathematical Society ◽  
Publishing Company ◽  
Foundations Of Mathematics ◽  
Reidel Publishing Company ◽  
Pure Mathematics ◽  
Iterative Conception ◽  
The University ◽  
Axiomatic Set Theory

The dense linear ordering principle

1997 ◽  
Vol 62 (2) ◽  
pp. 438-456 ◽  
Author(s):  
David Pincus
Keyword(s):  
Prime Ideal ◽  
Set Theory ◽  
Ordered Set ◽  
Axiom Of Choice ◽  
Linear Ordering ◽  
Independence Result ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Prime Ideal Theorem ◽  
The Way

AbstractLet DO denote the principle: Every infinite set has a dense linear ordering. DO is compared to other ordering principles such as O, the Linear Ordering principle, KW, the Kinna-Wagner Principle, and PI, the Prime Ideal Theorem, in ZF, Zermelo-Fraenkel set theory without AC, the Axiom of Choice.The main result is:Theorem. AC ⇒ KW ⇒ DO ⇒ O, and none of the implications is reversible in ZF + PI.The first and third implications and their irreversibilities were known. The middle one is new. Along the way other results of interest are established. O, while not quite implying DO, does imply that every set differs finitely from a densely ordered set. The independence result for ZF is reduced to one for Fraenkel-Mostowski models by showing that DO falls into two of the known classes of statements automatically transferable from Fraenkel-Mostowski to ZF models. Finally, the proof of PI in the Fraenkel-Mostowski model leads naturally to versions of the Ramsey and Ehrenfeucht-Mostowski theorems involving sets that are both ordered and colored.

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