The independence of the Prime Ideal Theorem from the Order-Extension Principle

1999 ◽  
Vol 64 (1) ◽  
pp. 199-215 ◽  
Author(s):  
U. Felgner ◽  
J. K. Truss
Keyword(s):  
Boolean Algebra ◽  
Prime Ideal ◽  
Partial Ordering ◽  
Linear Ordering ◽  
Extension Principle ◽  
Generic Extension ◽  
Direct Verification ◽  
The Family ◽  
Order Extension ◽  
Prime Ideal Theorem

AbstractIt is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel–Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a ‘generic’ extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure.

On the strength of the Sikorski extension theorem for Boolean algebras

1983 ◽  
Vol 48 (3) ◽  
pp. 841-846 ◽  
Author(s):  
J.L. Bell
Keyword(s):  
Boolean Algebra ◽  
Prime Ideal ◽  
Extension Theorem ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
The Axiom Of Choice ◽  
Universe Of Sets ◽  
The Universe ◽  
Boolean Extension ◽  
Prime Ideal Theorem

The Sikorski Extension Theorem [6] states that, for any Boolean algebra A and any complete Boolean algebra B, any homomorphism of a subalgebra of A into B can be extended to the whole of A. That is,Inj: Any complete Boolean algebra is injective (in the category of Boolean algebras).The proof of Inj uses the axiom of choice (AC); thus the implication AC → Inj can be proved in Zermelo-Fraenkel set theory (ZF). On the other hand, the Boolean prime ideal theoremBPI: Every Boolean algebra contains a prime ideal (or, equivalently, an ultrafilter)may be equivalently stated as:The two element Boolean algebra 2 is injective,and so the implication Inj → BPI can be proved in ZF.In [3], Luxemburg surmises that this last implication cannot be reversed in ZF. It is the main purpose of this paper to show that this surmise is correct. We shall do this by showing that Inj implies that BPI holds in every Boolean extension of the universe of sets, and then invoking a recent result of Monro [5] to the effect that BPI does not yield this conclusion.

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.

Normal Operators on the Banach Space Lp(-∞,∞). Part I

1961 ◽  
Vol 13 ◽  
pp. 505-518 ◽  
Author(s):  
Gregers L. Krabbe
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Abelian Subalgebra ◽  
The Family ◽  
Normal Operators ◽  
Image Position

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that(1)

scholarly journals On the prime ideal theorem

1968 ◽  
Vol 20 (1-2) ◽  
pp. 233-247 ◽  
Author(s):  
Takayoshi MITSUI
Keyword(s):  
Prime Ideal ◽  
Prime Ideal Theorem

The prime ideal theorem in non-commutative arithmetic

1982 ◽  
Vol 181 (2) ◽  
pp. 143-170 ◽  
Author(s):  
Colin J. Bushnell ◽  
Irving Reiner
Keyword(s):  
Prime Ideal ◽  
Prime Ideal Theorem
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