## MATHEMATICAL LOGIC QUARTERLY (2007)

### BibTeX

@MISC{Journal07mathematicallogic,

author = {A Journal and Mathematical Logic and John L. Bell and M Lq},

title = {MATHEMATICAL LOGIC QUARTERLY},

year = {2007}

}

### OpenURL

### Abstract

The axiomofchoice and the law of excluded middle in weak set theories

### Citations

150 |
Intuitionistic Type Theory, Bibliopolis
- Martin-Löf
- 1984
(Show Context)
Citation Context ... ∗ : (∀x ∈ a)[x ⊆ b ∧∃y(y ∈ x)] ⇒ (∃f : a −→ b)[Ext(f) ∧ (∀x ∈ a)(f(x) ∈ x)] is derivable in WST from EAC. InWST + Pow, EAC and EAC ∗ are equivalent. 5. A version of the axiom of choice considered in =-=[9]-=- is what we shall call Rep(resentatives): Eq(s, a) ⇒∃f[f : a −→ a ∧ (∀x ∈ a)(xsf(x)) ∧ (∀x ∈ a)(∀y ∈ a)[xsy ⇔ f(x) =f(y)]]. Rep asserts that unique representatives can be chosen from the equivalence c... |

26 |
Axiom of choice and complementation
- Diaconescu
- 1975
(Show Context)
Citation Context ...ice (AC) has a somewhat ambiguous status. On the one hand, in intuitionistic set theory, or the local set theory associated with a topos [2] it can be shown to entail the law of excluded middle (LEM) =-=[3, 6]-=-. On the other hand, under the “propositions-as types” interpretation which lies at the heart of constructive predicative type theories such as that of Martin-Löf [10], the axiom of choice is actually... |

12 |
Choice implies excluded middle
- Goodman, Myhill
- 1978
(Show Context)
Citation Context ...ice (AC) has a somewhat ambiguous status. On the one hand, in intuitionistic set theory, or the local set theory associated with a topos [2] it can be shown to entail the law of excluded middle (LEM) =-=[3, 6]-=-. On the other hand, under the “propositions-as types” interpretation which lies at the heart of constructive predicative type theories such as that of Martin-Löf [10], the axiom of choice is actually... |

11 |
About effective quotients in constructive type theory
- Maietti
(Show Context)
Citation Context ...m of choice is actually derivable (see, e. g. [12]), and so certainly cannot entail the law of excluded middle. This incongruity has been the subject of a number of recent investigations, for example =-=[7, 8, 10, 13]-=-. What has emerged is that for the derivation of LEM from AC to go through it is sufficient that sets (in particular, power sets), or functions, have a degree of extensionality which is, so to speak, ... |

9 | Can you add power-sets to Martin-Löf ’s intuitionistic set theory
- Maietti, Valentini
- 1999
(Show Context)
Citation Context ...m of choice is actually derivable (see, e. g. [12]), and so certainly cannot entail the law of excluded middle. This incongruity has been the subject of a number of recent investigations, for example =-=[7, 8, 10, 13]-=-. What has emerged is that for the derivation of LEM from AC to go through it is sufficient that sets (in particular, power sets), or functions, have a degree of extensionality which is, so to speak, ... |

7 |
Notes on constructive set theory. Institut Mittag-Leffler (Royal Swedish Academy of Sciences) technical report number 40, 2001. Available at
- Aczel, Rathjen
(Show Context)
Citation Context ...xiom of extensionality 1) and supports only minimal set-theoretic constructions. WST may be considered as a fragment both of (intuitionistic) ∆0-Zermelo set theory and Aczel’s constructive set theory =-=[1]-=-. In particular, WST is, like constructive type theories, too weak to allow the derivation of LEM from AC. But we shall see that, as with constructive type theories, beefing up WST with extensionality... |

6 |
On The Axiom of Extensionality -Part I
- Gandy
- 1956
(Show Context)
Citation Context ...restricted if it contains only restricted quantifiers. ∗ e-mail: jbell@uwo.ca 1) Set theories (with classical logic) lacking the axiom of extensionality seem first to have been extensively studied in =-=[4, 5, 11]-=-. 2) While the ordered pair 〈u, v〉 could be defined in the customary way as {{u}, {u, v}}, here it is taken as a primitive operation – as it is in type theory – both for reasons of simplicity and to e... |

5 |
100 years of Zermelo’s axiom of choice: what was the problem with it? Comput
- Martin-Löf
(Show Context)
Citation Context ...e law of excluded middle (LEM) [3, 6]. On the other hand, under the “propositions-as types” interpretation which lies at the heart of constructive predicative type theories such as that of Martin-Löf =-=[10]-=-, the axiom of choice is actually derivable (see, e. g. [12]), and so certainly cannot entail the law of excluded middle. This incongruity has been the subject of a number of recent investigations, fo... |

5 |
More on the axiom of extensionality
- Scott
- 1966
(Show Context)
Citation Context ...restricted if it contains only restricted quantifiers. ∗ e-mail: jbell@uwo.ca 1) Set theories (with classical logic) lacking the axiom of extensionality seem first to have been extensively studied in =-=[4, 5, 11]-=-. 2) While the ordered pair 〈u, v〉 could be defined in the customary way as {{u}, {u, v}}, here it is taken as a primitive operation – as it is in type theory – both for reasons of simplicity and to e... |

5 |
The law of excluded middle and the axiom of choice
- Tait
- 1994
(Show Context)
Citation Context ...der the “propositions-as types” interpretation which lies at the heart of constructive predicative type theories such as that of Martin-Löf [10], the axiom of choice is actually derivable (see, e. g. =-=[12]-=-), and so certainly cannot entail the law of excluded middle. This incongruity has been the subject of a number of recent investigations, for example [7, 8, 10, 13]. What has emerged is that for the d... |

5 |
Neuer Beweis für die Möglichkeit einer Wohlordnung
- Zermelo
- 1908
(Show Context)
Citation Context ...aA, Weinheim www.mlq-journal.orgsMath. Log. Quart. 54, No. 2 (2008) / www.mlq-journal.org 201 6. Finally, consider the following versions of AC which are closely related to that introduced by Zermelo =-=[14]-=- (see also [10]), namely ACZ: [(∀x ∈ a)∃y(y ∈ x) ∧ (∀x ∈ a)(∀y ∈ a)[∃z(z ∈ x ∧ z ∈ y) ⇒ x = y]] ⇒∃u(∀x ∈ a)∃!y(y ∈ x ∧ y ∈ u), EACZ: [(∀x ∈ a)∃y(y ∈ x) ∧ (∀x ∈ a)(∀y ∈ a)[∃z(z ∈ x ∧ z ∈ y) ⇒ x ≈ y]] ⇒... |

4 | Extensionality versus constructivity
- Valentini
- 2000
(Show Context)
Citation Context ...m of choice is actually derivable (see, e. g. [12]), and so certainly cannot entail the law of excluded middle. This incongruity has been the subject of a number of recent investigations, for example =-=[7, 8, 10, 13]-=-. What has emerged is that for the derivation of LEM from AC to go through it is sufficient that sets (in particular, power sets), or functions, have a degree of extensionality which is, so to speak, ... |

1 |
Toposes and Local Set Theories: An Introduction (Clarendon
- Bell
- 1988
(Show Context)
Citation Context ...Co. KGaA, Weinheim In constructive mathematics the axiom of choice (AC) has a somewhat ambiguous status. On the one hand, in intuitionistic set theory, or the local set theory associated with a topos =-=[2]-=- it can be shown to entail the law of excluded middle (LEM) [3, 6]. On the other hand, under the “propositions-as types” interpretation which lies at the heart of constructive predicative type theorie... |