## List of Symbols 4 (2008)

### BibTeX

@MISC{Coumans08listof,

author = {Dion Coumans},

title = {List of Symbols 4},

year = {2008}

}

### OpenURL

### Abstract

### Citations

27 |
Consequences of the axiom of choice
- Howard, Rubin
- 1998
(Show Context)
Citation Context ...ns ‘open subset of N ’ and ‘closed subset of N ’, which would be equivalent from a classical point of view. In classical mathematics, the Axiom of Choice has a large number of equivalent formulations =-=[2]-=-. In the last chapters we study two of these and see what becomes of them if we try to interpret them intuitionistically. In Chapter 6 we start from the statement: every equivalence relation has a set... |

26 |
Zermelo’s Axiom of Choice, its origins, development, and influence
- Moore
- 1982
(Show Context)
Citation Context ...s, if ZF is consistent then both ZF+AC and ZF+¬AC are consistent. Cohen used a new method for his proof, called ‘forcing’, which turned out to be a very fruitful method to obtain independence results =-=[1]-=-. 1 The Axiom of Choice is in fact equivalent to the Well-Ordering Principle. 61.2 Overview This thesis is written from an intuitionistic point of view. A basic introduction to intuitionism is given ... |

10 |
The Foundations of Intuitionistic Mathematics
- Kleene, Vesley
- 1965
(Show Context)
Citation Context ...n is even ] ∨ ∀n ∈ N [ n = k99 → n is odd ]. But this is reckless. Brouwer proved the Fan Theorem using his Principle of Bar Induction. For a full explanation of his argument the reader could consult =-=[3]-=- or [4]. 2.6.2 Unrestricted Fan Theorem Up to now we have only spoken about decidable bars. Leaving out the adjective ‘decidable’ in the formulation of the Fan Theorem gives us a more general form of ... |

4 | fan theorem as an axiom and as a contrast to Kleene’s alternative - Veldman - 2005 |

4 | R.: Classical and constructive hierarchies in extended intuitionistic analysis
- Moschovakis
(Show Context)
Citation Context ...ve: ∀α∈N ¬¬∃γ ∈A [ IEγ = IEα ]? (6.25) Inspired by an idea of Solovay, Moschovakis has proven ∀α∈N ¬¬∃β ∈ N [ IEα = Dβ ], (6.26) using both Markov’s Principle and Brouwer’s Principle of Bar Induction =-=[8]-=-. Under the assumption of Markov’s Principle (6.25) and (6.26) are equivalent. We will prove this equivalence first, after which we present Moschovakis’ proof of (6.26). Definition 6.21. Markov’s Prin... |

3 | A peculiar connection between the Axiom of Choice and predicting the future
- Hardin, Taylor
(Show Context)
Citation Context ...ulting from the guesses of the prisoners and the actual sequence are in the same equivalence class, they differ only in finitely many positions. This means at most finitely many prisoners guess wrong =-=[7]-=-. 6.2.1 Equivalence relation on C Let us investigate the equivalence relation that helped out the prisoners (see (6.8)). What can we say intuitionistically about the existence of a set of representati... |

1 |
Intuïtionistische wiskunde, collegedictaat FNWI, Radboud Universiteit Nijmegen
- Veldman
- 2006
(Show Context)
Citation Context ...R(α, γ|α). 16It takes a bit more effort to defend this axiom. For the justification of this axiom and a thorough explanation of all choice axioms introduced in this section the reader is referred to =-=[4]-=-. GAC1,1 is the strongest axiom we have introduced in this section. Both AC0,0 and GAC1,0 are a consequence of GAC1,1. 2.6 Fans We have seen two examples of spreads up to now: N and C. There is an imp... |

1 |
Axiomatische verzamelingsleer, collegedictaat FNWI, Radboud Universiteit
- Veldman
- 2006
(Show Context)
Citation Context ...tion defined in (6.1) cannot be Lebesgue measurable. Lebesgue himself did not accept the Axiom of Choice and for him it remained possible that every bounded subset of the reals is Lebesgue measurable =-=[6]-=-. In this chapter we study several equivalence relations and see what we can say about the existence of a set of representatives. We begin by considering some classes of relations on the set N of the ... |

1 |
Hamel bases and measurability
- Nadkarni, Sunder
- 2004
(Show Context)
Citation Context ...ncomplete iff there exists x ∈ R such that, for all n ∈ N, for all α0, . . . , αn ∈ N × C satisfying ∀i, j [ i ̸= j → h(αi) # h(αj) ] and for all q0, . . . , qn ∈ Q, q0h(α0) + . . . + qnh(αn) # x. In =-=[9]-=- M.G. Nadkarni and V.S. Sunder mention a connection between Vitali independent sets and Hamel independent sets. We will give a connection between the weakly Hamel independent functions and the weakly ... |