## Typical ambiguity: trying to have your cake and eat it too

Venue: | the proceedings of the conference Russell 2001 |

Citations: | 8 - 2 self |

### BibTeX

@INPROCEEDINGS{Feferman_typicalambiguity:,

author = {Solomon Feferman and John Heywood},

title = {Typical ambiguity: trying to have your cake and eat it too},

booktitle = {the proceedings of the conference Russell 2001},

year = {},

pages = {2--5}

}

### OpenURL

### Abstract

Would ye both eat your cake and have your cake?

### Citations

210 |
Non-Well-Founded Sets
- Aczel
- 1988
(Show Context)
Citation Context ... satisfactory solution of the problem posed by naive category theory in that form. At first sight, one direction in which such a solution might be sought is to make use of theories like that of Aczel =-=[1]-=- as exposited in [4], in which the Foundation Axiom is replaced by the Anti-Foundation Axiom AFA. That certainly allows many examples of self membership such as a ∈ a and (a, b) ∈ a, and so on, with i... |

102 |
Vicious Circles
- Barwise, Moss
- 1996
(Show Context)
Citation Context ...on of the problem posed by naive category theory in that form. At first sight, one direction in which such a solution might be sought is to make use of theories like that of Aczel [1] as exposited in =-=[4]-=-, in which the Foundation Axiom is replaced by the Anti-Foundation Axiom AFA. That certainly allows many examples of self membership such as a ∈ a and (a, b) ∈ a, and so on, with interesting applicati... |

90 |
A language and axioms for explicit mathematics
- Feferman
- 1975
(Show Context)
Citation Context ...nal set theory, in which the system is extended further by variables for a partial combinatory algebra over the universe. 11 Another is to use one of the systems of Explicit Mathematics introduced in =-=[7]-=- and studied in a number of publications since then. There the operational structure in the form of a partial combinatory algebra on the universe of all individuals is taken as basic. For a system of ... |

20 |
Categorical Foundations and the Foundations of Category Theory
- Feferman
- 1977
(Show Context)
Citation Context ...s literally true, and not reinterpreted according to context. For a discussion of the desiderata for that kind of foundation, see my paper ”Categorical foundations and foundations of category theory” =-=[8]-=-. We have yet to obtain a satisfactory solution of the problem posed by naive category theory in that form. At first sight, one direction in which such a solution might be sought is to make use of the... |

16 |
On the consistency of a slight (?) modification of Quine’s New Foundations
- Jensen
- 1969
(Show Context)
Citation Context ...n might be sought is via some form of stratified theories like Quine’s NF. Though that is still not known to be consistent, the system NFU with urelements allowed was shown to be consistent by Jensen =-=[10]-=-, and he also showed how it can be beefed up to include, conservatively, ZFC. However, the formalism of NFU is not as it stands suitable to define {x|x is a relational structure of signature σ} where ... |

11 |
Set-theoretical foundations of category theory
- Feferman
- 1969
(Show Context)
Citation Context ...conservative extension of ZF. Proof. This is by a straightforward extension of the result of Montague and Vaught [14] which in turn modifies Levy’s well known reflection principle argument; see also (=-=[5]-=-: sec. 2). We give here a brief sketch of the ideas involved. Note that for conservation we need only show how, given any finite set of axioms of ZF/U<ω involving only symbols Ui for i = 0, . . . , n ... |

7 | STS: A structural theory of sets, L
- Baltag
- 1999
(Show Context)
Citation Context ...establish existence 12 According to one of the referees, the work of Barwise and Moss on SEC0 has been continued by Alexandru Baltag in his doctoral thesis, and one publication resulting from that is =-=[2]-=-. I do not know how relevant it may be to the questions dealt with here. 13 The proof makes use of the existence of two inaccessible cardinals, an assumption which, as remarked above, is regarded as i... |

6 |
One Universe as a Foundation for Category Theory
- MacLane
- 1969
(Show Context)
Citation Context ...9 Large categories have as surrogates subcategories of U0, and functor categories for them simply sit as sets in V . On the face of it, this would seem to make it similar to the suggestion of MacLane =-=[12]-=- according to which one universe suffices. But there is an essential difference here in the further use of the reflection Axiom V to insure that there is nothing special about the choice of the univer... |

5 |
Some formal systems for the unlimited theory of structures and categories
- Feferman
- 1974
(Show Context)
Citation Context ...n one can beef it up to obtain a version of it conservative over ZFC. I carried this out in an unpublished MS, “Some formal systems for the unlimited theory of structures and categories” (abstract in =-=[6]-=-). 13 This system indeed serves to literally verify examples like 4.1-4.4. But it has other defects as a proposed foundation of naive category theory. One is that there is no way that one can establis... |

2 |
Locally small categories and the foundations of mathematics
- MacLane
- 1961
(Show Context)
Citation Context ... property of all small categories also applies to the superlarge category B A . 7. Discussion: Axiomatic foundations of category theory One of the usual foundations of category theory, due to MacLane =-=[11]-=-, takes as its setting the language of the Bernays-Gödel theory of sets and classes; this allows us to talk about two kinds of categories, those that are small, i.e. are sets, and those that are large... |

1 |
Familiar Quotations, Fifteenth edition. Edited by
- Bartlett
- 1980
(Show Context)
Citation Context ...e process of separating out the possible meanings of ambiguous expressions. Ambiguity is typical, or systematic, if the process of disambiguation can 1 Ascribed to John Heywood (c. 1497 – c. 1580) in =-=[3]-=- at Heywood 13.s2 S. Feferman be carried out in some uniform way. In natural languages this is determined by specification of context. For example, the words ‘I’,‘you’,‘here’,‘now’,‘this’, ‘my’, and ‘... |

1 |
Collected Works, Volume IV. Correspondence A-G. Edited by S. Feferman et al
- Gödel
(Show Context)
Citation Context ... German original of this letter is in the Bernays archives at the ETH in Zürich; the full letter is to be found, along with much else between Gödel and Bernays, in the collection of correspondence in =-=[9]-=-.)s10 S. Feferman defined to be {x ∈ U + |ϕ(x)}. This leads to the disambiguation convention: (Dis 3) A ∈ B means A ∈ B + . To take the simplest example, 4.1, P is defined to be the class of structure... |