display all the ideas for this combination of philosophers
12 ideas
13451 | The two best understood conceptions of set are the Iterative and the Limitation of Size [Rayo/Uzquiano] |
Full Idea: The two best understood conceptions of set are the Iterative Conception and the Limitation of Size Conception. | |
From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2) |
9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members. | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'? |
9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'? | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence. |
9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this). | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4) | |
A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong. |
9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it. | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set. |
9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe? | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change? |
13452 | Some set theories give up Separation in exchange for a universal set [Rayo/Uzquiano] |
Full Idea: There are set theories that countenance exceptions to the Principle of Separation in exchange for a universal set. | |
From: Rayo,A/Uzquiasno,G (Introduction to 'Absolute Generality' [2006], 1.2.2) |
9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
Full Idea: The set theorist cannot tell us anything about the true relationship of membership. | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it. |
9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world. | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5) | |
A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects. |
18151 | Could we replace sets by the open sentences that define them? [Chihara, by Bostock] |
Full Idea: Chihara proposes to replace all sets by reference to the open sentences that define them. | |
From: report of Charles Chihara (Ontology and the Vicious Circle Principle [1973]) by David Bostock - Philosophy of Mathematics 9.B.4 | |
A reaction: This depends on predicativism, because that stipulates the definitions will be available (cos if it ain't definable it ain't there). Chihara went on to define the open sentences in terms of the possibility of uttering them. Cf. propositional functions. |
9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed. | |
From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5) | |
A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack. |
8758 | We could talk of open sentences, instead of sets [Chihara, by Shapiro] |
Full Idea: Chihara's programme is to replace talk of sets with talk of open sentences. Instead of speaking of the set of all cats, we talk about the open sentence 'x is a cat'. | |
From: report of Charles Chihara (Constructibility and Mathematical Existence [1990]) by Stewart Shapiro - Thinking About Mathematics 9.2 | |
A reaction: As Shapiro points out, this is following up Russell's view that sets should be replaced with talk of properties. Chihara is expressing it more linguistically. I'm in favour of any attempt to get rid of sets. |