display all the ideas for this combination of texts
10 ideas
17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
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? |
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. |
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. |