display all the ideas for this combination of texts
7 ideas
14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
A reaction: They charge that this leads to circularity, as Infinity depends on the empty set. |
14240 | The empty set is something, not nothing! [Oliver/Smiley] |
Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage. |
14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ¬∃xUx are no less concise, and certainly less roundabout. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) |
14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything. | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2) | |
A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics. |
14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint). | |
From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2) |
7785 | The use of plurals doesn't commit us to sets; there do not exist individuals and collections [Boolos] |
Full Idea: We should abandon the idea that the use of plural forms commits us to the existence of sets/classes… Entities are not to be multiplied beyond necessity. There are not two sorts of things in the world, individuals and collections. | |
From: George Boolos (To be is to be the value of a variable.. [1984]), quoted by Henry Laycock - Object | |
A reaction: The problem of quantifying over sets is notoriously difficult. Try http://plato.stanford.edu/entries/object/index.html. |
10699 | Does a bowl of Cheerios contain all its sets and subsets? [Boolos] |
Full Idea: Is there, in addition to the 200 Cheerios in a bowl, also a set of them all? And what about the vast number of subsets of Cheerios? It is haywire to think that when you have some Cheerios you are eating a set. What you are doing is: eating the Cheerios. | |
From: George Boolos (To be is to be the value of a variable.. [1984], p.72) | |
A reaction: In my case Boolos is preaching to the converted. I am particularly bewildered by someone (i.e. Quine) who believes that innumerable sets exist while 'having a taste for desert landscapes' in their ontology. |