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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.
Gist of Idea
Maybe we can treat the empty set symbol as just meaning an empty term
Source
Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2)
Book Ref
'Metaphysics (Philosophical Perspectives 20)', ed/tr. Hawthorne,John [Blackwell 2006], p.131
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.
| 14234 | If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley] |
| 14237 | We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley] |
| 14239 | The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley] |
| 14240 | The empty set is something, not nothing! [Oliver/Smiley] |
| 14241 | We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley] |
| 14242 | Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley] |
| 14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |
| 14245 | Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley] |
| 14246 | If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley] |
| 14247 | Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley] |