Monday, July 08, 2019

Null Set?

The null set can be represented as {}.

What is this: {{}}?

The set of the null set? The set of the concept of the null set? I'm just asking.



At 8:37 AM, Blogger Allen Hazen said...

It is the set (which can also be called {x:x={}}, or {x:x={y:~y=y}} ) whose only member is the null set. There is nothing nonsensical about this: the null set is, after all, something (i.e. Ex(x={}) ) and not, properly speaking, nothing: containing the null set (and nothing else), therefore, is not the same as containing nothing!
Mathematically it is convenient to have, in addition to all the sets with members, another perfectly good set with no members: the null set: without it various definitions and proofs would need extra, trivial but annoying, clauses or paragraphs. Depending on one's conception of the metaphysical nature of sets, it may or may not be ... odd. (Frege, for example, conceived of sets as extensions of concepts, and so found a null set -- the extension of any concept that has no application -- perfectly natural. David Lewis, in his lovely little monograph Parts of Classes (1991) presents a conception of set which does not naturally give a null set, but gives an artificial but simple interpretation of the term "{}" which allows him to argue for the truth of the standard axioms which do imply its existence.) What I think is important to remember is that this makes no mathematical difference: set theory can be reformulated in such a way as to preserve its mathematical content but avoid postulation of the null set (or, for that matter, unit sets, which also seem unintuitive on some ways of thinking about the metaphysics of sets). I argued this in a short note, "Small sets," in Philosophical Studies, vol. 63 (1991), pp. 119-123, and Alex Oliver and Timothy Smiley have recently published a longer (and more sophisticated) defence of the same idea in "Cantorian set theory," Bulletin of Symbolic Logic, vol. 24 (2018), pp. 393-451.

At 9:37 AM, Blogger Horace Jeffery Hodges said...

Thank you, Mr. Hazen, for taking the time and trouble to explain so clearly, fully, and carefully.

Jeffery Hodges

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