## Oops

I’ve been cleaning up my “Indefinite Divisibility” paper from last year. One of my arguments in it concerned **supergunk**: X is supergunk iff for every chain of parts of X, there is some y which is a proper part of each member of the chain. I claimed that supergunk was possible, and argued on that basis against absolutely unrestricted quantification. I even thought I had a kind of consistency proof for supergunk: in particular, a (proper class) model that satisfied the supergunk condition as long as the plural quantifier was restricted to set-sized collections. Call something like this **set-supergunk**.

Well, I was wrong. I’ve been suspicious for a while, and I finally proved it today: set-supergunk is impossible. So I thought I’d share my failure. In fact, an even stronger claim holds:

**Theorem.** If is atomless, then has a countable chain of parts such that nothing is a part of each of them.

*Proof.* Since is atomless, there is a (countable) sequence . For each positive integer , let be . Then let be the sum of . Note that the ’s are a countable chain. Note also that each is part of .

Now suppose that is a part of each . In that case, is part of each . But since is disjoint from , this means that is disjoint from each , and so by the definition of a mereological sum, is disjoint from each . This is a contradiction.

Written by Jeff

January 15, 2010 at 6:22 pm

Posted in Logic, Metaphysics

Tagged with absolutely everything, gunk, mereology, ontology

### One Response

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Thanks for writing this.

Perhaps another definition of hypergunk that might be in a similar ballpark:

a is a piece of hypergunk iff whenever the xx form a well-ordered chain of a’s parts there are yy which form a well-ordered chain of a’s parts that are isomorphic to the successor of the xx.

You can state this rigorously in second order logic. (Or even plural logic, I think, if you look at the appendix to parts of classes.)

It’s also inconsistent if you assume the plural quantifiers are ranging over all pluralities (you get a kind of Burali-Forti paradox) but not if they only range over set sized pluralities so it might tie in with the indefinite extensibility stuff.

Andrew BaconJanuary 18, 2010 at 4:57 am