# Speak with the vulgar.

Think with me.

## Oops

with one comment

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 $x_0$ is atomless, then $x_0$ has a countable chain of parts such that nothing is a part of each of them.

Proof. Since $x_0$ is atomless, there is a (countable) sequence $x_0 > x_1 > x_2 > \dots$. For each positive integer $k$, let $y_ k$ be $x_{k-1} - x_ k$. Then let $z_ k$ be the sum of $y_ k, y_{k+1}, y_{k+2}, \dots$. Note that the $z_ k$’s are a countable chain. Note also that each $z_ k$ is part of $x_{k-1}$.

Now suppose that $z_\omega$ is a part of each $z_ k$. In that case, $z_\omega$ is part of each $x_ k$. But since $x_ k$ is disjoint from $y_ k$, this means that $z_\omega$ is disjoint from each $y_ k$, and so by the definition of a mereological sum, $z_\omega$ is disjoint from each $z_ k$. This is a contradiction.

Written by Jeff

January 15, 2010 at 6:22 pm

## Indefinite divisibility

If you’re interested, I’ve written a short paper on my nominalistic indefinite extensibility arguments. (This is also my way of making good on my offer in the comments to discuss a sort of consistency result for supergunk—it’s in the appendix.)

Written by Jeff

February 17, 2009 at 8:49 pm

## Getting in touch with the universe

In my last post I argued that the set-theoretic problems with “absolutely everything” carry over even for those who don’t believe in sets, by appealing to the possibility of “supergunk”. There’s another route to the same conclusion by way of some principles about contact. I think it’s kind of neat.

Let’s take contact to be a two-place relation between objects; it is reflexive (we count overlap as contact), symmetric, and monotonic: if $X$ touches a part of $Y$, then $X$ touches $Y$. These are all standard so far.

The following additional principles seem jointly possible:

1. A pretty weak separation principle: if $X$ and $Y$ don’t touch, then there is some further $Z$ that doesn’t touch either of them. (Think of $Z$ as being located between $X$ and $Y$, keeping them apart.)

2. A very strong distribution principle: if $X$ touches the fusion of the $\phi$’s, then $X$ touches some $\phi$. (Since the last post, I’ve switched from plural quantification to schemes, since I think it helps avoid some issues.) We might call this contact supervenience: what touches the whole touches some part.

The finite version of distribution is completely tame and standard: if $X$ touches $Y + Z$, then $X$ touches $Y$ or $X$ touches $Z$. It’s very hard to imagine the finite version failing. It turns out that the general version can fail, though. For instance, none of the intervals $\left[\frac{1}{n}, 1\right]$ touches the interval $\left[-1, 0\right]$; but their fusion does (under ordinary topology). But this is pretty counterintuitive (John Hawthorne has written a whole paper about the principle’s failure). And so, even if it turns out that actually contact doesn’t supervene on parts, it still strikes me as a way things could have been.

But these two principles together give rise to another extensibility argument. Suppose that something doesn’t touch $X$. Given any $\phi$’s that don’t touch $X$, their fusion doesn’t touch $X$ by (2), and so by (1) there is some further thing that doesn’t touch $X$. So the $\phi$’s, whatever they may be, don’t exhaust the things that don’t touch $X$: the non-$X$-touchers are indefinitely extensible. Thus, in a world where (1) and (2) hold, it doesn’t make sense to talk about absolutely everything there is.

To sum up: (1) and (2) are jointly possible; therefore, generality absolutism is possibly false. Since generality absolutism isn’t contingent, generality absolutism is actually false.

Written by Jeff

December 6, 2008 at 7:59 pm

## All things great and small

This is a blog-sized summary of a paper I’m working on.

For more than a century now, there’s been a problem with “everything”. Here’s a simple version: say you have all of the sets. Then there ought to be a set of just those things—a set $X$ that contains all the sets. But in that case $X$ is a member of itself, which no set can be. Paradox!

In 1906 Bertrand Russell writes,

[T]he contradiction results from the fact that…there are what we may call self-reproducing processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property.

Michael Dummett (1993) calls properties like this indefinitely extensible—the main example is “set”, but related paradoxes also show up for “cardinal number”, “order-type”, “property”, and “proposition”. Because of this a lot of philosophers are driven to conclude that we can’t speak intelligibly of all the sets (cardinals, properties, etc.). Whenever we think we’ve caught them all, another pops up to defy us. And if we can’t talk about every set, then we also can’t talk about plain everything—since that would have to include all the sets.

This kind of argument leaves open an escape to somebody with enough nerve: one way out is to deny outright that there are any sets (cardinals, properties, etc.). This is kind of an attractive view anyway, since sets are a lot spookier than, say, tables and chairs and galaxies and electrons—even without the paradoxes. The strong-nerved people who deny the existence of such things are called nominalists (contrasted with platonists or realists).

I have a way to close of the nominalists’ escape route. What we need is a new indefinitely extensible property that isn’t “abstract” (like “set”, etc.): instead, it applies to concrete, material objects. (Even nominalists don’t want to deny those!) I don’t claim that there actually are any such things, though: instead I claim that there could be. This is enough, because it would be very odd if it turned out that “absolutely everything”-talk was intelligible just by luck. The people who think it makes sense to talk that way think that it necessarily makes sense to talk that way. If they’re right, then it shouldn’t even be possible for something to be the way I suggest.

Here’s the idea. Material things could be made of atoms: they might have smallest parts that cannot be divided any further. Alternatively, they could be made of “atomless gunk” (David Lewis’s term (1991)): any piece of it contains ever-smaller bits. Inside our “atoms” we find protons, in the protons we find quarks, and it never stops. Gunk has a long pedigree as a theory of how the world is—and even if it happens to be false about our world, it sure seems like a way a world could possibly be.

But gunk doesn’t by itself give us what we need: it could be that the parts of a gunky material object eventually run out. If you follow finite chains of decreasing objects, there is always something further down—but if you follow infinite chains, you may succeed in getting all the way to the bottom, with nothing smaller below. But also, (it seems) that might not happen. As you go further and further down to smaller and smaller parts, there are always smaller parts further on. An object with parts like this I’ll call supergunk.1

More precisely, an object $X$ is hypergunk iff it satisfies the following condition:

• For any parts of $X$, the $x$’s, such that each $x$ is a part of or has as a part each of the $x$’s, there is something that is a proper part of each of the $x$’s.

From this condition it follows that “part of $X$” is an indefinitely extensible property: $X$ is indefinitely divisible. So if there’s trouble for the sets, there is just as much trouble for supergunk. And it sure seems like there could be supergunk (even if there isn’t any in the actual world). So the nominalist has a problem with “everything”, too.

1. Daniel Nolan (2004) describes something he calls “hypergunk”, but unfortunately that’s a bit different. ↩

Written by Jeff

November 16, 2008 at 12:00 am