[FOM] Finite Set Theory

[Harvey Friedman]

On 2/19/06 3:59 PM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

Friedman wrote:

>> 1. A set is a "bunch of things" arranged in any order whatsoever,
>> with no repetition. The only thing that matters about a set is
>> what is in it, and what is not in it. It doesn't matter how one
>> describes what is in it. There may be many ways to describe the
>> same set.
> 
> You have told *what matters about a set* in completely clear terms,
> but you haven't said what a set *is*.  There is a hidden assertion
> that there exist abstract objects with the behavior you describe.

My recollection is that Piaget's experiments with young children show that
at an early stage, pinpointed by his work, this kind of thinking is entirely
incorporated. 

Even more severe difficulties of the same or related nature are involved in
all reasoning. What is a chair? Table? person? Clock? House? Bank account?
Natural number? Zero? Computer? Dog? Conversation? Telephone? Automobile?
Insult? Lecture? Family? Family tree? List? Computer program? Food? Steak?
Vegetable? Metal? Futures contract? Stock? Stock market? Window? Classroom?
Professor? Law? Constitution? Court? Government? War? Peace? Conflict?
Crime? Shirt? Clothes? Door? Child? Children? Contract? Argument?
Mathematics? Philosophy? School? Parent? FOM? f.o.m.?
 
>... are a grammatical
> mirage.

What is a grammatical mirage? Can you explain this as well as I have
explained the emptyset and singletons?
> 
> Implicit in your post is the question "if set theory is nonsense,
> why is it so easy for ordinary people to understand how it works?"

Basic set theory does not appear to be nonsense to anyone. It is implicit in
everyone's organization of the world. Would you be more congenial to lists?

> My answer to that is that the behavior you've described is very
> familiar to ordinary people.  It's just how nouns behave, or maybe
> I should say noun phrases.

What is a noun? What is a phrase? What is a noun phrase? Can you explain
this as well as I have explained the emptyset and singletons?

>The supposed "set of all mammals"
> functions in exactly the same way as the word "mammal",

No. "set of all mammals" refers to an object, whereas "being a mammal" or
"mammalness" is a property. "set consisting of zero" refers to an object,
whereas "being zero" or "zeroness" is a property. This is taught in school
throughout the world.

> So the
> behavior of your fictional sets is unmysterious and that's
> probably why people have an easy time of picking it up (at least
> for finite sets).

In direct analogy: "So the behavior of your fictional chairs is unmysterious
and that's probably why people have an easy time of picking it up (at least
for the most common kinds of chairs".

> This suggestion implies a possible route to rehabilitating set
> theory, 

It requires no rehabilitation, since everyone finds it so plain and simple
and convenient. The same is true of lists.

>by identifying sets with linguistic descriptions in some
> way.  

Cumbersome, messy, and unproductive. Besides what is a "linguistic
description"? Can you explain this as well as I have explained the emptyset
and singletons?

>There could be other ways of doing this too.  I find it
> unbelievable that if this were done carefully the result wouldn't
> be predicative.

Why make it absurdly complicated for no good reason, as it was originally
completely understood? Oh, I guess you think that predicativity is simpler
than sets with at most 1 element? Let's have a contest to see how our non
mathematical friends absorb predicativity.

> Any attempt to justify set theory by saying *what
> sets are* (e.g., a "bunch of things") seems not only sure to
> fail in the sense of being, literally, nonsense.

Makes complete sense to me, at least in the elementary contexts we are
talking about. Makes sense to any non mathematician I know. Makes sense to
children. And, I am sure that it makes sense to you.

>It is also
> clear that were any such attempt to succeed, it would succeed
> all too well and would in fact justify naive set theory, which
> is inconsistent.

Not at all. The natural continuation of my explanation in
http://www.cs.nyu.edu/pipermail/fom/2006-February/009891.html

is the hereditarily finite sets. Or in computer science environments, the
hereditarily finite lists, which are so convenient, transparent,
unmysterious, and useful, there.

There is no trace of a paradox whatsoever. And it can be reasonably argued
that: there is no good reason to avoid a finitely generated infinite set
like Zermelo's infinite set

{emptyset,{emptyset},{{emptyset}},...}

and that the basic picture can be reasonably assumed to be lifted. E.g.,
power set is not a problem for finite sets, so it can be argued, why should
we think that it has any problem for infinite sets?

It is of course trivial to complain about what 0,1,2,3 is, and of course,
what 0,1,2,3,... is.

And it is of course trivial to complain about infinite sets, and power sets
of infinite sets. 

It is also trivial to defend against the complaints, etcetera.

Harvey Friedman