Pardon that this is probably a dumb question but I'm half asleep right now.

As I understand it, basically all mathematical objects are reducible to sets, yes? E.g. a group is a special kind of set defined in relation to certain operations thereupon meeting certain rules?

And, as I understand it less clearly, the objects of modern physical theories are reducible to some kind of mathematical objects or another, yes? E.g. (and this is getting in over my head here) quantum fields are defined in terms of some kind of special unity group?

Is it consequently possible to e.g. describe an electron in terms of sets? Not that it would be useful for practical science purposes, but in principle?

## Particles as sets?

**Moderators:** gmalivuk, Moderators General, Prelates

### Particles as sets?

Forrest Cameranesi, Geek of All Trades

"I am Sam. Sam I am. I do not like trolls, flames, or spam."

The Codex Quaerendae (my philosophy) - The Chronicles of Quelouva (my fiction)

"I am Sam. Sam I am. I do not like trolls, flames, or spam."

The Codex Quaerendae (my philosophy) - The Chronicles of Quelouva (my fiction)

- doogly
- Dr. The Juggernaut of Touching Himself
**Posts:**5431**Joined:**Mon Oct 23, 2006 2:31 am UTC**Location:**Lexington, MA-
**Contact:**

### Re: Particles as sets?

Sure, but sets don't have nice properties. Their only property is the number of things in them. It is a lossy reduction.

LE4dGOLEM: What's a Doug?

Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.

Or; Is that your eye butthairs?

Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.

Or; Is that your eye butthairs?

### Re: Particles as sets?

Also the sets involved would be absurdly complicated.

For instance, real numbers are usually defined as either equivalence classes of Cauchy sequences of rational number or of Dedekind (sp?) cuts or real numbers. If we go with the Dedekind cut definition, then every real number is actually a subset of the rational numbers.

The rational numbers are usually defined as equivalence classes of fractions of integers, with positive integers. So each rational number is really a subset of Z cross N.

The integers can be constructed as equivalence classes of ordered pairs of natural numbers. So each integer is really a subset of N cross N.

The natural numbers (inc zero) can be constructed as nested sets with 0 = {}, 1 = {0} = {{}}, 2 = {1}={{{}}}, etc. Or other similar constructions.

So a real number is a subset of a set consisting of subsets of the cartesian product of (subsets of N cross N) and N. Where N is a set consisting of nested sets terminating with the empty set.

That is just to get to a number like Sqrt(2).

To get complex numbers, add another layer, vector spaces, another layer, a manifold several more layers, function on a manifold, several more layers, by the time you get to gauge theory or some other mathematical description of an electron. The stack of sets is absurd.

For instance, real numbers are usually defined as either equivalence classes of Cauchy sequences of rational number or of Dedekind (sp?) cuts or real numbers. If we go with the Dedekind cut definition, then every real number is actually a subset of the rational numbers.

The rational numbers are usually defined as equivalence classes of fractions of integers, with positive integers. So each rational number is really a subset of Z cross N.

The integers can be constructed as equivalence classes of ordered pairs of natural numbers. So each integer is really a subset of N cross N.

The natural numbers (inc zero) can be constructed as nested sets with 0 = {}, 1 = {0} = {{}}, 2 = {1}={{{}}}, etc. Or other similar constructions.

So a real number is a subset of a set consisting of subsets of the cartesian product of (subsets of N cross N) and N. Where N is a set consisting of nested sets terminating with the empty set.

That is just to get to a number like Sqrt(2).

To get complex numbers, add another layer, vector spaces, another layer, a manifold several more layers, function on a manifold, several more layers, by the time you get to gauge theory or some other mathematical description of an electron. The stack of sets is absurd.

- Eebster the Great
**Posts:**3086**Joined:**Mon Nov 10, 2008 12:58 am UTC**Location:**Cleveland, Ohio

### Re: Particles as sets?

Nicias wrote:The natural numbers (inc zero) can be constructed as nested sets with 0 = {}, 1 = {0} = {{}}, 2 = {1}={{{}}}, etc. Or other similar constructions.

You want to construct the natural numbers in such a way that |n| = n, so a more typical construction would be 0={}, 1={0}, 2={0,1}, 3={0,1,2}, etc.

- thoughtfully
**Posts:**2253**Joined:**Thu Nov 01, 2007 12:25 am UTC**Location:**Minneapolis, MN-
**Contact:**

### Re: Particles as sets?

It's worth pointing out that the quest to ground all mathematics in Set Theory ultimately failed.

- gmalivuk
- GNU Terry Pratchett
**Posts:**26453**Joined:**Wed Feb 28, 2007 6:02 pm UTC**Location:**Here and There-
**Contact:**

### Re: Particles as sets?

thoughtfully wrote:It's worth pointing out that the quest to ground all mathematics in Set Theory ultimately failed.

Where does that page talk about set theory?

- thoughtfully
**Posts:**2253**Joined:**Thu Nov 01, 2007 12:25 am UTC**Location:**Minneapolis, MN-
**Contact:**

### Re: Particles as sets?

You're right. Whitehead and Russell were attempting to unify all of mathematics (or at least point in a direction they believed would work), but not using Set Theory, which I was mistaken about. Later mathematicians projected Set Theory onto their work. Gödel did come along and crash the party, though.

- gmalivuk
- GNU Terry Pratchett
**Posts:**26453**Joined:**Wed Feb 28, 2007 6:02 pm UTC**Location:**Here and There-
**Contact:**

### Re: Particles as sets?

I'm still not sure what you think that page or unification mean.

Solving or proving all of math isn't the same as unifying it.

Solving or proving all of math isn't the same as unifying it.

### Who is online

Users browsing this forum: No registered users and 8 guests