Particles as sets?

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Pfhorrest
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Particles as sets?

Postby Pfhorrest » Thu Apr 19, 2018 7:29 am UTC

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?
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Re: Particles as sets?

Postby doogly » Thu Apr 19, 2018 12:02 pm UTC

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

Postby Nicias » Thu Apr 19, 2018 1:31 pm UTC

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.

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Re: Particles as sets?

Postby Eebster the Great » Fri Apr 20, 2018 12:38 am UTC

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.

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Re: Particles as sets?

Postby thoughtfully » Sat May 26, 2018 9:39 am UTC

It's worth pointing out that the quest to ground all mathematics in Set Theory ultimately failed.
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Re: Particles as sets?

Postby gmalivuk » Sat May 26, 2018 2:58 pm UTC

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?
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Re: Particles as sets?

Postby thoughtfully » Sat May 26, 2018 4:20 pm UTC

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.
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Re: Particles as sets?

Postby gmalivuk » Sun May 27, 2018 3:01 am UTC

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.
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