Can you define a set by what it excludes?

[G037H3]

The theory of sets which all have a finite number of elements is simple, and doesn't need the idea of "classes".

Classes were defined to sort out the paradoxes that arise with infnite sets. The earliest record of such a paradox (quoted by St Paul in the Bible) is the statement "A Cretan told me that all Cretans are liars." That can't be either true or false, but it looks like a meaningful English sentence, so what's going on here?

At a more abstract level, consider whether "the set of all sets that are not members of themselves" is or is not a set. If it is a set, then by definition it isn't. If it isn't a set, then by definition it is.

One solution to these issues is to carefully define the concept of a set, such that statements like the above are not true or false, but meaningless. The consequence of this is that the idea of "the set of all sets" is meaningless. So you need a new concept called a "class", such that "the class of all sets" is meaningful. See Russell and Whitehead's "Principia Mathematica" for a very great deal more on this. (Warning: it takes them about 1000 pages to get as far as proving from first principles that 1+1=2)

I'll probably read that book sometime in the next year, after I learn a lot more formal logic.

I already know about the universal set issues (in fact, the book I'm using tries to use the label of a 'universal set' U that contains all possible answers to a problem, when really they mean the 'universe').

The solution to issues such as the above is solved by metalogic, obviously.

I stated an issue with 'depth' from before; this is because I was thinking in terms of CS. For instance:

>>> a_list = ['a', 'b', 'c']
>>> a_list.extend(['d', 'e', 'f'])  
>>> a_list
['a', 'b', 'c', 'd', 'e', 'f']
>>> len(a_list)                     
6
>>> a_list[-1]
'f'
>>> a_list.append(['g', 'h', 'i'])  
>>> a_list
['a', 'b', 'c', 'd', 'e', 'f', ['g', 'h', 'i']]
>>> len(a_list)                     
7

As you can see, when using append, the length is of objects at depth 1, the list ['g', 'h', 'i'] is only viewed as a single object. Before knowing of this, I applied similar thinking to the definition of a set within a set.

 

[Jimmy Snyder]

There is a notation (in fact more than one) in common use for defining a set by what it excludes. A = U \ B, where U is the universal set, and B is the set of things to be excluded defines A by what it excludes. An even simpler example is the empty set which can be defined as the set that excludes everything.

 

[Studiot]

I don't know why G037H3 is struck out since this is my first viewing of this thread.

So here is a simple example about your difficulty with the question

Is {R} \equiv R ? (Note I have used the identity symbol, not equals.)

Consider a heap of auto components. An auto contains 1 each of these components in a specific order.

Compare the above question about the reals with the question

Is {one auto} identical to that heap of components?

@Jimmy Snyder

An even simpler example is the empty set which can be defined as the set that excludes everything.

This definition runs into immediate difficulty because the complement of this empty set is the set of everything, which is of course a set of all sets, which some exclude as not a valid set.

 

[disregardthat]

This definition runs into immediate difficulty because the complement of this empty set is the set of everything, which is of course a set of all sets, which some exclude as not a valid set.

The complement with respect to what?