Is everything in the Naive Set Theory included in the Axiomatic Set Theory?

Question

I am trying to understand if Naive Set Theory (NST) should be understood as a "core" for Axiomatic Set Theory (AST).

Is everything (all data) included in NST included in AST?
Would NST be the "software core" of AST?

Answer 1

If anything, it’s probably the other way around. Axiomatic set theory posits some very specific set existence principles, whereas the Comprehension schemata of Naive set theory form a larger range of sets using more general notions of predicate definitions.

It might be useful to think of AST as defining a minimal base set of objects satisfying “pure” set theory, and Naive set theory as patching Set theory to define sets within natural language operating conditions.

This patch doesn’t always work as expected, and sometimes unless you weaken your background operating logic the patch breaks the system.

Answer 2

Short Answer

The relationship between naive and formal set theory is one primarily concerned with foundationalism. How do you know that the set theory you're doing is logically consistent? In fact, Russell's paradox exposed naive set theory as being inconsistent. Uh oh. To deal with that consistency, set theoreticians began devising rules so that the paradox could be avoided. This led to the infamous ZF(C) axioms of formal theory (note objection below and see MathOverflowSE: Can we prove set theory is consistent?).

Long Answer

Wir müssen wissen.
Wir werden wissen.

(We must know.
We will know.)

— David Hilbert

When one does naive set theory, one says a set is a collection of objects. Some objects fit in others. But are there any circumstances that this causes problems in reasoning? The answer is a definitive answer. If one asks, "can I put the collection in itself", one goes down a road and arrives at Russell's paradox. It turns out that if you define a set to exclude itself, you have a problem. Formally:

R:={x:x∉x} → (R∈R ⇔ R∉R)

which says that if you define a set such that it contains all things that don't contain themselves, then if you consider whether or not the set is a member of itself, it creates the contradiction. Suddenly, mathematicians realized they needed to start figuring out what logical presumptions it takes to avoid these sticky-wickets.

If you want to use a computer metaphor, think of a functional API called APISets. Let's say APISets has a function call called DefineMember() and you make a call:

DefineMember(R,x='red')

Now, no problems, right?

R.Contains('red apple') ⇒ T
R.Contains('human blood') ⇒ T
R.Contains('blue paint') ⇒ F

But what happens when you do this?

DefineMember(R,'x∉x') // Note this is a recursive definition!!! Oh, boy.

What is the result of this call?

R.Contains(R)

Uh, oh. Problems. So, to avoid problems, the method of DefineMember must have exclusions and other rules when interacting with related data and methods. These are what the axioms are.

See Also

PhilSE: What is Naive Set Theory?
PhileSE: Is a set containing itself already a paradox?