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

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?

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– J D
Commented Oct 31, 2020 at 4:34
– J D
Commented Oct 31, 2020 at 4:37
• How about you give the possible duplicate a read, and if it doesn't answer your quesitons or point you in the direction, let us know.
– J D
Commented Oct 31, 2020 at 4:38
• It does not answer my question but currently I recognize no way of editing my question; perhaps I should just avoid that "software core" comparison. Thank you. Commented Oct 31, 2020 at 4:40
• The answer to "everything" is a trivial no. Naive set theory is informal, and while its intuitions motivate axiomatic theories none of them can capture them all. For one thing, intuitions are incoherent, so one has to choose what to capture, and informal theories mix base level and meta level arguments, while axiomatic theories can not do that. As for the "core", what that is is too vague and subjective to say anything cogent. Commented Oct 31, 2020 at 4:51

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.

• Fascinating +1. Do you have any references that might introduce me to your definition of "pure set theory"?
– J D
Commented Oct 31, 2020 at 19:33
• Thanks @JD, Philip Welch has some great notes online (people.maths.bris.ac.uk/~mapdw/current-axiomatic-set-theory.pdf) - AST (particularly theories of the form ZFC + Φ) gives us a formal framework to discuss theories of the set- theoretic hierarchy V, in contrast with Descriptive set theory whose main aim is to consider sets of particular mathematical interest (eg. the Reals, analytical functions etc). The intuition behind V is a compositional hierarchy of sets extending the Empty set as the main starting point - a minimal definition of an interesting basic collection of sets. Commented Oct 31, 2020 at 20:42
• Awesome! I wanted to compliment your use of the phrase "more general notions of predicate definitions" because it hadn't occurred to me that one might actually use FOPC to provide a rigorous analysis between the essence of more and less formal predications of set theory. That's thought provoking. Time to move you up the ladder.
– J D
Commented Nov 1, 2020 at 0:36
• I think Alan Weir made an interesting observation that our naive concept of Set is basically the same thing as a Natural Kind or the Extension of a concept - so really, that’s exactly what FOPC was made to do! Commented Nov 1, 2020 at 13:09

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?).

Wir müssen wissen.
Wir werden wissen.

(We must know.
We will know.)

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.

• Not a very good answer, since we can't prove ZFC is consistent without assuming the consistency of even stronger systems. Commented Oct 31, 2020 at 16:51
• @user4894 Not a very good criticism: 1) Nowhere did I assert that (you did actually read the post, right?) 2) If he doesn't know the difference between naive and formal models, how in Zeus' name am I supposed to explain metalanguages, model theory, and Goedelian objections? (You understand he is a beginner, right?). Questions, comments, and feedback always welcomed! ; )
– J D
Commented Oct 31, 2020 at 19:05
• I think I see where you read an implication not present. I'll clarify.
– J D
Commented Oct 31, 2020 at 19:22
• Also put in a link to a MathOverflow article I scanned. He shouldn't have problems getting into turnstiles now, right? :D Feel free to tweak my language if you find my clarification misses the mark. I'd rather have a good answer than hoard credit for contributions.
– J D
Commented Oct 31, 2020 at 19:26
• @user4894 Better yet, why not provide a good answer so I can learn from your interpretation? I'm here to learn, after all! :D
– J D
Commented Oct 31, 2020 at 19:31