What is Naive Set Theory?

Question

Is naive set theory, simply set theory that has been left unformalised as the entry in Wikipedia suggests?

However, the SEP, in its entry on inconsistent mathematics, suggests that:

It should also be noted that Brady's construction of naive set theory opens the door to a revival of Frege-Russell logicism, which was widely held, even by Frege himself, to have been badly damaged by the Russell Paradox. If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is reducible to logic via the naive comprehension schema. The only change needed is a move to an inconsistency-tolerant logic.

One should note here that the denial of Explosion, allows the assumption of Russells set - the set of all sets that are not members of themselves, and the universal set - the set of all sets without leading to paradox.

Does naive set theory when assuming a paraconsistent logic:

a. Has a simpler axiomatisation than ZFC?

b. What is the status of Choice in Naive Set Theory

c. Can successfully push through Freges Logicist programme?

Answer 1

You can see in SEP :

Set theory

and also

The early development of set theory

Alternative Axiomatic Set Theories

The "standard" book is Paul Halmos, Naive Set Theory (1960).

From Wikipedia :

"Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally, in natural language."

But you must face the same problems; you need to introduce axioms in order to :

(i) avoid paradoxes

(ii) have at your disposal enough resources to build all the set you need.

Of course, the "standard" approach is with first-order classical logic, but (as you said) you can change it...

Answer 2

One interpretation I've seen of Naive Set theory gives it a little more formal structure than just "set theory done in natural language" (this is the treatment given in, for example, Ross Brady's "The Simple Consistency of a Set Theory Based on the Logic CSQ"). When we want to form a naive set theory in a first order language, we make appeal to a schema of Comprehension axioms which define the set membership operator. That is, our set theory is at root just FOL with the axioms:

∃y.∀x(x ∈ y ↔ ϕx )

(where ϕ is each (definable/applicable) first order predicate applying to variable x, assuming y not free in ϕ)

This will look pretty similar to, for example, Frege's basic law V, but with the distinction that we're making reference to a schema of axioms rather than a single axiom with a second order quantifier, and defining the membership operator rather than giving criteria for set identity.

So for question 1, hell yeah this has a simpler axiomatization than ZFC. And for 3, the principle is certainly amenable to considering as a kind of basic extension of first order classical logic.

Choice might be a little trickier, partly because the mathematics involved in this set theory becomes heavily parasitic on the kinds of ϕs we allow our axiom schema to range over. Therein is the difficulty; how do you formally justify what kinds of exclusions to make in order to avoid the collapse into inconsistency? Classically we certainly can't allow the Russell set to be introduced (let ϕ be ¬(x ∈ x)), so we'd need to restrict the range of axioms somehow.

There are some interesting lines of thought we might go down here that would involve reversing the traditional dependence of model or proof theory in formal logic on set theoretic foundations, but the work to be done is definitely of a philosophically formal character!