ChatGPT: Is it possible to axiomatically construct a set theory in which not all subsets of a countable set are countable?
Yes, it is possible to construct a set theory in which not all subsets of a countable set are countable, depending on what axioms you adopt — especially regarding the Axiom of Choice (AC) and Power Set Axiom, and how you interpret "countable."
Let's unpack this carefully.
First, some context and definitions:
A set is countable if it's finite or has the same cardinality as the natural numbers N.
The set N is countable (obviously).
In standard ZFC (Zermelo–Fraenkel with Choice), there exist uncountable subsets of N only if we're talking about subsets of the power set of N — i.e., the power set P(N) is uncountable.
But every individual subset of N is countable (being a subset of a countable set).
What does your question really ask?
You’re asking whether it’s possible to axiomatize a set theory where:
N is countable, but
Not all of its subsets are countable, meaning some subsets of a countable set (like N) are uncountable.
This would mean that the set of all subsets of N includes uncountable subsets, despite N being countable.
In ZFC, this is not the case — all subsets of N are themselves at most countable, even though P(N) (the full power set) is uncountable.
The answer is YES, but with caveats.
You can build alternative set theories (often called non-classical set theories) where such a phenomenon could occur, but they require changing foundational assumptions.
Some options:
1. Dropping the Axiom of Choice (ZF instead of ZFC):
In ZF (without Choice), it's possible that certain subsets of a countable set are not explicitly constructible or not provably countable.
Without AC, "every subset of a countable set is countable" is not provable.
There are models of ZF in which there exists a Dedekind-infinite set (i.e., infinite in the sense of injections) that is not countably infinite in the usual sense.
So some subsets of N might not be countable in such a model.
Still, each such subset is at most countable in cardinality — it's just that you can't construct a bijection with N for it.
2. Alternative Set Theories (e.g., NFU, Constructive Set Theory, etc.):
In New Foundations with Urelements (NFU) or Constructive Zermelo–Fraenkel (CZF) set theory, different behavior can emerge.
For instance, in Constructive Set Theory, not all subsets of N are necessarily sets in the traditional sense — so some may not be countable (constructively).
These theories reject or weaken the Power Set Axiom and/or the Axiom of Choice, allowing unusual behaviors with subsets of countable sets.
3. Nonstandard Models / Forcing / Relative Consistency Results:
You can use forcing (from model theory) to construct models of ZF (or ZFC) in which the set of reals is countable, or in which we can’t well-order all subsets of N.
In these models, N remains countable, but P(N) can behave strangely.
You can force models in which certain subsets of N are not countable in the model, even though they are externally.
4. Constructive or Predicative Mathematics:
In predicative mathematics or intuitionistic logic, certain subsets of N might not be countable because they aren't constructible in a specific sense. You can even say that not all subsets exist as completed totalities — so uncountability emerges for a different reason.
Summary:
Yes — in certain axiomatic systems that differ from classical ZFC, it's possible to construct a theory in which some subsets of a countable set (like N) are uncountable, especially if you reject or modify:
The Axiom of Choice,
The Power Set Axiom,
Or classical logic itself.
Would you like to explore an example of such a set theory or a model where this happens?
Awesome — let's dive into a concrete example. One of the most interesting and accessible examples comes from constructive set theory, especially when paired with intuitionistic logic and a rejection of the full Power Set Axiom. A good starting point is:
Example: Subsets of N in Constructive Zermelo–Fraenkel Set Theory (CZF)
In CZF, we do not assume:
The Power Set Axiom, and
The Law of Excluded Middle (so we use intuitionistic logic instead of classical logic).
As a result, the landscape of subsets of N changes dramatically.
What's Different in CZF?
In CZF:
A set of natural numbers A⊆NA⊆N is only a set if it can be explicitly constructed — e.g., by giving a rule or generating procedure.
The total collection of all subsets of N, P(N), does not exist as a set — it's only a proper class (or a "collection").
So while each individual subset may still be countable, there isn't a set of all of them, and not all of them are constructible.
Uncountable collections of subsets of N can exist as classes that do not have a well-defined cardinality.
Key Consequence:
There exist subsets of N that:
Are not enumerable in the constructive sense,
Cannot be shown to be countable (i.e., no constructively valid bijection to N),
And yet are subsets of a countable set.
So, from a classical ZFC perspective: "Of course they're countable!"
But from a constructive CZF viewpoint, these subsets are not countable, since there’s no constructive method to enumerate them.
Example of a Non-countable Subset in CZF
Let’s consider the set:
A={n∈N∣P(n)}
Where P(n) is some non-decidable property — like:
"The Turing machine with code n halts on input n" (i.e., the Halting Problem).
In classical logic, this still defines a subset of N, even if we can't compute it. But in constructive logic, we can’t prove that such a set is countable because we lack a constructive method to list its elements.
Hence, in CZF:
The set A exists as a subclass of N,
But it is not countable — because we can't construct a surjection from N to A,
Even though it’s a subset of a countable set.
The Technical Language
In CZF, “countable” typically means "inhabited set with a surjection from N" (constructively given). So if such a function cannot be constructed, the set is not countable — even if it’s a subset of N.
This creates a model where:
N is countable
Not all subsets of N are countable (in the constructive sense)
First, some context and definitions:
What does your question really ask?
The answer is YES, but with caveats.
1. Dropping the Axiom of Choice (ZF instead of ZFC):
Summary:
What's Different in CZF?
Key Consequence:
Example of a Non-countable Subset in CZF
The Technical Language
Not all subsets of N are countable (in the constructive sense)