Which of the following correctly states the relationship between ℕ (the natural numbers) and P(ℕ)?
AThey have the same cardinality, since both are infinite sets
BP(ℕ) is larger, but it is still countably infinite
CP(ℕ) is uncountably infinite — strictly larger than ℕ — and has the same cardinality as ℝ
DP(ℕ) is exactly twice as large as ℕ, since each natural number either is or is not in a subset
By Cantor's theorem, there is no surjection from any set onto its power set, so |P(ℕ)| > |ℕ|. Since ℕ is countably infinite, P(ℕ) is uncountably infinite. Moreover, each subset S ⊆ ℕ corresponds to an infinite binary sequence (1 if n ∈ S, 0 if not), and these sequences biject with real numbers via binary expansion. Options A and B err by treating all infinities as equal; option D confuses the binary encoding with a size ratio.
Question 2 Multiple Choice
Set A = {1, 2, 3}. How many elements does P(A) contain, and does A itself appear as an element of P(A)?
A7 elements; A does not appear in P(A) because A is not a proper subset of itself
B8 elements; yes, A ∈ P(A) because A ⊆ A
C6 elements; only proper subsets are included in P(A)
D8 elements; no, A ∉ P(A) because A is the original set, not one of its own subsets
|P(A)| = 2³ = 8, including ∅, the three singletons, the three pairs, and {1,2,3} itself. Crucially, every set is a subset of itself (A ⊆ A), so A ∈ P(A). This is not the same as A ∈ A (self-membership, barred by the axiom of regularity). The power set contains all subsets — proper and improper — including ∅ and A itself.
Question 3 True / False
The power set of ℕ has the same cardinality as the set of real numbers ℝ.
TTrue
FFalse
Answer: True
Each subset S ⊆ ℕ encodes as an infinite binary sequence: position n is 1 if n ∈ S, 0 otherwise. Infinite binary sequences are exactly binary expansions of real numbers in [0,1], giving a bijection between P(ℕ) and (essentially) ℝ. So |P(ℕ)| = 2^{ℵ₀} = |ℝ|. The power set axiom, applied to ℕ, delivers the continuum.
Question 4 True / False
The power set axiom in ZFC specifies which subsets of a given set exist by providing a rule for constructing them.
TTrue
FFalse
Answer: False
The power set axiom only asserts existence — that the collection of all subsets exists as a set. It says nothing about what those subsets are or how to construct them. The axiom of separation is what lets you identify specific subsets via a property. The power set axiom's controversial impredicative character comes precisely from collecting all subsets at once without specifying them.
Question 5 Short Answer
Why does the power set axiom produce strictly larger sets for infinite inputs, while the axiom of separation cannot?
Think about your answer, then reveal below.
Model answer: The axiom of separation can only extract a subset from a set you already have — the result is never larger than the input. The power set axiom collects all subsets into a new set, which by Cantor's theorem is always strictly larger than the original. It's generative rather than selective.
Separation gives {x ∈ A : φ(x)}, which is at most as large as A. The power set gives P(A), whose cardinality exceeds |A| by Cantor's diagonal argument: no function from A to P(A) can be surjective. This is what allows ZFC to produce uncountable sets (P(ℕ)) from countable ones (ℕ), and sets larger than ℝ by iterating the operation.