A student claims: 'ℵ₁ is just another name for the cardinality of the real numbers — they're defined to be the same thing.' What is wrong with this claim?
ANothing is wrong — ℵ₁ and |ℝ| are equal by definition in standard set theory
Bℵ₁ is defined as the smallest uncountable cardinal; whether |ℝ| = ℵ₁ is the continuum hypothesis, which is independent of ZFC
CThe student has the direction reversed — |ℝ| = ℵ₀ and ℵ₁ is strictly larger than the reals
Dℵ₁ is not well-defined without additional axioms beyond ZFC
ℵ₁ is defined purely as the cardinal successor of ℵ₀ — the smallest uncountable cardinal, constructed as the cardinality of the set ω₁ of all countable ordinals. The cardinality of the reals is 2^{ℵ₀}, which is provably uncountable but whose position in the aleph hierarchy is not determined by ZFC alone. The continuum hypothesis (CH) is precisely the claim that 2^{ℵ₀} = ℵ₁. Gödel and Cohen together proved CH is independent of ZFC — it can neither be proved nor disproved. Defining ℵ₁ as |ℝ| conflates a definition with an open mathematical question.
Question 2 Multiple Choice
What role does the axiom of choice play in the relationship between the aleph numbers and all infinite cardinalities?
AThe axiom of choice defines what the aleph numbers are — without it, ℵ₀ does not exist
BUnder the axiom of choice, every infinite cardinal equals ℵ_α for some ordinal α, making the alephs a complete account of all infinite cardinalities
CThe axiom of choice is needed only to define ℵ₁ and beyond; ℵ₀ exists without it
DThe axiom of choice guarantees that 2^{ℵ₀} = ℵ₁
The axiom of choice (AC) guarantees that every set can be well-ordered. This implies that every infinite cardinal is comparable to the ordinals and therefore equals ℵ_α for some ordinal α. Without AC, there can be infinite sets whose cardinality is incomparable to any aleph — 'wild' cardinals that fall outside the hierarchy entirely. With AC, the alephs exhaust all infinite cardinalities: every infinite set has a cardinality that appears somewhere in the sequence ℵ₀, ℵ₁, ℵ₂, .... Option D is false — AC says nothing about where 2^{ℵ₀} sits in the aleph hierarchy.
Question 3 True / False
ℵ₁ is defined as the cardinality of the real number line.
TTrue
FFalse
Answer: False
ℵ₁ is defined as the smallest uncountable cardinal — the cardinality of the set ω₁ of all countable ordinals. The cardinality of the real line is 2^{ℵ₀}, the cardinality of the power set of ℕ. Whether 2^{ℵ₀} = ℵ₁ is the continuum hypothesis, which is independent of ZFC. This is one of the most important conceptual distinctions in set theory: the cardinal successor operation (giving the next cardinal) and the power set operation (giving the cardinality of all subsets) are different constructions that happen to produce the same value only if the continuum hypothesis holds — which we cannot prove or disprove.
Question 4 True / False
Under the axiom of choice, every infinite cardinal is equal to ℵ_α for some ordinal α, so the aleph sequence contains all infinite cardinalities.
TTrue
FFalse
Answer: True
This is one of the most important consequences of the axiom of choice. AC implies the well-ordering theorem: every set can be well-ordered. Well-ordered sets have cardinalities that are aleph numbers (since every well-ordered cardinal is an initial ordinal, and initial ordinals are indexed by the aleph sequence). Therefore, assuming AC, no infinite cardinality 'falls between' alephs or is incomparable to them — the aleph hierarchy is complete and total. Without AC, this fails: there can be infinite cardinals that are incomparable and do not appear in the hierarchy.
Question 5 Short Answer
What is the conceptual difference between ℵ₁ (the cardinal successor of ℵ₀) and 2^{ℵ₀} (the power set cardinal of ℵ₀), and why does the distinction matter?
Think about your answer, then reveal below.
Model answer: ℵ₁ is defined by the successor operation on cardinals: it is the smallest cardinal strictly greater than ℵ₀, constructed as the cardinality of all countable ordinals. 2^{ℵ₀} is defined by the power set operation: it is the cardinality of the set of all subsets of a countably infinite set (equivalently, the cardinality of the real numbers). These are two genuinely different mathematical operations — successor and power set — that produce different cardinals in general. Whether they happen to produce the same cardinal (ℵ₁ = 2^{ℵ₀}) is the continuum hypothesis, proved by Gödel and Cohen to be independent of ZFC. The distinction matters because conflating them assumes the answer to an open (undecidable) question.
The continuum hypothesis is undecidable precisely because the successor operation and the power set operation are conceptually distinct. You can build models of set theory where 2^{ℵ₀} = ℵ₁ (Gödel's constructible universe L) and models where 2^{ℵ₀} = ℵ₂, ℵ₃, or even larger alephs (Cohen's forcing models). In all these models, ℵ₁ is still the smallest uncountable cardinal; it's just that the power set of ℕ lands at different positions in the aleph hierarchy depending on the model.