Practice: Russell's paradox and why sets need axioms

Russell's paradox and why sets need axioms

Warm-upHuman

A village barber is supposed to shave exactly those villagers who do not shave themselves. What does the barber argument actually prove?

Choose one

No village can have such a barber.The barber must be a woman, or live outside the village.Logic is inconsistent.The barber both shaves and does not shave himself.

Russell's paradox and why sets need axioms

CoreHuman

Which axiom of ZF set theory is responsible for blocking Russell's paradox?

Choose one

Separation:

\{x \in A : P(x)\}

exists only inside an existing set

A

Foundation, which implies that no set is an element of itselfExtensionality: sets with the same elements are equalThe axiom of choice

Russell's paradox and why sets need axioms

CoreHuman

Let $A = \{\varnothing, \{\varnothing\}\}$ . How many elements does $R_A = \{x \in A : x \notin x\}$ have?

Russell's paradox and why sets need axioms

CoreHuman

In the proof of Theorem 4 we have $x \in R_A \iff (x \in A \text{ and } x \notin x)$ . We suppose $R_A \in A$ and put $x = R_A$ . Why does the condition “ $x \in A$ ” drop out?

Choose one

Because we assumed

R_A \in A

, so “

R_A \in A

” is true.Because every set is an element of

A

.Because of the axiom of foundation.Because

R_A

is defined without mentioning

A

.

Russell's paradox and why sets need axioms

CoreHuman

On the objects $a, b, c$ , a relation $E$ holds exactly for: $a \mathrel{E} a$ , $a \mathrel{E} b$ , $b \mathrel{E} c$ , $c \mathrel{E} c$ , $c \mathrel{E} a$ . List the elements of the Russell set $D = \{x : \text{not } (x \mathrel{E} x)\}$ .

Russell's paradox and why sets need axioms

CoreHuman

Which statement best describes the relationship between Russell's paradox and the liar paradox (“this sentence is false”)?

Choose one

They are analogous — both lead to a statement equivalent to its own negation — but Russell's is a rigorous deduction from a precise principle about sets.They are the same argument in different words.They have nothing in common.Separation solves the liar paradox too.

Russell's paradox and why sets need axioms

CoreFormal

In Mathlib's model of ZFC, which Lean statement says “there is no set of all sets”?

Choose one

¬ ∃ U : ZFSet, ∀ x, x ∈ U∀ U : ZFSet, U ∉ U¬ ∃ U : ZFSet, U ∈ U∃ U : Set α, ∀ x : α, x ∈ U

Russell's paradox and why sets need axioms

StretchComputational

A relation on the objects 0, 1, ..., n-1 is given as a set rel of pairs (x, y), meaning “x E y”. Write russell_set(rel, n), returning the set of x with (x, x) not in rel, and witnesses(rel, n), returning the list of all r whose members {x : (x, r) in rel} are exactly the Russell set. Theorem 2 predicts that witnesses always returns [].

Runs entirely in your browser, in an isolated worker. Nothing is sent to a server.