Question 1
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Key Idea
A subset is a set whose members all fit inside another set.
Logic and Foundations Quiz
Logic, proof, and limits of formal reasoning.
Answer five questions about Logic and Foundations and get instant feedback.
Question 2
What axiom says you can pick one item from each set, even if there are infinitely many sets?
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Key Idea
It is equivalent to surprising results like the well-ordering theorem, and it can even imply counterintuitive things like the Banach-Tarski paradox, where a ball can be split and reassembled into two.
Question 3
A step-by-step build-up of a conclusion from earlier steps
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Key Idea
In proof theory, a derivation is often pictured as a tree: the leaves are assumptions or axioms, and the root is the final conclusion you have justified.
Question 4
If a set has $n$ elements, then the power set has $2^n$ elements.
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Key Idea
Every element either appears or does not appear in a subset, so $n$ independent yes-no choices give $2^n$ subsets, for example 3 elements yield 8.
Question 5
What theorem shows that any strong enough consistent system misses some arithmetic truths?
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Key Idea
Gödel builds a self-referential arithmetic sentence $G$ that effectively says "I am not provable here", so if the system is consistent then $G$ is true but unprovable.
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