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The Philosophy of Set Theory

Description: The Philosophy of Set Theory Quiz
Number of Questions: 15
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Tags: set theory mathematical philosophy foundations of mathematics
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What is the Zermelo-Fraenkel set theory?

  1. A set of axioms used to define the concept of a set

  2. A method for constructing sets

  3. A theory of the relationship between sets and other mathematical objects

  4. A set of rules for manipulating sets

Show answer
Correct Option: A
Explanation:

The Zermelo-Fraenkel set theory is a set of axioms that are used to define the concept of a set. It is the most widely accepted set theory and is used as the foundation for most of modern mathematics.

What is the axiom of extensionality?

  1. Two sets are equal if and only if they have the same elements

  2. Two sets are equal if and only if they have the same number of elements

  3. Two sets are equal if and only if they are constructed in the same way

  4. Two sets are equal if and only if they are both empty

Show answer
Correct Option: A
Explanation:

The axiom of extensionality is one of the axioms of Zermelo-Fraenkel set theory. It states that two sets are equal if and only if they have the same elements.

What is the axiom of regularity?

  1. Every set is a member of another set

  2. Every set is a subset of another set

  3. Every set is a proper subset of another set

  4. Every set is a disjoint set

Show answer
Correct Option: A
Explanation:

The axiom of regularity is one of the axioms of Zermelo-Fraenkel set theory. It states that every set is a member of another set.

What is the axiom of choice?

  1. For any collection of non-empty sets, there exists a function that selects an element from each set

  2. For any collection of non-empty sets, there exists a set that contains exactly one element from each set

  3. For any collection of non-empty sets, there exists a set that contains all of the elements from all of the sets

  4. For any collection of non-empty sets, there exists a set that is disjoint from all of the sets

Show answer
Correct Option: A
Explanation:

The axiom of choice is one of the axioms of Zermelo-Fraenkel set theory. It states that for any collection of non-empty sets, there exists a function that selects an element from each set.

What is the continuum hypothesis?

  1. The cardinality of the set of real numbers is equal to the cardinality of the set of natural numbers

  2. The cardinality of the set of real numbers is equal to the cardinality of the set of integers

  3. The cardinality of the set of real numbers is equal to the cardinality of the set of rational numbers

  4. The cardinality of the set of real numbers is equal to the cardinality of the set of irrational numbers

Show answer
Correct Option: A
Explanation:

The continuum hypothesis is a statement in set theory that states that the cardinality of the set of real numbers is equal to the cardinality of the set of natural numbers.

What is the Löwenheim-Skolem theorem?

  1. Any first-order theory that has a model has a model of any infinite cardinality

  2. Any first-order theory that has a model has a model of any finite cardinality

  3. Any first-order theory that has a model has a model of any countable cardinality

  4. Any first-order theory that has a model has a model of any uncountable cardinality

Show answer
Correct Option: A
Explanation:

The Löwenheim-Skolem theorem is a theorem in set theory that states that any first-order theory that has a model has a model of any infinite cardinality.

What is the Gödel-Bernays set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  3. A set theory that is based on the Morse-Kelley set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The Gödel-Bernays set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the Gödel-Bernays set theory.

What is the von Neumann-Bernays-Gödel set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the Morse-Kelley set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The von Neumann-Bernays-Gödel set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the von Neumann-Bernays-Gödel set theory.

What is the Morse-Kelley set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The Morse-Kelley set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the Morse-Kelley set theory.

What is the Tarski-Grothendieck set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  4. A set theory that is based on the Morse-Kelley set theory

Show answer
Correct Option: A
Explanation:

The Tarski-Grothendieck set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the Tarski-Grothendieck set theory.

What is the Zermelo-Fraenkel-Skolem set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The Zermelo-Fraenkel-Skolem set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the Zermelo-Fraenkel-Skolem set theory.

What is the New Foundations set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The New Foundations set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the New Foundations set theory.

What is the Quine set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The Quine set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the Quine set theory.

What is the Scott-Potter set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The Scott-Potter set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the Scott-Potter set theory.

What is the Zermelo-Fraenkel-Mostowski set theory?

  1. A set theory that is based on the Zermelo-Fraenkel set theory

  2. A set theory that is based on the Gödel-Bernays set theory

  3. A set theory that is based on the von Neumann-Bernays-Gödel set theory

  4. A set theory that is based on the Tarski-Grothendieck set theory

Show answer
Correct Option: A
Explanation:

The Zermelo-Fraenkel-Mostowski set theory is a set theory that is based on the Zermelo-Fraenkel set theory. It is a conservative extension of the Zermelo-Fraenkel set theory, which means that all of the theorems that can be proved in the Zermelo-Fraenkel set theory can also be proved in the Zermelo-Fraenkel-Mostowski set theory.

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