What set is in another class of infinity that is bigger than N?

Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter9: Real Numbers And Right Triangles
Section9.3: The Pythagorean Theorem
Problem 21E
icon
Related questions
Question
**Question:**

What set is in another class of infinity that is bigger than ℕ?

**Explanation:**

In the theory of infinite sets, the natural numbers (denoted by ℕ) form an infinite set. However, there are larger classes of infinity. For example, the set of real numbers (denoted by ℝ) is a larger infinity than the set of natural numbers. This concept was first introduced by the mathematician Georg Cantor, and it is explored in detail within the framework of set theory.

Cantor showed that the cardinality (size) of the set of real numbers is strictly greater than the cardinality of the natural numbers. This means there are different "sizes" or "types" of infinity. The cardinality of the natural numbers is denoted by ℵ₀ (aleph-null), and the cardinality of the real numbers is denoted by c (the cardinality of the continuum). Cantor’s diagonal argument provides a proof that ℵ₀ < c, establishing that there are more real numbers than natural numbers, even though both sets are infinite.
Transcribed Image Text:**Question:** What set is in another class of infinity that is bigger than ℕ? **Explanation:** In the theory of infinite sets, the natural numbers (denoted by ℕ) form an infinite set. However, there are larger classes of infinity. For example, the set of real numbers (denoted by ℝ) is a larger infinity than the set of natural numbers. This concept was first introduced by the mathematician Georg Cantor, and it is explored in detail within the framework of set theory. Cantor showed that the cardinality (size) of the set of real numbers is strictly greater than the cardinality of the natural numbers. This means there are different "sizes" or "types" of infinity. The cardinality of the natural numbers is denoted by ℵ₀ (aleph-null), and the cardinality of the real numbers is denoted by c (the cardinality of the continuum). Cantor’s diagonal argument provides a proof that ℵ₀ < c, establishing that there are more real numbers than natural numbers, even though both sets are infinite.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Finite State Machine
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Holt Mcdougal Larson Pre-algebra: Student Edition…
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
College Algebra
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning