Natural numbers

Definition

Natural numbers are the set of positive integers starting from 1 and going upwards (1, 2, 3, ...). They are fundamental in mathematics for counting and ordering, serving as the basis for more complex number systems. Their significance spans various concepts, including set size comparisons, countability, well-ordering properties, and the establishment of total orders.

5 Must Know Facts For Your Next Test

1. Natural numbers are denoted by the symbol $$ extbf{N}$$ and start from 1, continuing indefinitely.
2. They are used to define countable sets, which can be put into a one-to-one correspondence with natural numbers.
3. Cantor's theorem shows that there are different sizes of infinity by demonstrating that the set of natural numbers is countably infinite.
4. The Well-Ordering Theorem states that every non-empty set of natural numbers has a least element, reinforcing their ordered structure.
5. Natural numbers form a total order under the standard 'less than' relation, meaning any two natural numbers can be compared.

Review Questions

• How do natural numbers relate to the concept of comparing set sizes and Cantor's theorem?
  • Natural numbers are essential in comparing set sizes because they help define what it means for a set to be countable. Cantor's theorem shows that while the set of natural numbers is countably infinite, there exist larger sets that cannot be matched with them. This reveals that not all infinities are equal in size and illustrates the distinction between countable and uncountable sets.
• Discuss how natural numbers support the properties of countable sets and what implications this has in mathematics.
  • Natural numbers serve as a benchmark for determining whether other sets are countable. A set is considered countable if its elements can be matched with natural numbers in a one-to-one manner. This property has significant implications in various areas of mathematics, including analysis and set theory, as it allows mathematicians to classify different types of infinities and establish relationships between sets.
• Evaluate the role of natural numbers in establishing total orders and well-orders within mathematical frameworks.
  • Natural numbers play a crucial role in creating both total orders and well-orders. They establish a total order through the standard 'less than' relationship, ensuring that every pair of natural numbers can be compared. The Well-Ordering Theorem specifically asserts that any non-empty subset of natural numbers has a least element, reinforcing their well-ordered nature. This property is foundational for many mathematical proofs and concepts related to induction and recursive definitions.

Related terms

Whole numbers: Whole numbers are a set that includes all natural numbers along with zero (0, 1, 2, 3, ...).

Cardinality: Cardinality refers to the size or number of elements in a set, which helps compare the sizes of different sets.

Ordinal numbers: Ordinal numbers indicate position or order in a sequence (1st, 2nd, 3rd, ...), unlike natural numbers which represent quantity.