The cardinality of infinite numbers is described by aleph numbers, which are used to measure the size of infinite sets.
Understanding Cardinality and Infinity
Cardinality, in simple terms, refers to the "size" of a set. For finite sets, this is just the count of elements. However, with infinite sets, things get more complex because all infinities are not created equal. The concept of cardinality helps differentiate the sizes of these infinite sets.
Aleph Numbers: Measuring Infinity
- Aleph-null (ℵ₀): This is the smallest infinite cardinal number, representing the cardinality of the set of natural numbers (1, 2, 3...). It can be thought of as the "smallest infinity".
- Higher Alephs (ℵ₁, ℵ₂, ...): The reference states "The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal." This means there exists a sequence of aleph numbers beyond aleph-null (ℵ₀), each representing a different and larger infinity. These higher alephs correspond to the cardinalities of different infinite ordinal numbers and thus, different, increasingly large infinities.
- The Continuum Hypothesis: The cardinality of the real numbers (all the numbers on the number line, including decimals and fractions) is denoted by 'c' and is greater than ℵ₀. The Continuum Hypothesis (CH) states that 'c' is equal to the next aleph number after ℵ₀, which is ℵ₁. However, this hypothesis cannot be proved nor disproved using standard mathematics.
Key Concepts and Examples
- Ordinal Numbers: These numbers represent the well-ordering of a set. For example, the first few ordinal numbers are 0, 1, 2, 3..., but after all the finite ordinals, comes ω (omega) and further infinities.
- Initial Ordinal: For each aleph number, the initial ordinal is the smallest ordinal that has that cardinality.
- Countable Infinity: A set is considered countably infinite if its elements can be put into a one-to-one correspondence with the natural numbers. The cardinality of a countably infinite set is ℵ₀. Examples include integers, rational numbers.
- Uncountable Infinity: Sets that are too large to be put into one-to-one correspondence with the natural numbers are uncountably infinite, such as the real numbers which have a cardinality of 'c' which, if we accept the continuum hypothesis (CH), is ℵ₁.
Table of Cardinalities
| Cardinality | Description | Examples |
|---|---|---|
| ℵ₀ | Countably infinite | Natural numbers, integers, rationals |
| c (or ℵ₁) | Uncountably infinite | Real numbers |
| ℵ₂, ℵ₃,... | Progressively larger infinities | Infinite ordinal numbers |
Summary
The cardinality of infinite numbers is described by aleph numbers, with ℵ₀ representing the smallest infinity and further alephs representing increasingly larger infinities. The set of all real numbers has a higher cardinality, usually denoted by 'c', the value of which is not completely specified but, if the continuum hypothesis is accepted, would be ℵ₁.
