In set theory, 2ω refers to the cardinality of the power set of the natural numbers, or the set of all subsets of the natural numbers. This notation emphasizes the relationship between ordinal numbers and cardinal numbers, showing how infinite sets can be described in terms of their sizes and structures. It highlights the distinction between countable and uncountable sets, where 2ω represents a larger infinity than that of the natural numbers, denoted by ω.
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2ω represents an uncountably infinite set, meaning it cannot be matched one-to-one with the natural numbers.
The notation is derived from considering the power set of ω, which is the first infinite ordinal.
2ω is a larger cardinality than ω, as shown by Cantor's theorem, which states that for any set, the power set has a strictly greater cardinality.
The cardinality 2ω is commonly used in discussions about the continuum hypothesis and its implications for set theory.
In terms of sizes, 2ω is equal to the cardinality of the real numbers, which is often denoted as c (the cardinality of the continuum).
Review Questions
How does 2ω illustrate the difference between countable and uncountable infinities?
2ω illustrates this difference by representing an uncountable infinity, specifically the size of the power set of natural numbers. While countable infinities like ω can be put into a one-to-one correspondence with natural numbers, 2ω cannot. This distinction is crucial in understanding different sizes of infinity and shows how power sets yield larger cardinalities than their original sets.
What role does Cantor's theorem play in understanding the significance of 2ω in set theory?
Cantor's theorem states that for any set S, its power set P(S) has a greater cardinality than S itself. Applying this theorem to ω shows that 2ω, representing the power set of natural numbers, is a larger cardinality than ω. This underscores the concept that there are infinitely many different sizes of infinity and highlights the profound implications for our understanding of set theory and infinite sets.
Evaluate the implications of 2ω regarding the continuum hypothesis and its relation to real numbers.
The implications of 2ω regarding the continuum hypothesis center on whether there is a cardinality between that of the integers (ω) and the real numbers (2ω). The continuum hypothesis posits that there is no such cardinality; if true, it implies that 2ω equals c, the cardinality of real numbers. This relationship invites deep discussions about foundational questions in mathematics, including how we understand infinity and structure within sets.