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April 3rd, 2019, 10:38 AM   #1
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An infinite set of finite sequences of combinations of a finite set of elements

An infinite set of finite sequences of combinations of a finite set of elements is an infinite uncountable set?
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April 3rd, 2019, 10:42 AM   #2
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How does the set get to be infinite if there are only a finite number of members available?
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April 3rd, 2019, 10:46 AM   #3
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No, because the set of hereditarily finite sets is countable.
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April 3rd, 2019, 10:48 AM   #4
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Quote:
Originally Posted by Germann View Post
An infinite set of finite sequences of combinations of a finite set of elements is an infinite uncountable set?
Say the finite set of elements has cardinality $n$.

There are $n$ distinct sequences of length 1.

There are $n^2$ sequences of length 2.

In general there are $n^k$ distinct sequences of length $k$.

The union of countably many finite sets is countable.
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