Dick said:You probably know how to show that the set of all infinite sequences of 0's and 1's is uncountable by a diagonal argument. Can you think of a bijection between infinite sequences of increasing natural numbers and infinite sequences of 0's and 1's?
mtayab1994 said:That's kind of what I was trying to tell him.
An increasing sequence of natural numbers is a sequence in which each term is larger than the previous one. For example, 1, 2, 3, 4, 5 is an increasing sequence of natural numbers.
To prove that increasing sequences of natural numbers are uncountable, we can use the diagonalization argument. This involves assuming that there is a countable number of terms in the sequence and then constructing a new term that is not in the sequence, thus showing that the sequence is actually uncountable.
This result is significant because it demonstrates that there are infinite sets that are larger than the set of natural numbers. It also has implications in other areas of mathematics, such as set theory and analysis.
One example of an increasing sequence of natural numbers that is uncountable is the sequence of powers of 2: 1, 2, 4, 8, 16, 32, ... This sequence is uncountable because it is infinite and each term is larger than the previous one.
Yes, there is a strong connection between these two concepts. Proving that increasing sequences of natural numbers are uncountable demonstrates that there are infinitely many numbers that cannot be counted, which is a fundamental characteristic of infinity.