Exploring Infinity: From Countable Integers to the Unbounded Real Numbers
Delve into the realm of infinity, where the concept transcends our finite understanding. Discover how the set of positive integers, even numbers, all integers, positive rational numbers, and real numbers each showcase a distinct level of infinity. Explore the notion of cardinality, bridging the gap between aleph-null and the uncountable real numbers, challenging the hypothesis of the Continuum. These intricate concepts provoke contemplation on the vastness and boundlessness of mathematical infinity.
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There are an infinite number of positive integers but is there a smaller infinity than this? Is there a bigger infinity? The infinity of positive integers is denoted aleph-null ( 0) If a set of numbers can be put in one-one correspondence with the positive integers, it has the same cardinality What about the even numbers?
Even numbers 1 2 2 4 3 6 4 8 etc.
All integers 1 0 2 1 3 -1 4 2 5 -2 etc.
Positive Rational Numbers 1/1 1/2 1/3 1/4 1/5 1/6 . 2/1 2/2 2/3 2/4 2/5 2/6 . 3/1 3/2 3/3 3/4 3/5 3/6 . 4/1 4/2 4/3 4/4 4/5 4/6 . 5/1 5/2 5/3 5/4 5/5 5/6 .
Real Numbers 1 0.a1a2a3a4a5a6a7 . 2 0.b1b2b3b4b5b6b7 .. 3 0.c1c2c3c4c5c6c7 4 0.d1d2d3d4d5d6d7 0.z1z2z3z4z5z6z7 . z1 a1 z2 b2 z3 c3 z4 d4 etc This number is not on the list. The number of reals has a cardinality bigger than aleph- null
Is there a cardinality between aleph-null and the number of reals? The Continuum hypothesis The answer can be yes or no !
