whole number Seth always be greater than natural numbers set justify this
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Why isn't the set of whole numbers bigger than the set of natural numbers, if the whole numbers include all of the natural numbers plus 0?
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5 Answers

Alan Bustany, Trinity Wrangler, 1977 IMO
Answered May 20, 2015
Originally Answered: Why isn't the set of whole numbers bigger than the set of natural numbers if the whole numbers include all of the natural numbers plus 0?
You have fallen foul of Bustany's Rule of Infinity:
Intuition and Infinity do not mix
Your intuition that there are elements of a set "left over" implying that the set must be bigger is faulty for infinite sets :-(. In fact it is a property of anyinfinite set that it can be put in one-to-one correspondence with a proper subset of itself. That is there are elements "left over" even when you compare an infinite set with itself! So much for intuition...
Cantor's genius with his diagonal argument was to show that no list of Real numbers (that is a pairing of real numbers with the Natural numbers) could ever cover all the Real numbers. Not just that he could come up with some list of Real numbers that was incomplete (as you did for the Whole numbers). That is, he showed that a bijection did not exist, whereas others have pointed out that it is pretty easy to show a bijection between Whole numbers, or all the Integers, and the Natural numbers.
Thus Cantor showed that the set of Real numbers was definitely bigger than the set of Natural numbers and not just that there could be some left over.
If you want to get further into infinity you should be aware that there is a one-to-one correspondence between the Natural numbers and the Rational numbers (fractions). Even though there are infinitely many fractions between any two separate points on the number line, which says that the Rational numbers are dense on the number line. Despite this density there are a lot more Irrational numbers somehow squeezed in amongst the Rationals to make up the totality of Real numbers. Intuition fails again :-(