Recently I have been thinking about infinity and more specifically, sets of an infinite size, or an infinite amount of elements in them. And I found this concept of a bijection where if each element in one infinite set can correspond to another element in another infinite set, without any overlap, then the two sets have the same 'cardinality' or the same number of elements within them. Using this method you can also prove that there are the same number of reals from 0-1 as there are 0-2. But can I prove that there are three times more even numbers than natural numbers by assigning three natural numbers to every even number? What is the flaw in this logic?
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2$\begingroup$ You have it backwards here -- you're proving that there are three times as many natural numbers as there are even numbers. Aside from that, your logic is sound. Infinite sets and our attempts to squeeze their cardinality into arithmetic gets non-intuitive. $\endgroup$– user694818Commented Jun 22, 2021 at 18:06
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$\begingroup$ that is a property of infinity sets, there are bijection into proper subset of them, but it doesn't mean that one has more than other $\endgroup$– janmarqzCommented Jun 22, 2021 at 18:07
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2$\begingroup$ “Three times more” doesn’t make sense in infinite sets. $|X|\times 3=|X|$ for any infinite set $X,$ so you can say “three times”, but it isn’t very interesting. The sets of primes, squares, even numbers, and powers of $2$ all have the same cardinality. $\endgroup$– Thomas AndrewsCommented Jun 22, 2021 at 18:09
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Even if your logic was right, you would show that there are three times as many natural numbers as even numbers.
Anyway, the cardinality of infinite sets is defined such that if there exists a bijection between a set $X$ and the natural numbers, $\mathbf{N}$, $|{X}| = |\mathbf{N}|$. Since there clearly exists such a bijection (e.g. to each even number, assign its half), the sets have equal cardinality (denoted $\aleph_0$). From then on, any other mapping is irrelevant, and with infinities, comparisons such as "three times as many" do not make sense anyway, as the basic arithmetic operations for infinite cardinalities do not coincide with what we are used to with natural numbers.
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$\begingroup$ ah, i see. That makes sense $\endgroup$ Commented Jun 22, 2021 at 18:59
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2$\begingroup$ "comparisons such as "three times as many" do not make sense anyway, as the basic arithmetic operations are ill-defined." This is not right, basic arithmetic certainly exists for infinite cardinalities and is well-defined. It just happens that multiplying an infinite cardinality by 3 will result in the original cardinality. $\endgroup$– VsotvepCommented Jun 22, 2021 at 19:29
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$\begingroup$ You are right. What I meant is that the ones we are used to with natural numbers are not, since, as mentioned in the wiki, the operations only coincide for finite cardinalities. Thanks for pointing this out! $\endgroup$ Commented Jun 22, 2021 at 19:48