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I just started learning about set theory and the countability of sets. And there is something my brain does not quite understand about infinities. It might be a dumb question, but here it goes:

You can prove that the integers are countable and have the same cardinality as the natural numbers by finding a bijection between the sets.

So now you can map each integer to a natural number. But to me it seemed logical that the cardinality of the integers are twice as large compared to the natural numbers since half the integers are just negative natural numbers.

So if the natural numbers are: $1, 2, 3, 4, 5,...$

And the integers are: $..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,...$

The natural numbers goes to infinity in the positive direction while the integers goes to infinity in both the positive and negative direction. So its infinity has to be twice as large. - "my brain thought"

Could someone explain to me where my intuition is wrong?

volticus
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  • Because we prove that they have the same cardinality. In addition $2 \cdot \aleph_0=\aleph_0$ – Mauro ALLEGRANZA Mar 06 '24 at 10:50
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    Because you use intuition from finite case. In infinite case we "count" sets using bijections. – zkutch Mar 06 '24 at 10:50
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    The shortcuts of intuition wrt inifine are well-known; see Galileo's paradox (1630 ca.) – Mauro ALLEGRANZA Mar 06 '24 at 10:52
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    @MauroALLEGRANZA You mean shortcomings, not shortcuts. – Jaap Scherphuis Mar 06 '24 at 10:54
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    @JaapScherphuis - thanks :-) – Mauro ALLEGRANZA Mar 06 '24 at 10:57
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    The integers do have twice the cardinality of the naturals. But, twice infinity is still infinity. – Gerry Myerson Mar 06 '24 at 11:07
  • Your intuition is assuming that direction is intrinsic information about sets, but there is no direction in a set. The integers are also $0$, $-1$, $1$, $-2$, $2$, etc. – Randy Marsh Mar 06 '24 at 11:17
  • Galileo had more or less the same question, with natural numbers vs. perfect squares instead of integers vs. natural numbers. Later on Bolzano and Cantor clarified matters in a certain way. – user14111 Mar 06 '24 at 11:32
  • The concept of cardinality doesn't always map nicely onto our intuitive notions of certain sets being 'larger' than others. Indeed, given that the natural numbers are a strict subset of the integers you can certainly say that there are 'more' integers than natural numbers, but that sense of 'more' is not captured by our mathematical definition of cardinality. Cardinality cares more about 'kinds' of infinity, and since 'twice' natural number infinity is still the same kind of infinity we say that the integers and the natural numbers have the same cardinality. – Bram28 Mar 13 '24 at 14:03

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