Lately I've been interested in building, formally, all sets of numbers, starting from ℕ, then ℤ, then ℚ, then ℙ, then ℝ, then iℝ, then ℂ.

The only book I have come up, so far, is "Foundations of analysis" by Edmund Landau, which seems to build them like this : ℕ, ℤ and ℚ, ℝ, ℂ. Is that one a good book? which other source would you recommend?

This would also include the mathematical operations each set has.

Also, I'm corious about why ℕ⊂ℤ⊂ℚ, (ℚ∪ℙ)⊂ℝ, ..., and so on.

Thanks in advance.

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    $\begingroup$ You could also have a look at John Conway's On Numbers And Games. $\endgroup$ – Gerry Myerson Nov 16 '17 at 6:05
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    $\begingroup$ What's $\Bbb P$? $\endgroup$ – user228113 Nov 16 '17 at 6:28
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    $\begingroup$ You acn see Jürg Kramer & Anna-Maria von Pippich, From Natural Numbers to Quaternions, Springer (2018). $\endgroup$ – Mauro ALLEGRANZA Nov 16 '17 at 7:19
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    $\begingroup$ @G.S, from context, I'd say the real algebraics. $\endgroup$ – Gerry Myerson Nov 16 '17 at 12:10
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    $\begingroup$ In my answer to Is the real number structure unique?, I posted a list of 9 textbooks each of whose primary purpose is the construction of various number systems. $\endgroup$ – Dave L. Renfro Nov 16 '17 at 13:02

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