What's a good source on how to construct all sets of numbers?

Asked Nov 16 '17 at 05:59

Active Nov 16 '17 at 06:10

Viewed 117 times

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.

edited Nov 16 '17 at 06:10

asked Nov 16 '17 at 05:59

Daniel Bonilla Jaramillo

1,155

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You could also have a look at John Conway's On Numbers And Games. – Gerry Myerson Nov 16 '17 at 06:05
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What's $\Bbb P$? – Nov 16 '17 at 06:28
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You acn see Jürg Kramer & Anna-Maria von Pippich, From Natural Numbers to Quaternions, Springer (2018). – Mauro ALLEGRANZA Nov 16 '17 at 07:19
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@G.S, from context, I'd say the real algebraics. – Gerry Myerson Nov 16 '17 at 12:10
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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. – Dave L. Renfro Nov 16 '17 at 13:02
ℙ = {x|x is an irrational number} – Daniel Bonilla Jaramillo Nov 16 '17 at 17:03
https://math.stackexchange.com/questions/450524/is-there-an-accepted-symbol-for-irrational-numbers – Daniel Bonilla Jaramillo Nov 16 '17 at 17:10
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The accepted answer to that question, Daniel, says "But if and when an alternative letter like ℙ or is used, it should be preceded by a clear statement as to the fact that it is being used to denote the set of irrational numbers." Moreover, you are not using it to represent the irrational numbers, but the real irrational numbers. E.g., $\sqrt{-1}$ is irrational. – Gerry Myerson Nov 16 '17 at 21:13
True that, thanks for the correction. – Daniel Bonilla Jaramillo Nov 17 '17 at 00:46

What's a good source on how to construct all sets of numbers?

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