Is this a correct way to describe number sets?

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Discussion Overview

The discussion revolves around the definitions and representations of various number sets, including natural numbers, integers, rational numbers, and irrational numbers. Participants explore how to accurately describe these sets using set notation and the implications of including or excluding certain elements, such as infinity.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants describe natural numbers as N={1,2,3,...} and integers as Z={..., -3, -2, -1, 0, 1, 2, 3,...} while rational numbers are expressed as Q=a/b, where a is an integer and b is a non-zero integer.
  • There is a question about how to define irrational numbers, with some suggesting it could be represented as the difference set of real numbers minus rational numbers.
  • One participant argues that irrational numbers can be denoted as ##\mathbb{R}\backslash\mathbb{Q}##, emphasizing that rational numbers include integers and natural numbers.
  • Another participant points out that including ##+ \infty## and ##-\infty## in definitions is problematic, as they are not considered numbers in the context of number sets.
  • There is a discussion about the nesting of number sets, with the relationship N⊂Z⊂Q⊂R being highlighted.
  • Some participants express uncertainty about the proper notation for rational numbers and the implications of infinity in their definitions.
  • The Peano Postulates are introduced as a way to define natural numbers, with some participants discussing their understanding and memorization of these axioms.
  • One participant mentions that there is no unique set satisfying the Peano Postulates, indicating a potential disagreement on this point.

Areas of Agreement / Disagreement

Participants generally express differing views on the definitions and representations of number sets, particularly regarding the inclusion of infinity and the proper notation for rational numbers. The discussion remains unresolved on several points, including the uniqueness of the set satisfying the Peano Postulates.

Contextual Notes

Some limitations in the discussion include the dependence on definitions of number sets, the implications of including or excluding infinity, and the unresolved nature of certain mathematical steps related to the Peano Postulates.

  • #31
micromass said:
But sure, ##0## is unnatural. I have 2 apples and give 2 apples to my brother. What I have left is nothing natural??

Also, axiomatic set theory is pretty clear on the issue that ##0## should be a natural number. You'd make a mess out of the theory otherwise.
But this argument is essentially practicability as it is to exclude units to be prime. I have no problem when people work with ##\mathbb{N}_0## though.
 
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  • #32
micromass said:
Meh, once you rigorously define those terms, we might find an answer. So far, nobody has really given a satisfactory definition.
ok boss..this is all way over my head. xD
 
  • #33
fresh_42 said:
But this argument is essentially practicability as it is to exclude units to be prime. I have no problem when people work with ##\mathbb{N}_0## though.

Well, clearly you don't think math should be elegant. I think elegance trumps everything else. And if you believe in elegance, ##0## should be a natural.
 
  • #34
number is undefined, so is point, proposition, true, false, set, element...o0):frown:
 
  • #35
micromass said:
Well, clearly you don't think math should be elegant. I think elegance trumps everything else. And if you believe in elegance, ##0## should be a natural.
Oh no. I won't turn into this ... Far too obvious!
 
  • #36
Compromise:

natural.png


Include 0 only half. Is that natural enough?
 
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  • #37
mfb said:
Compromise:

View attachment 106007

Include 0 only half. Is that natural enough?
We have to ask the Indians. As far as I know they have the copyright on the ##0##. I'm sure we've had numbers when we lived in Africa, but it took a civilization to use the ##0##. Probably some bookies ...
 
  • #38
fresh_42 said:
We have to ask the Indians. As far as I know they have the copyright on the ##0##. I'm sure we've had numbers when we lived in Africa, but it took a civilization to use the ##0##. Probably some bookies ...

Some tribes count as ##1,2,3,\text{many}##. So I deny that the number ##10## is very natural.
 
  • #39
micromass said:
Some tribes count as ##1,2,3,\text{many}##. So I deny that the number ##10## is very natural.
It's just another name for 2. :wink:
 
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