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8 Some important sets
Note: In this section, a, b, and c are natural numbers, and r and s are real numbers.
- Natural numbers are used for counting. A blackboard bold capital N () often represents this set.
- Integers appear as solutions for x in equations like x + a = b. A blackboard bold capital Z () often represents this set (for the German Zahlen, meaning numbers, because I is used for the set of imaginary numbers).
- Rational numbers appear as solutions to equations like a + bx = c. A blackboard bold capital Q () often represents this set (for quotient, because R is used for the set of real numbers).
- Algebraic numbers appear as solutions to polynomial equations (with integer coefficients) and may involve radicals and certain other irrational numbers. A blackboard bold capital A () or a Q with an overline often represents this set.
- Real numbers include the algebraic numbers as well as the transcendental numbers, which can’t appear as solutions to polynomial equations with rational coefficents. A blackboard bold capital R () often represents this set.
- Imaginary numbers appear as solutions to equations such as x2 + r = 0 where r > 0.
- Complex numbers are the sum of a real and an imaginary number: r + si. Here both r and s can equal zero; thus, the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. A blackboard bold capital C () often represents this set.
9 Paradoxes
We referred earlier to the need for a formal, axiomatic approach.
What problems arise in the treatment we have given?
The problems relate to the formation of sets.
One's first intuition might be that we can form any sets we want, but this view leads to inconsistencies.
For any set we can ask whether x is a member of itself.
Define
- Z = {x : x is not a member of x}.
Now for the problem: is Z a member of Z?
If yes, then by the defining quality of Z, Z is not a member of itself, i.e., Z is not a member of Z.
This forces us to declare that Z is not a member of Z.
Then Z is not a member of itself and so, again by definition of Z, Z is a member of Z.
Thus both options lead us to a contradiction and we have an inconsistent theory.
Axiomatic developments place restrictions on the sort of sets we are allowed to form and thus prevent problems like our set Z from arising.
(This particular paradox is Russell's paradox.)
The penalty is a much more difficult development.
In particular, it is problematic to speak of a set of everything, or to be (possibly) a bit less ambitious, even a set of all sets.
In fact, in the standard axiomatisation of set theory, there is no set of all sets.
In areas of mathematics that seem to require a set of all sets (such as category theory), one can sometimes make do with a universal set so large that all of ordinary mathematics can be done within it (see universe (mathematics)).
Alternatively, one can make use of proper classes.
Or, one can use a different axiomatisation of set theory, such as W. V. Quine's New Foundations, which allows for a set of all sets and avoids Russell's paradox in another way.
The exact resolution employed rarely makes an ultimate difference.
10 See also
11 References
- Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. BooksEnthsiast.com (Springer-Verlag edition).
