E PLURIBUS UNUM
A set is a Many which allows itself to be thought of as a One.
Gesammelte Abhandlungen (1883)
AXIOMS FOR SET THEORY
- AXIOM OF EXTENSIONALITY
Two sets are equal if and only if they have the same members.
- AXIOM OF UNORDERED PAIRS
The pairing of two sets is also a set.
- AXIOM OF SUBSETS
Any subset of a set is also a set.
- AXIOM OF THE SUM SET OR UNION
The union of sets is also a set.
- AXIOM OF THE POWER SET
The set of all subsets of a set is also a set.
- AXIOM OF INFINITY
There exists an infinte set.
- AXIOM OF REPLACEMENT
The image of a set is also a set.
- AXIOM OF REGULARITY
No set may be a member of itself.
- AXIOM OF CHOICE
For any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets.
-PHILIP J. DAVIS & REUBEN HERSH
The Mathematical Experience (1981)
The most famous is Russell's paradox. Most sets, it would seem,
are not members of themselves - for example, the set of walruses
is not a walrus, the set containing only Joan of Arc is not Joan
of Arc (a set is not a person) - and so on. In this respect, most
sets are rather "run-of-the-mill". However, some "self-swallowing" sets
do contain themselves as members, such as the
set of all sets, or the set of all things except Joan of Arc, and
so on. Clearly, every set is either run-of-the-mill or self-swallowing,
and no set can be both. Now nothing prevents us from
the set of all run-of-the-mill sets. At first, R
might seem a rather run-of-the-mill invention - but that opinion
must be revised when you ask yourself, "Is R itself a run-of-the-mill set
or a self-swallowing set?" You will find that the answer
is: "R is neither run-of-the-mill nor self-swallowing, for either choice
leads to paradox." Try it!
-DOUGLAS R. HOFSTADTER
Gödel, Escher, Bach (1979)
The overall image he had of this activity was of two spheres, one
expanding outwards towards infinity, and the other contracting in
towards zero. The large one grew by continually doubling its
size, the smaller shrank by repeatedly halving its size ... and
they seemed to be endlessly drawing apart. But with a sudden
feeling of freedom and air, Vernor had the conviction that the two
spheres were on a direct collision course - that somehow the
sphere expanding outwards and the sphere contracting inwards
would meet and merge at some attainable point where Zero was
Infinity, where Nothing was Everything.
Spacetime Donuts (1981)
Thus while the eternal non-being leads toward the fathomless, the
eternal being conducts to the boundary.
Tao Te Ching (604-666)