lpetrich
Contributor
I had earlier posted here on God Paradoxes, and I have come across another one: omniscience and infinite sets.
The world of infinite sets deserves some explaining, to give some background information for this paradox. It was first worked out in detail in the late 19th cy. by mathematician George Cantor, and though it provoked a lot of controversy, it is now generally accepted. That is because of all the paradoxes it has.
Galileo discovered one of these, that a proper subset of an infinite set can have the same cardinality or number of elements or members as that set itself. In particular, he noted that there is the same number of squares of positive integers as of positive integers themselves. For finite sets,
card(proper subset of set S) < card(S)
But for infinite sets,
card(proper subset of set S) <= card(S)
Georg Cantor discovered that there is an infinite hierarchy of infinities, and he called the infinite cardinalities aleph-0, aleph-1, aleph-2, etc. But of these, he could only identify aleph-0, the cardinality of the natural numbers (either nonnegative or positive integers). It is also the cardinality of the integers, the rational numbers, the algebraic numbers, the computable numbers, and the definable numbers. Even though:
(positive integers) < (nonnegative integers) < (integers) < (rational numbers) < (real algebraic numbers) < (real computable numbers) < (real definable numbers)
But GC showed that any attempt to match real numbers with integers would always leave some left over. That is his famous diagonal argument. Thus, there are more real numbers than integers, or
C = card(real numbers) > aleph-0
He thought that C = aleph-1, the "continuum hypothesis", but he could not prove it. Later mathematicians discovered that the usual axioms of set theory, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC), are consistent with that hypothesis and its negation, with both C = aleph-1 and C != aleph-1.
When checking over GC's work, Bertrand Russell discovered another paradox of infinite sets. The set of "normal" sets N, those that are not members of themselves. Is N a member of itself? If it is, then it isn't, and if it isn't, then it is. This paradox thus related to the liar paradox and to Goedel's paradox.
An infinite sequence of infinite sets can be constructing by taking "power sets" of sets. A power set of a set is the set of all subsets of that set, including that set itself. For finite sets,
card(power set of A) = 2^(card(A))
GC himself proved that a power set has a cardinality greater that that of that set itself. He did it by considering the set of all elements that are matched to subsets that do not contain them. By considering whether or not some element is matched to it, he discovered a liar-paradox sort of contradiction. This result is not only valid for finite sets, but also for infinite sets. Repeating the power-set operation thus gives an infinite series of sets, each one larger than all those before it. Power sets starting with the natural numbers have cardinalities sometimes called the beth numbers:
Now for the omniscience paradox.
Since there is no largest set, there can be no such thing as complete omniscience. That is because if one takes the set of everything that one knows about, then one can construct a larger set from it by taking its power set.
The world of infinite sets deserves some explaining, to give some background information for this paradox. It was first worked out in detail in the late 19th cy. by mathematician George Cantor, and though it provoked a lot of controversy, it is now generally accepted. That is because of all the paradoxes it has.
Galileo discovered one of these, that a proper subset of an infinite set can have the same cardinality or number of elements or members as that set itself. In particular, he noted that there is the same number of squares of positive integers as of positive integers themselves. For finite sets,
card(proper subset of set S) < card(S)
But for infinite sets,
card(proper subset of set S) <= card(S)
Georg Cantor discovered that there is an infinite hierarchy of infinities, and he called the infinite cardinalities aleph-0, aleph-1, aleph-2, etc. But of these, he could only identify aleph-0, the cardinality of the natural numbers (either nonnegative or positive integers). It is also the cardinality of the integers, the rational numbers, the algebraic numbers, the computable numbers, and the definable numbers. Even though:
(positive integers) < (nonnegative integers) < (integers) < (rational numbers) < (real algebraic numbers) < (real computable numbers) < (real definable numbers)
But GC showed that any attempt to match real numbers with integers would always leave some left over. That is his famous diagonal argument. Thus, there are more real numbers than integers, or
C = card(real numbers) > aleph-0
He thought that C = aleph-1, the "continuum hypothesis", but he could not prove it. Later mathematicians discovered that the usual axioms of set theory, the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC), are consistent with that hypothesis and its negation, with both C = aleph-1 and C != aleph-1.
When checking over GC's work, Bertrand Russell discovered another paradox of infinite sets. The set of "normal" sets N, those that are not members of themselves. Is N a member of itself? If it is, then it isn't, and if it isn't, then it is. This paradox thus related to the liar paradox and to Goedel's paradox.
An infinite sequence of infinite sets can be constructing by taking "power sets" of sets. A power set of a set is the set of all subsets of that set, including that set itself. For finite sets,
card(power set of A) = 2^(card(A))
GC himself proved that a power set has a cardinality greater that that of that set itself. He did it by considering the set of all elements that are matched to subsets that do not contain them. By considering whether or not some element is matched to it, he discovered a liar-paradox sort of contradiction. This result is not only valid for finite sets, but also for infinite sets. Repeating the power-set operation thus gives an infinite series of sets, each one larger than all those before it. Power sets starting with the natural numbers have cardinalities sometimes called the beth numbers:
- Beth-0 = aleph-0
- Beth-1 = cardinality of the real numbers C.
- Beth-2 = cardinality of all the functions from real numbers to real numbers. The functions need not be continuous, and the continuous ones have cardinality beth-1.
Now for the omniscience paradox.
Since there is no largest set, there can be no such thing as complete omniscience. That is because if one takes the set of everything that one knows about, then one can construct a larger set from it by taking its power set.
for n = 1, 2, 3, ... converges to some limit A if for every e > 0, there is some N such that |a
