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proof of pseudoparadox in measure theory
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(Proof)
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Since this paradox depends crucially on the axiom of choice, we will place the application of this controversial axiom at the head of the proof rather than bury it deep within the bowels of the argument.
One can define an equivalence relation on
by the condition that if and only if is rational. By the Archimedean property of the real line, for every
there will exist a number
such that . Therefore, by the axiom of choice, there will exist a choice function
such that
if and only if .
We shall use our choice function to exhibit a bijection between and . Let be the “wrap-around function” which is defined as when and
when . Define
by
From the definition, it is clear that, since
and
,
. Also, it is easy to see that maps into . If
, then
. On the other hand, if , then
. Since is strictly negative, . Since , .
Next, we will show that is injective. Suppose that
and . By what we already observed, , so is a non-negative rational number and
. There are 3 possible cases: 1)
In this case,
implies that
, which would imply that . 2)
In this case,
implies
which, in turn, implies that , which is impossible if both and belong to . 3)
In this case,
implies that
, which would imply that . The only remaining possibility is that , so
implies that .
Next, we show that is surjective. Pick a number in . We need to find a number
such that
. If
, we can choose
. If
, we can choose
.
Having shown that is a bijection between and , we shall now complete the proof by examining the action of . As we already noted, is a rational number. Since the rational numbers are countable, we can arrange them in a series
such that no number is counted twice. Define
as
It is obvious from this definition that the are mutually disjoint. Furthermore,
and
where is the translate of by .
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"proof of pseudoparadox in measure theory" is owned by rspuzio.
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(view preamble)
Cross-references: translate, mutually disjoint, series, countable, action, complete, surjective, implies, rational number, injective, negative, strictly, maps, clear, bijection, choice function, number, line, real, Archimedean property, rational, equivalence relation, argument, proof, axiom, application, place, axiom of choice, paradox
This is version 7 of proof of pseudoparadox in measure theory, born on 2004-09-25, modified 2005-08-04.
Object id is 6233, canonical name is ProofOfPsuedoparadoxInMeasureTheory.
Accessed 1195 times total.
Classification:
| AMS MSC: | 28E99 (Measure and integration :: Miscellaneous topics in measure theory :: Miscellaneous) |
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Pending Errata and Addenda
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