FOM: Brief history of the negation of the axiom of choice

[adib benjebara]

Brief history of the negation of the axiom of choice  

The story begins with Cantor who was working on the functions theory and
noticing there could not exist any bijection between N and R.
A tremendous work made him able to give shape to the set theory, being
encouraged by Dedekind and
criticised by Kronecker . On about 1900, it happened to him to hit on two
problems.
The first is a paradox which make it impossible
for the set of all sets to exist. The second , called continuum problem, is
about the proof 
that there does not exist a cardinal number  
between that of N and of R. Indeed, Cantor was  
reasoning by taking for obvious what will later be called the Axiom of
choice as he was taking for granted a total order relation between cardinal
numbers .This last issue made him exhausted. Zermelo  was trying to make the
theory of Cantor formalized...
So on 1904, he defines the Axiom of choice as being the assumption that the
Cartesian product of an infinite family of (non empty) sets is always
different from the empty set. Zermelo and Russell gave at about the same 
time another statement for the Axiom which is that there exists a function
which associates to each  set of the infinite family one element of the set.
The Axiom was called at the beginning the  
multiplicative Axiom. Controversies started between   mathematicians,
particularly, the French  mathematicians Hadamard, Borel and Lebesque, while
Zermelo was going on with his formalization work. The  polish mathematician
Sierpinski undertook the  identification of the theorems which need the
axiom.
The German mathematician Fraenkel used the axioms of Zermelo to define as
early as 1922 a model where the negation of the axiom of choice is an axiom.

Polish mathematicians like Tarski, Mostowski, Lindenbaum studied around the
thirties the negation of the axiom of choice. However, tragic deaths (of
young set theorists) happened after Banach pointed out unrealistic
consequences of the axiom. The French
mathematician and logician Herbrand died; Ramsay, a disciple of Russell,
also.
Lindenbaum died during the second world war and some papers of Mostowski
were
lost at that time.
Meanwhile , the reference to Zorn's lemma was taking the place of the
reference to the axiom.

Mostowski was the one who studied most the particular cases of the axiom :
denumerable or non denumerable family and above all, the case where all the
sets
have the same number of elements n, whose notation is Cn. He made about it a
work remarkably difficult to access.
In the fifties, the Swiss Specker and the French Fraissé studied also the
negation of the axiom of choice. But the coming of forcing on 1963 produced
a
fashion phenomena ( with some lobbies )which relegated Fraenkel Mostowski,
as they were called
henceforth, as of secondary importance.

However, the use of the axiom for the definition of infinite sums and
products
was not drawing attention, although it was mentioned in the “Principia
Mathematica” of Russell and Whitehead before the first world war. However,
this
use of the axiom was evoked in a remarkable paper, from Sierpinski, on 1918,
which did not get the echo it deserved.

Gauntt, in the seventies, was the only one trying to complete the work of
Mostowski on the Cn. May be a contribution from Truss should be also
mentioned.

The continuum problem is not an interesting one, because, with the axiom of
choice not being true, there is no total ordering between the cardinal
numbers.

Mr Andreas Blass proved on 1999 that we can assume CC(n) , n from 2 to m, m
included,
with CC(n) the countable axiom of choice for families of sets of n elements.

Adib Ben Jebara    ajebara at excite.com
http://homestead.dejanews.com/user.ajebara/axiom.html





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