[FOM] Banach Tarski Paradox/Line

Alasdair Urquhart urquhart at cs.toronto.edu
Wed Nov 30 09:35:40 EST 2011

Walter Taylor wrote:

"That is indeed amusing.  I also heard, possibly on this list, that one
of them, either Banach or Tarski, I forget which, was partly motivated
by the intent to convince the math world that AC was in fact unreasonable.
The math world by and large ignored this feature, clutching BT to its 
bosom in a similar way to the Cantor set, space-filling curves and others,
(in spite of the different logical footing).  So Banach (or Tarski)'s 
hopes came to nothing, and AC was retained."

I am very sceptical about these claims.  The paper itself shows
no indication of any such doubts about the Axiom of Choice.
Banach and Tarski remark only, in the introduction, that
the theorem "seems perhaps paradoxical."*  That is all
they say.  The article is easily available online at
the "Home Page of Stefan Banach":


It's in French.

As for the authors' philosophical views, Banach was a mathematician,
and I seriously doubt whether he had any philosophical scruples about
the Axiom of Choice.  On the contrary, the Polish school of topology
is notable for its free use of set-theoretical methods.

I find it harder to pin down Tarski's views on set theory.  He certainly
wrote quite a few papers on the subject, but when he expresses
philosophical opinions, they tend towards physicalism and nominalism.
So, it would seem that his attitude to set theory was somewhat
formalistic, but I am not clear on this.  In any case,
from a nominalistic point of view, the other axioms of set theory
are equally dubious and there is no special reason to single out
the Axiom of Choice.

* "Now, we do not know how to prove either of these two theorems without
appealing to the axiom of choice: neither the first, which seems perhaps
paradoxical, nor the second, which is fully in accord with intuition."

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