I precise for the possible down voters that I'm not student in maths I'm learning chemistry, and my friend is learning litterature, and we were speaking about BT paradox, my friend discovers this paradox three days ago and he asks me this question :
Is the BT paradox a proof that 1+1=1 ?
I'm really scared about his question I don't know how to answer him as well as possible. For me this paradoxe is rather a demonstration that $\infty+\infty=\infty$ but I have absolutely no idea what to say to answer his question and don't know if I understand this paradox as well as I think it.
So if you have an explenation with the less possible formulas, my friend will be happy !
Thank you
whilst contradiction does have a technical application in mathematics, the word paradox (to date!) remains informal. the use of the word may be seen as akin to putting an exclamation mark - it draws our attention to something that may evoke a sense of surprise or unease about the consistency of our assumptions or rules of inference.
our modern mathematics experienced what the poet might call its first, fine, careless rapture in the couple of centuries between the arrival of Newton and the departure of Gauss. since then unease has been experienced more frequently, and has in fact been a great stimulus to the development of rigor in deduction. it has also generated controversies from time to time - a thing one might, a priori, have thought impossible in a purely deductive science.
as you correctly suggest, much of this unease has been the symptom of a long and challenging process of developing the correct intuition of various aspects of the infinite.
if we allow that mathematics incorporates some of the features of artistic creativity, then the Banach-Tarski construction is certainly a masterpiece - a perfectly-formed miniature. if you want a less technical comparison you could do worse than think of what have become known as Zeno's paradoxes, after the ancient Greek philospher known as Zeno of Elea.
however one does not need to be a philosopher - a bright five-year-old can ask awkward questions. for example "if a space is composed of zero-dimensional points, how can it have a measurable volume?"
of course, we can wave our hands and mutter mysterious imprecations about "uncountable infinities", just as your average dinner party guest will dismiss Zeno with the remark "of course the poor dear had no knowledge of convergent sequences". but the question remains and is impressively simpler than any of the answers on offer.
informally, we might say that the Banach-Tarski construction places an exclamation mark at the junction of two major pillars of modern math - Lebesgue measure theory and the axiom of choice. as Thomas Jech writes:
"The Axiom Of choice is different from the ordinary principles accepted by mathematicians. And this was one of the sources of the objections to the Axiom Of Choice, as late as in the 1930's. The other source of objections is the fact that the Axiom Of Choice can be used to prove theorems which are to a certain extent 'unpleasant', and even theorems which are not exactly in line with our 'common-sense' intuition." The Axiom Of Choice (1973), p. 2
we should not be too influenced by preconceived notions about what mathematics is. whatever we say about its subject matter, the nature of mathematics as an activity of living creatures in a particular environment is an empirical phenomenon, and cannot be decided a priori. we live and learn.
recall that Dedekind, whose technique of "cuts" provided new clarity in thinking about real irrationals, soon realized that an uncountable aggregate has the paradoxical feature that all but a very small subset of its elements can never be named, even in principle, due to the finite/countable restrictions on language and symbolism. so what is the nature of these shadowy entities? if we eliminated them from consideration we would lose a very basic tool - the intermediate value theorem for continuous functions.
and what is logic?