# metadata.dc.description[].value
Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Juan Carlos Naranjo del Val

# metadata.dc.description.abstract[].value
[en] The Banach-Tarski paradox asserts that it is possible to duplicate the sphere and the closed ball by only breaking each object into a finite number of pieces, and using displacements of space such as rotations and translations, reconstruct two copies of the original. The result seems impossible, and that is why it is known as the Banach-Tarski paradox. In this project we study this paradox starting from the axioms that it uses, particularly the axiom of choice. Then we study some properties of the groups to reach a weaker but also surprising theorem, called the Hausdorff paradox, and finally we will reach the Banach-Tarski paradox.
In addition, we will see the minimum number of pieces needed to duplicate the sphere and the closed ball by doing an explicit construction of the sets, using the axiom of choice. To finish the main chapter, we will see that the paradox is not a result that can be done in our physical world, and we will see the importance of the axiom of choice. Finally, we will give a visual way to duplicate the hyperbolic plane.

# metadata.dc.rights[].value
cc-by-nc-nd (c) Ignasi Guillén Mola, 2019

# metadata.dc.source[].value
Treballs Finals de Grau (TFG) - Matemàtiques