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After graduating, Padoa became a secondary school teacher, teaching in Pinerolo, Rome and Cagliari. From 1909 he taught at the Technical Institute in Genoa.
Padua belonged to Peano's school of mathematical logic, popularising this type of work. He gave important lectures at the International Congress of Philosophy in Paris in 1900.
He discovered an important method in the theory of definition which became more important when model theory was developed and Tarski proved Padoa's method.
In 1934 Padoa was awarded the mathematics prize of the Accademia dei Lincei.
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