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| The "Gabriel's Horn" is an ideal object that can be finite and infinite at the same time. |
We are used to thinking that the "infinite" is a well-defined concept like some of our everyday ideas of "here", "there", etc. But, if that were the case, we would not have so many paradoxes arising from this field of science that Gauss referred to as the "queen of sciences".
Of course, George Cantor did a great contribution fixing the traditional and loose understanding of the infinite, especially introducing a scale of infinitudes when he proved that there is more than one kind of infinite. Since then, many other mathematicians and philosophers had been kept busy untangling some paradoxes that the new scale of the infinite had brought.
Among them, Jorge Luis Borges surfaced and worked literally with great success the problem of random infinite series without a first and the last term.
To learn about Borges' unusual understanding of the infinite follow this series of posts beginning with the post Nobody understand the infinite so well as Borges.
To learn about the unusual paradox of how can an "object" can be finite and infinite at the same time follow this article Top myths about the infinite about Torricelli's trumpet also called Gabriel's horn.