Testbrett

In the late 19th century Cantor discovered the existence of uncountable numbers, and then went on to define a whole hierarchy of infinite cardinal numbers, in other words there are "different levels of infinity". It is natural to ask if finitary and countably infinite combinatorial objects have uncountable analogues. It turns out that the answer is yes. We will focus on one such key combinatorial property, the tree property. A classical result from graph theory (König's infinity lemma) shows the existence of this property for countable trees. More precisely, the lemma says that every infinite, finitely branching tree has an infinite branch. We will discuss what happens in the case of uncountable trees.

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