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Contrary to the popular belief that "infinity is not a number; it's a concept," numbers that are not finite do exist. Mathematicians call them transfinite numbers. Just like ordinary numbers, some transfinite numbers are larger than others. This can be thought of as there being different levels of infinity, where some infinities are "more infinite" than others. If we draw a family tree in which every generation has finitely many offspring, and every chain of descendants is finite, then it is clear that we cannot have infinitely many family members. In the realm of the transfinite, things are not as intuitive: If we draw a family tree in which every generation has offspring at a certain level of infinity, and every chain of descendants is at that same level of infinity, it is possible (though not necessary) that the total number of family members is at a higher level of infinity.
Lynne C. Yengulalp
Primary Advisor's Department
Stander Symposium poster
Lian, Chester E., "An Aronszajn Tree" (2013). Stander Symposium Posters. 290.