4/18/2023 0 Comments Law of infinitesimals mathThe decades-long quest for a more complete axiomatic system, one that could settle the infinity question and plug many of the other holes in mathematics at the same time, has arrived at a crossroads. It “must be either true or false,” the mathematical logician Kurt Gödel wrote in 1947, “and its undecidability from the axioms as known today can only mean that these axioms do not contain a complete description of reality.” The continuum hypothesis asserts that there is no infinity between the smallest kind - the set of counting numbers - and what it asserts is the second-smallest - the continuum. Beyond the continuum lie larger infinities still - an interminable progression of evermore enormous, yet all endless, entities. As incomprehensible as it may seem, endlessness comes in many measures: For example, there are more points on the number line, collectively called the “continuum,” than there are counting numbers. “How can I stay in any field and continue to prove theorems if the fundamental notions I’m using are problematic?” asks Peter Koellner, a professor of philosophy at Harvard University who specializes in mathematical logic.Ĭhief among the holes is the continuum hypothesis, a 140-year-old statement about the possible sizes of infinity. But for the stewards of math’s logical underpinnings, their presence raises concerns about the foundations of the entire enterprise. Most mathematicians simply ignore the holes, which lie in abstract realms with few practical or scientific ramifications. In the course of exploring their universe, mathematicians have occasionally stumbled across holes: statements that can be neither proved nor refuted with the nine axioms, collectively called “ZFC,” that serve as the fundamental laws of mathematics.
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