Če se ekonomisti prerekamo glede realnosti prepostavk o obnašanju ekonomskih subjektov in gospodarstva kot celote, in če fiziki ne vedo odgovorov na številna vprašanja glede univerzuma, so matematiki neenotni glede definicije neskončnosti. Tako zelo, da so nedavno na Harvardu organizirali simpozij na to temo. Ampak enotnosti niso dosegli. Ena izmed strani bo morala popustiti. To pa je že stvar osebnega prestiža.

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. Most mathematicians simply ignore the holes, which lie in abstract realms with few practical or scientific ramifications. But for the stewards of math’s logical underpinnings, their presence raises concerns about the foundations of the entire enterprise.

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Chief among the holes is the continuum hypothesis, a 140-year-old statement about the possible sizes of infinity.

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The 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. During a recent meeting at Harvard organized by Koellner, scholars largely agreed upon two main contenders for additions to ZFC: forcing axioms and the inner-model axiom “V=ultimate L.”

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According to the researchers, choosing between the candidates boils down to a question about the purpose of logical axioms and the nature of mathematics itself. Are axioms supposed to be the grains of truth that yield the most pristine mathematical universe? In that case, V=ultimate L may be most promising. Or is the point to find the most fruitful seeds of mathematical discovery, a criterion that seems to favor forcing axioms? “The two sides have a somewhat divergent view of what the goal is,” said Justin Moore, a mathematics professor at Cornell University.

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At the recent Harvard meeting, researchers from both camps presented new work on inner models and forcing axioms and discussed their relative merits. The back-and-forth will likely continue, they said, until one or the other candidate falls by the wayside. Ultimate L could turn out not to exist, for example. Or perhaps Martin’s maximum isn’t as beneficial as its proponents hope.

As many of the mathematicians pointed out, the debate itself reveals a lack of human intuition regarding the concept of infinity. “Until you further investigate the consequences of the continuum hypothesis, you don’t have any real intuition as to whether it’s true or false,” Moore said.

Mathematics has a reputation for objectivity. But without real-world infinite objects upon which to base abstractions, mathematical truth becomes, to some extent, a matter of opinion—which is Simpson’s argument for keeping actual infinity out of mathematics altogether. The choice between V=ultimate L and Martin’s maximum is perhaps less of a true-false problem and more like asking which is lovelier, an English garden or a forest?

“It’s a personal thing,” Moore said.

Vir: Scientific American