mymarks
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What Do Gödel’s Incompleteness Theorems Truly Mean? | Quanta Magazine
https://www.quantamagazine.org/what-do-godels-incompleteness-theorems-truly-mean-20260518/
It is easy to lose one’s sense of wonder at the fact that such a blindingly obvious set of axioms — the Peano axioms for arithmetic (the set of rules about the natural numbers 0, 1, 2, 3 … closely related to the system that Gödel used in his proof, such as the rule, “Every number has a successor”) — is essentially incomplete and undecidable, meaning that all axiomatizable consistent extensions are incomplete and undecidable. Hold on to that wonder! The incompleteness theorems teach us that when it comes to our attempt to master the conceptual order, whether it be in mathematics or, for that matter, in any other domain, we will always fail — and indeed, in this case more than any other, we should be glad to have failed, for failure was clearly the more interesting, the more profound, outcome.
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Zermelo-Fraenkel set theory with the axiom of choice,” or ZFC
https://www.quantamagazine.org/why-maths-final-axiom-proved-so-controversial-20260429/
It is a paradox without resolution: The foundations of mathematics are as universal, as solid as anything humanity knows, a core part of nearly every mathematical truth. And yet they remain simply what we choose to believe.