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With an ATP, a programmer can code in all the rules, or axioms, and then ask if a particular conjecture follows those rules.
He also showed that no candidate set of axioms can ever prove its own consistency.
HOW GÖDEL’S PROOF WORKSNATALIE WOLCHOVERJULY 14, 2020QUANTA MAGAZINE
Gödel’s main maneuver was to map statements about a system of axioms onto statements within the system — that is, onto statements about numbers.
HOW GÖDEL’S PROOF WORKSNATALIE WOLCHOVERJULY 14, 2020QUANTA MAGAZINE
It would mean that there exists a sequence of formulas built from these axioms that proves the formula that means, metamathematically, “This set of axioms is consistent.”
HOW GÖDEL’S PROOF WORKSNATALIE WOLCHOVERJULY 14, 2020QUANTA MAGAZINE
We’ve learned that if a set of axioms is consistent, then it is incomplete.
HOW GÖDEL’S PROOF WORKSNATALIE WOLCHOVERJULY 14, 2020QUANTA MAGAZINE
By the first theorem, this set of axioms would then necessarily be incomplete.
HOW GÖDEL’S PROOF WORKSNATALIE WOLCHOVERJULY 14, 2020QUANTA MAGAZINE
It is a generally accepted axiom that a public man cannot afford to be modest in these go-ahead days of "boom."
THE DOCTOR OF PIMLICOWILLIAM LE QUEUX
This truth is as old as Homer, and its proofs are as capable of demonstration as a mathematical axiom.
By this, OLeary understood that he was definitely adopted by virtue of the axiom of what was his was theirs.
THE WOMAN GIVESOWEN JOHNSON
That was an axiom on which was founded a vigorous war against all capillary adornments.

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