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I. J. Kennedy
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The fact that Fermat's Last Theorem is false over the $p$-adics shows that it cannot be proved using arguments using congruences.

The fact that the Steiner-Lehmus TheoremSteiner-Lehmus Theorem is false over the complexes shows that it cannot be proved using what John Conway calls "equality-chasing" arguments.

This one is probably not explainable at the freshman level but the fact that the Paris-Harrington theorem and the Robertson-Seymour graph minor theorem are not provable in first-order Peano arithmetic shows that some kind of "infinitary" reasoning or sophisticated induction is needed to prove them.

The fact that Fermat's Last Theorem is false over the $p$-adics shows that it cannot be proved using arguments using congruences.

The fact that the Steiner-Lehmus Theorem is false over the complexes shows that it cannot be proved using what John Conway calls "equality-chasing" arguments.

This one is probably not explainable at the freshman level but the fact that the Paris-Harrington theorem and the Robertson-Seymour graph minor theorem are not provable in first-order Peano arithmetic shows that some kind of "infinitary" reasoning or sophisticated induction is needed to prove them.

The fact that Fermat's Last Theorem is false over the $p$-adics shows that it cannot be proved using arguments using congruences.

The fact that the Steiner-Lehmus Theorem is false over the complexes shows that it cannot be proved using what John Conway calls "equality-chasing" arguments.

This one is probably not explainable at the freshman level but the fact that the Paris-Harrington theorem and the Robertson-Seymour graph minor theorem are not provable in first-order Peano arithmetic shows that some kind of "infinitary" reasoning or sophisticated induction is needed to prove them.

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Timothy Chow
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The fact that Fermat's Last Theorem is false over the $p$-adics shows that it cannot be proved using arguments using congruences.

The fact that the Steiner-Lehmus Theorem is false over the complexes shows that it cannot be proved using what John Conway calls "equality-chasing" arguments.

This one is probably not explainable at the freshman level but the fact that the Paris-Harrington theorem and the Robertson-Seymour graph minor theorem are not provable in first-order Peano arithmetic shows that some kind of "infinitary" reasoning or sophisticated induction is needed to prove them.