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We have a statement that we want to prove. For this, if we assume that the statement is true and find no contradiction, can we conclude that the statement is indeed true? (Because if the statement was not true, we would have reached a contradiction).

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    No. A statement can be false without leading to any contradictions. That Trump is now president is false, it is not contradictory. "This sentence is true" does not lead to any contradictions either, that does not make it true.
    – Conifold
    Commented Dec 10, 2022 at 0:34
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    @Amin, what category of statements are you referring to? Commented Dec 10, 2022 at 1:40
  • @KristianBerry Mathematical statements
    – user63754
    Commented Dec 10, 2022 at 3:50
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    @Amin, the further we go in mathematics, it seems to have turned out that the best we can hope for is "relative consistency." AKA conditional/hypothetical consistency. But so suppose we start with ZFC and face the choice of tacking on a single uncountable inaccessible, some specific number of those, or a proper class of them. As far as we know, there's no contradiction derivable from ZFC + any of those add-ons, but the add-ons aren't directly consistent with each other. Though I suppose you might take that for a metamathematical argument against the method of arbitrary axioms... Commented Dec 10, 2022 at 4:37
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    No, "if P, tnen Q" and Q do not imply P. Commented Dec 10, 2022 at 9:06

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That depends on your ability to find contradictions. But Suppose you have a mathematical conjecture that works for the first 1,000,0000,0000,0000,...,0000,000,000,000 examples and then suddenly fails. You might "thoroughly" test it and not find a contradiction and it still turns out to be false. All you would know is that the statement is true for the points that you've tested.

Though to proof, without a doubt, that it is actually true, you'd need to test EVERY possible option. Then you know that every possible option works or would have found a contradiction. Though depending on the problem that is something that is physically impossible to do.

Though depending on the problem it is sometimes easier to prove the opposite. Like suppose the statement is false and if you find a contradiction then it must be true.

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