# If we assume that a statement is true and find no contradiction, can we conclude that the statement is indeed true?

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).

• 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. Commented Dec 10, 2022 at 0:34
• @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
• @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
• No, "if P, tnen Q" and Q do not imply P. Commented Dec 10, 2022 at 9:06