Improve this answer. Timothy Chow Timothy Chow Add a comment. Willie Wong Willie Wong If you are going to mark it, might as well actually mark it. People already do this for "positive" assumptions that they need to use repeatedly in the same paper. That said, the whole point of this question is to pindown an acceptable adjective in such cases where such an adjective is desirable.
How often is the truth state of an assumption known? Youvan Youvan 5 5 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Now live: A fully responsive profile. Create a free Team What is Teams? Learn more. Ask Question. Asked 4 years, 1 month ago. Active 4 years, 1 month ago. Viewed 5k times. Sorry if this is nonsense - just trying to clear out some confusions.
Winter Stephen Stephen 3, 1 1 gold badge 13 13 silver badges 29 29 bronze badges. Your question is a very important one which is easily "swept under the rug". You are justified in questioning the method.
The invocation of the law of the excluded middle that is often associated with a proof by contradiction is about proving a proposition from its double negation. Hence I think the link I've provided will give a fuller answer to your question. Show 13 more comments. Active Oldest Votes. Andreas Blass Andreas Blass Isn't an axiom by definition true, and so all statements deduced from them are also true in the system formed by the axioms?
I'm not sure I understand what is a 'false' axiom. If someone were to take theorems about the natural numbers and claim that they're also true about the real numbers, I'd tell him that he's wrong because that axiom, true of the natural numbers, is false of the real numbers. More generally, axioms are barring errors true in the situation that they're intended to describe, but they can be false in other situations. If the axioms were inconsistent, they'd be false in all situations.
If we consider other means of knowing things, the justification is a lot weaker. For example, a clear intuition of the cumulative hierarchy may well allow us to know that ZFC is consistent.
Show 17 more comments. But in other logics, like minimal logic or intuitionistic logic, that might or might not be the case. In any case, that last example was merely to show that a contradiction can be 'caused' by the current assumptions being impossible in conjunction , rather than necessarily being due to some single wrong assumption. That's why I think my answer best addresses your question, since we often do proof by contradiction within some context under some other assumptions, yet it's valid to conclude the negation of the innermost assumption.
In most modern foundations for mathematics, such statements are impossible to make, and so that is why we can still afford to have LEM in modern mathematics. But if we want to permit this kind of statements, then we can't have LEM for them.
Add a comment. Morgan Rodgers Morgan Rodgers Good arguments for this include that 1 LEM is usually not necessary most theorems can be proven constructively ; 2 even if that fails, any classical statement using LEM can be translated into a constructive statement that is proven without LEM, or simply proven with an additional hypothesis that LEM holds; 3 LEM is undesirable in logic because, unlike the other laws of inference, it is not simply "true by definition".
But yeah, really just trying to say that whether using LEM is the logically appropriate way to approach a proof is more a question for philosophers and logicians. I would also say that the other answers support this. Mathematicians in most areas of mathematics are happy to take the fact that it is true a sufficient justification for using it.
What if it is neither true or false? If we were formally proving by contradiction that Sally had paid her ticket, we would assume that she did not pay her ticket and deduce that therefore she should have got a nasty letter from the council. However, we know her post was particularly pleasant this week, and contained no nasty letters whatsoever. This is a contradiction, and therefore our assumption is wrong. In this example it all seems a bit long winded to prove something so obvious, but in more complicated examples it is useful to state exactly what we are assuming and where our contradiction is found.
They can be put into what is called i rreducible form , which is where the numerator top number and denominator bottom number have no common factors other than 1, i. Then we try to arrive at a contradiction. Proving something by contradiction can be a very nice method when it works, and there are many proofs in mathematics made easier or, indeed, possible by it. However, it is not always the best way of approaching a problem.
This would have been much quicker than going through the whole proof by contradiction. Even more importantly it was, in fact, a step in the above proof. Having just warned you of the dangers of blindly trying to prove things by contradiction, we end with one of the nicest proofs - by contradiction or otherwise - I know. This is Euclid's proof that there are infinitely many prime numbers, and does indeed work by contradiction.
We can prove this by, in fact, contradiction.
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