Mfw "proof by contradiction"

mfw "proof by contradiction"

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Why?

>proof
Why not?

Ok, so you wanna prove P->Q:

If you prove the opposite of (P->Q) is always false, then P->Q itself is always true.

So what's wrong with that user?

Some people agree that proving that something doesn't work one way doesn't mean it works the opposite, but other ways. However in our time proof by contradiction is known as valid.

>All functions are continuous.
Constructive mathematics was a mistake.

It works like: You prove the opposite is false, it doesn't mean the premise is true, it just states that the opposite is false and has nothing to do with the original premise.

idiot

When you are proving something cannot exist, proof by contradiction is assuming it can exist, then proving a logical inconsistency. Therefore, there was a poor assumption somewhere, which is that the thing can exist. Therefore, it cannot exist.

It is not doing a mythbuster approach saying "well, we couldn't do it this way so it must be impossible."

What?

If you reject proof by contradiction, then you reject the law of excluded middle (and thus the axiom of choice as well). Mathematics without lem is called constructive mathematics, and in certain models of constructive mathematics, all functions are continuous.

A statement being a contradiction means it's false under all possible conditions.

Therefore the opposite of that given statement would be a Tautology which means that it's true under all possible conditions.

What's the 'opposite' of P->Q? P->Q'?

>certain models
AKA, also rejecting the set theoretical definition of a function. Also they don't "reject it", they just don't take it as an axiom, law of thought.

( P -> Q ) is equivalent to ( !P v Q )
(You can get the CNF from the truth table if you're not sure about it.)

Therefore the opposite of P -> Q:
!( P -> Q ) is equivalent to !( !P v Q )
which is equivalent to (P ^ !Q) according to DeMorgan's law.

TL;DR
(P ^ !Q)

Well, yes. Rejecting an axiom means that you don't accept/use it. It doesn't mean that you assume that it's incorrect or whatever you're trying to imply it does.

Yea, and still, most of real anal is perfectly constructiblr.

>which is that the thing can exist.
what if one of your axioms that underlie what you mean by e.g "existing" are "wrong"?

What axiom underlying existing? Are you talking math anymore or philosophy with what it means to exist?

I have nothing concrete.
But when you say "Assume that X is Y", you are working in some axiomatic system, and what if the contradiction arises due to a problem with the system you are working in?

>Proof requires induction
DROPPED

Did we come to any conclusions then?

Everything requires induction

That's literally a good thing

>X=>B
>-X=>B
Where X is unknown.

P -> Q / ~P | Q / ~(~P | Q ) / P & ~Q
programming.dojo.net.nz/study/truth-table-generator/index

The explanations of proof-by-contradiction in here are neglecting the distinction between that and proof-by-negation, which is only fine if you're working in brainlet classical logic.

There is no difference, and classical logic is suspect as well.

>can't show surjection iff right inverse
>perfectly constructible
sure bud

This was amazing. So original. Who came up with it? This is real math. And the originator deserves credit.

Oh, probably not a jew, lets ignore this type of proof. Not suspect at all.