Problem with logic

[Anonymous No.8636881]

How can proof by contradiction be a valid method? In systems like ZFC that are not known to be consistent then how do we know that if a sentence S is true then its negation Z must be false? I mean, the definition of consistency is that the theory does not contain contradictions. So if we do not know that ZFC is consistent (unprovable) then how can we assume that it can't contain contradictions? For all we know, ZFC could contain contradictions. Many.

Just because Z is false, that does not mean S is true. If this was true for all Z then ZFC would be consistent, which is an unprovable statement inside ZFC. And if ZFC is the foundation for all of mathematics does this not mean that the method of proof by contradiction is invalid for all areas of mathematics, unless reformulated in smaller, self contained theories that can be proven to be consistent?

[Anonymous No.8636909]

>self contained theories that can be proven to be consistent

these are not very powerful

[Anonymous No.8636911]

>>8636881
Not all logics allow proof by contradiction. It basically boils down to whether you take the Law of the Excluded Middle.

>For all we know, ZFC could contain contradictions

In order to use ZFC, you have to assume it's consistent.

[Anonymous No.8636942]

>>8636909
>these are not very powerful

But they would logically allow us to use the proof by contradiction method, which is very powerful. For example, Euclid's axiom for geometry have been prove to be consistent. But in the modern geometry I was taught at university we defined almost everything in terms of sets, which the only undefined object being "points". Not only that but we constantly used arguments involving sets to prove even elementary theorems. That is not okay. I have used proof by contradiction many times when working with sets and only now I realize that I am fucking retarded for not realizing this sooner.

>>8636911
>In order to use ZFC, you have to assume it's consistent.

Not really. I have been using ZFC for years and I have never made any assertion regarding its consistency. I just keep using it. We all do. We can't assume it is consistent.

>Not all logics allow proof by contradiction.

Then how do we know ZFC isn't one of these logics? It could be. We don't know.

I think we need a new axiom. The axiom of determinacy together with ZF (ZFAD) makes it possible to prove that ZF is consistent so I think we should take the axiom of determinacy into ZFC and just not use it. That would yield ZFCD but AD contradicts the axiom of choice so we would have to work as such:

Use proofs by contradiction when using only ZF. Do not use that method when using the axiom of choice, which we will use whenever convenient. And never use the axiom of determinacy, despite assuming it in our theory.

Does that sound good?

[Anonymous No.8636966]

>>8636881
you're conflating "proof of ~P" with "P is false". if you're concerned about inconsistency of your system, then nothing in specific about proof by contradiction should be troubling you, as opposed to using the system at all.