Consistency saves us some time.
An inconsistent thing is garbage. It can prove anything and thereby effectively proves nothing (The Principle of Explosion).
Euclidean Geometry and non-Euclidean Geometry are defined to be inconsistent with each other.
But that is okay because….?
Any system that contains an inconsistency is inconsistent.
If a system is capable of describing an inconsistency; it is inconsistent. A consistent system cannot describe an inconsistency.
If your language is inconsistent then anything built from that language is also inconsistent.
Which is a fair representation: Language can prove anything you want it to.
There's no wand that magically makes everything work out.
Any system capable of describing an inconsistency is inconsistent.
Which leaves the entirety of mathematics as being inconsistent.
Again – you can prove pretty much anything you like within mathematics – which is what you would expect of an inconsistent system.
If you start with the idea of inconsistency then everything that follows must be inconsistent.
Don't start with the concept of inconsistency.
Take a step back. Look at all the relationships between things. Don't try to label anything. Just look at the network of relationships.
There is no pattern of relationships that precludes some other pattern. One pattern of relationships is as valid as any other pattern of relationships.
Some patterns might have more significance to us than other patterns – But no pattern is inherently right or wrong. They are just patterns interacting.
And that is it. That is everything. Everything else is just a complicated way of stating a given pattern. There isn't anything that doesn't resolve to a pattern of relationships. And the only way of defining one set of relationships – is another set of relationships.