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" Godel destroyed the presumption of reducibility forever."

By reducibility Godel meant the rules of arithmetic in addition to two truth values plus two logical operators, (addition and subtraction) should obey the conditions of completeness and consistency of formal logic.

His proof proved otherwise. Even for simple arithmetic that only recognized addition and subtraction as logical operators some valid formulas could not be identified as theorems and existing theorems could not be determined to be valid formulas according to the rules of arithmetic.

Quoted from you

"I presume you mean that all of its true theorems can be proved."

Exactly the opposite.

Godel proved that any logic that contained quantifiers, (For some, For Every) would be incomplete and inconsistent.

Godels proof is so powerful that it blew the bottom out of mathematical logic and consigned to a sidebar in the history of Philosophy.

If logic can't construct a coherent account of the underlying rules of addition and subtraction then it can't account for anything in the world. (Paraphrasing Wittgenstein)

"I presume you mean that all of its true theorems can be proved."

Exactly the opposite.

Are you sure this is what you intended to say? That reducability (or completeness) means NOT "all true theorems can be proved"? You're aware the Godel's proof is called the incompleteness proof?

"I presume you mean that all of its true theorems can be proved."

A point of clarification is due here because I didn't read this sentence carefully enough.

All theorems of a logical system are valid formulas according to the rules of the formal system.

All valid formulas render true and only true valuations according to the rules of the formal system.

Therefore all theorems render true valuations according to the rules of the formal system.

The theorems of logic can only determine that the rules of logic have been obeyed. They say nothing about the world

So by definition all theorems render a true valuation.

Godel proved that any logic that contained quantifiers, (For some, For Every) would be incomplete and inconsistent.

No, he didn't. That's an easy assumption to make, and I made it when I was younger, much to my eventual embarassment. Universal and particular Quantifiers can be used to state the axioms of plain geometry, for an obvious, not to say, classical example. Quantifiers don't necessarily imply specifiable, logically tractable sets--barbers in villages that don't shave themselves come to mind.

When you try to run the proof over a continuous domain, you find that you have no contained sets to map to the Godel strings that represent particular theorems+theirProofs. You need sets to complete the proof, and continuous domains of discourse are not about discrete sets of things, they are about continuous things. You could just try to run Godel's proof over plane geometry, instead of arguing with me about it.

Godels proof is so powerful that it blew the bottom out of mathematical logic and consigned to a sidebar in the history of Philosophy.

Um, well, it blew the bottom out of the grand project to formalize all of mathematics. As far as I can tell meta-mathematics is still alive and kicking. It's a great proof, but it ain't a universal solvent.