I have a degree in computer science. I'm well versed in Gödel, Turing, and Church.

I understand your concept when you say "formal system". What I'm saying is the formal system you are proposing is already been tried and it is a repeated FAIL.

If you are trying to say, I'm designing a formal system. As such, things which are true and things which are false EXIST, but you can't through logic, mathematics or programming PROVE these things to be true or false. Then yes that statement is true.

But I you say, "We should do this! It is good!" and you assert that, I cannot disprove such a thing formally, than you can't prove such a think formally either.

You can say, "This is an experiment and it will prove itself over time. Just like a program either shows itself to "halt" or "not halt" given enough time, even though no one one can formally prove what it will do."

So I'm saying, "I HAVE SEEN this program run before. It halts. It halted the first time. It halted the second time. It halts every time because there are no changes to it or its inputs. Get over yourself.

It's been done, I've seen it. It fails.
Because you don't understand how it could fail doesn't mean it will stop failing.

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There is an old American bit of sarcasm that goes:

If you are not a liberal when you a young, you have no heart.
If you are not a conservative when you are old, you have no brain.

Congratulations, you've discovered your heart.
Keep on searching, perhaps you'll discover...