Eliminating Undecidability and Incompleteness in Formal Systems (As simple as possible)

[Speakpigeon]

Hello?! We're talking about logic here. You know? Logic? Nah. You don't.

But the most important question is: What do YOU mean by 'logic'?

https://en.wikipedia.org/wiki/Logic

There is no universal agreement as to the exact scope and subject matter of logic

So many different conceptions

• The tool for distinguishing between the true and the false (Averroes)

• The science of reasoning, teaching the way of investigating unknown truth in connection with a thesis (Robert Kilwardby).

• The art whose function is to direct the reason lest it err in the manner of inferring or knowing (John Poinsot).

• The art of conducting reason well in knowing things (Antoine Arnauld).

• The right use of reason in the inquiry after truth (Isaac Watts).

• The Science, as well as the Art, of reasoning (Richard Whately).

• The science of the operations of the understanding which are subservient to the estimation of evidence (John Stuart Mill).

• The science of the laws of discursive thought (James McCosh).

• The science of the most general laws of truth (Gottlob Frege)

LOL.
I can see why you should be so confused. Bloody Hell, it's all expressed in good English!
So you're stuck, hey? You can 't see the commonality. Well, if you did understand logic, you would.
What strikes me in all these definitions is the expression in so many different ways of a shared notion of logic. That's to be expected since there is just one human logic, i.e. the logic of human reasoning, and more fundamentally the logic of the human brain.
You, on the other hand, what strikes you is the incomprehensible differences. That too should be expected. Like mathematicians and computer scientists, you think of logic as arbitrary. Each system of logic its own rules. Well, sure, but then its definitely not the logic of human reasoning.
So, in effect, like mathematicians and computer scientists, you are equivocating on the word "logic". You use the word "logic" as in the phrase "The logic of the quirky arbitrary formal system I'm spending my time on", or "The logic of the nice machine that does all the hard work for me and saves me from any thinking at all". This is pathetic and disingenuous.
This is also polluting the conversation. It's no longer possible to have any conversation about logic as the logic of human reasoning because there is a plague of self-appointed experts on logic who don't do anything like thinking about the logic of human reasoning but nonetheless insist on calling the terms of the debate.
Sorry, love, I'm not interested in the least in the logic of your fridge or the logic of the mechanical device you use to toast your bread by exposure to electrically heated wire coils. Not even in the logic of the nice little machines you call computers. Because, you see, whatever you do, the only thing you really need to use in all circumstances is you own brain. And nobody is going to do it for you.
You can't use it? Fine. Ask your nice little computer to tell you what is the notion of logic common to all the definitions you just quoted above. Go on, it's a challenge. Are you game for it?
I'll be not waiting.
EB

[Logik]

the logic of human reasoning, and more fundamentally the logic of the human brain.

THE logic? Which one is THE logic? The human brain invented so many logics...

Ask your nice little computer to tell you what is the notion of logic common to all the definitions you just quoted above. Go on, it's a challenge. Are you game for it?

That which is common to all logic is the machine which generates it. You call it 'mind' - I call it a Turing machine

My 'little computer' is my own mind.

The logic of the quirky arbitrary formal system I'm spending my time on", or "The logic of the nice machine that does all the hard work for me and saves me from any thinking at all". This is pathetic and disingenuous.

And this would be a strawman...

[Speakpigeon]

the logic of human reasoning, and more fundamentally the logic of the human brain.

THE logic? The human brain invented so many logics...

"The logic" as in "the logic of the human brain".
Yeah, BUT, there are so many human brains, yes?
I guess you're terminally lost in translation. Even though I use your own language. The logic? What could that possibly be!!!
EB

[Logik]

I guess you're terminally lost in translation. Even though I use your own language. The logic? What could that possibly be!!!

Whatever gave you the idea that English is 'my own' language'?

I SPEAK English. I don't THINK in English.

Thought is my first language. And from what I've seen it's not a language you speak.

"The logic" as in "the logic of the human brain".

https://en.wikipedia.org/wiki/Circular_definition