A_Seagull » February 15th, 2018, 5:54 pm wrote:Because "We know we can not fully model the arithmetic of the positive integers with any consistent set of axioms." is not the same as what Gödel proved!

That's exactly what Gödel proved. The fact that you're thrown off by a simple restatement/reframing of the theorem is a clue that your knowledge in this area is not up to snuff. You should take note of that yourself. Apply some introspection, don't just toss out weak objections. Either that, or explain why you think I've got it wrong. If you did that, then I'd learn something. If you don't, you haven't made your point.

You are confusing proof by contradiction, which Gödel's proof isn't; with contrapositive. If every complete system is inconsistent, then no consistent system is complete. Once again you're weakening your argument by pretending to understand more than you do. I'll recommend some better tactics in a moment.

Generally considered? Not something subject to investigation or proof? Gödel proved the completeness of first-order predicate logic. That is, in first-order predicate logic, a proposition is provable if and only if it's true in every model. That's his slightly less famous

completeness theorem.

https://en.wikipedia.org/wiki/G%C3%B6de ... ss_theorem. It may have been "generally considered," but I doubt anyone regarded that as the last word on the matter, absent proof.

What's the difference between first-order predicate logic and number theory? Number theory has induction. Once you have induction, you lose either completeness or consistency. Logicism is busted. A lot of people were very upset about this in 1931, but most of them have gotten over it by now.

You've convinced me -- absent your specific, detailed, point-by-point refutation of my remarks -- that you're weak on the incompleteness theorem and the meaning and context of the subject in general.

But there's no shame in that. The problem is that doubling down on your misunderstanding is a poor tactic. What you should do here (IMO of course) is to gracefully retreat. Admit that yes, if we include the whole of number theory under logic, then your thesis is false. Gödel proved that, and there's no point in your objecting that it's a proof by contradiction or that I phrased it differently than it was expressed in the original German. No win possible for you there, you need to retreat.

A much better strategy would be to agree that (as Gödel showed) math

isn't logic; but that certain special cases of your thesis are much more tenable.

For example if by logic you mean basic propositional logic, the Boolean logic of AND, OR, and NOT, then your thesis is entirely correct. A computer can easily do propositional logic. Computers are based on propositional logic. We all know this.

Now the question is, what

other realms can you extend your thesis to? What about modal logic, paraconsistent logic, and all the other arcane logics studied by the specialists. That's why I asked you what you mean by logic. Or more accurately what you mean by "logical processes," which is how you put it.

I thought you would go in that sensible direction: admitting that you can't apply your thesis to math as a whole; but that you can restrict your thesis to more defensible domains. That might be a productive discussion.