Can you point me to a system of logic that is both able to make substantial claims and for which every statement in that system can be proven. Mathematics says

Can you point me to a system of logic that is both able to make substantial claims and for which every statement in that system can be proven. Mathematics says you cannot (the incompleteness theorems), so there will always be a fundamental truth you will need to subscribe to. This indicates that those claiming 'rationality' almost certainly have a belief that is irrational even according to themselves (if they are honest enough to admit it). Here is an example. Suppose we are arguing about the need for a certain tax. The point of contention is whether the tax will majnly benefit the rich or the poor. You start out with an argument: even if the tax benefits those not in poverty, the effects will trickle down. Despite your opinions on trickle down economics, the argument was unsound at the very first words. 'Even if' is a form of argument via excluded middle. It presupposes that either tbe tax will help those in poverty or it will not. The law of excluded middle forms unsound logical systems. There are ways around it but i guarantee you you will not see this in 'rational' discourse In the context of these everyday absurdities and the fact most logical systems are -- by necessity -- incomplete, its almost inavariably more interesting to talk about the unproven statements (the ideology) than the system of logic surrounding it. In other words, the ideology is ultimately what matters, even to rational thinkers.