Boolean and ZFC logic are constructed from fundamentally different foundations. Modern (20th century) mathematics appears to be dominated by the ZFC approach. I am not sure it is understood the two approaches are incompatible. Is this incompatibility studied?
Es posible debo seguir profundizando y actuálizando me en Matemáticas pero según lo que he estudiado lo métodos matemáticos no se contradicen por diversas vías si son correctas se llega a los mismos resultados
Boolean logic is the propositional calculus and it can be built within the system ZFC.
If you refer to infinitely many propositions, then you need an axiomatic metatheory that defines the necessary infinity (if you're a platonist or you have some other philosophical commitment you can avoid it).
The system ZFC is good for this metatheory, and other class theories, arithmetical backgrounds (first or second order), categorical foundations (and more options) are acceptable as well.
Balazs Inotai yes, I was not familiar with the term propositional calculus and it does look like quite close to what I would define as Boolean logic, but it certainly cannot be built from ZFC.
It is incorrect to apply infinite sets to boolean logic as it is not founded in sets (sets are a higher level concept within boolean logic - and only ever finite or unbounded - so incomplete with respect to infinity).
Boolean logic must exclude what it defines as infinity to preserve logical consistency. This also requires great care with predicate logic, particularly with the term "all" given that boolean objects must always be incomplete with respect Boolean infinity. Care must also be taken with unbounded inductive reasoning for the same reason. The sum of an unbounded number of rationals is always rational, but the same is not true of an infinite sum. Boolean logic is clear about why as it has a clear defintion of infinity.
This is why Bolean logic is incompatible with ZFC. Boolean logic cannot allow the concept of an infinite set, and it certainly does not have any place in replacing the foundation. Both infinite and set are non-foundational concepts. Once consistently defined they are not allowed together. Indeed no infinite object is allowed if logic is correctly constructed from two base boolean concepts (generally 0 and 1 or true and false). Infinity may only be considered as a limiting case, and only with analytic care since we are treading beyond the true/false certainty (but incompleteness) of logic. In the limit we reach continuity where 0 and 1 are no longer distinguishable but this is not helpful within logic. Logic must carefully preserve incompleteness to preserve true and false.
Well, if the theory of Boolean logic is consistent,
then there is a model of this theory in any model of ZFC (Completeness Theorem), and you can prove theorems to Boolean logic in ZFC:)
Balazs Inotai no, that is not true. Boolean logic comes before not after ZFC, and ZFC is inconsistent by the rules of Boolean logic. You cannot define logic until you define true and false. Boolean logic does that. Thanks for the comments. I can see the whole of mathematics has been re-written with this error. If I look up propositional calculus it quickly jumps to ZFC language without properly recognising the jump or inconsistency. I will write a short paper on Boolean logic and how it is formed to explain. The ZFC definition of complete does not hold in consistently formed boolean logic. Boolean logic is always incomplete but by the Boolean definition of completeness.
I don't define logic, I just derive formulas from the ZFC, and then define logic inside ZFC with the Completeness Theorem, and then speak about Boolean logic or, say, Calabi-Yau manifolds.
It is a way of doing mathematics and widely accepted.
You can say that Boolean logic comes first, without any ZFC as a metatheory.
Then you reconstruct mathematics inside Boolean logic, and then it would be an other kind of foundations of math.
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