I would like to do mathematics so that I can trust the evidence of results rather than just suggestions about the results that might apply. This involves formulating maths into explicit substitution rules on finite symbol sequences (so something that everyone can agree on at the beginning). Because I could not find any decent text on this subject (quite expressive), I thought I could do it with theories on sets of materials that I know little, to begin with, so at least I know where to start. However, I have encountered problems, the main one (which I see) is currently that I can not find a practical set of rules of evidence for FOL.
In fact, my idea was obviously to use FOL (special case of my set theory), but to formulate the arguments of the mathematicians (who are generally claimed to be of the set theory), I would need rules of evidence to be able to transfer their arguments. However, I immediately observed that mathematicians seemed to assume certain schemes of first-order propositions that I call here "clearly valid", without reference to rules of evidence. Since I did not find any set of appropriate rules of evidence in which the evidence would be obvious, my next idea was to do enough with some un-known rules known to get something more practical by a meta-argument. However, I did not succeed either.
We can see that most mathematicians do not specify at all that their proof would be anything other than the respect of certain cultural norms, but with the exception of certain logicians who seem to claim to be able to formally prove their propositions by an essentially model theory. himself or completeness. Indeed, I have especially observed that, in many books on mathematical logic, a writer applied few rules of evidence to prove the entirety of the evidence, and then referred to it in the evidence. However, I do not see that the naive idea "to use Hilbert's deduction until completeness, then all the intuitive proofs used by mathematicians to be available" really works. In fact, I have not found any way to use completeness at all.
To consider this technically, suppose that I manage to formulate the integrated FOL into (my set theory) with appropriate short hands to make it manageable. It seems to me that it is usually quite complicated to work with theories embodied in set theory, but suppose that for the moment I can prove the completeness of the embedded FOL.
It does not say anything about my theory of outer sets yet, so what I think I have to do next is to prove in my meta theory an integration theorem that every formula is provable in my theory of sets. external if its built-in version is provable – the embedded version representing the provability of the same formula but in the theory of built-in sets (special case of the inner FOL) in which completeness applies.
First, while apparently formal, it challenges the idea that set theory was the foundation – in fact, meta-theory is used much more than basic theory. Instead, it is used enough to show arguments about the foundation.
I do not think it's a good idea to suddenly decide that the foundation is set theory within set theory and not even try to show anything in external theory – in Indeed, I believe that this kind of "transfer" of radical problems under the carpet eventually leads to even more dips, so we never do anything to a fixed theory really. In addition, I think that any kind of non-recoverable embalming makes the theory quite unusable being so complex that it is necessary to work at different levels, even to present a simple argument.
Suppose then that we accept that we are using more meta theory (this is not a real formality problem, because the necessary arguments in the meta-theory are always realizable with conclusions about symbol sequences). Then I could probably handle the embedding theorem.
However, the real complication I think actually occurs even after that. In fact, the goal was to be able to use completeness to prove things, but I have not found any (general) way of transferring anything into outer set theory from what now appears to be technically n ' is an incoherent intuition of the functioning of certain models. . It is obvious that the accounting completeness is sufficient and that the necessary set theory does not need to be so strong, but that does not seem to make things easier.
In fact, I would say that if we can deduce from the completeness that the "obviously valid" proposition schemes can be proven, then one should be able to write factually this argument in a way that symbolic, which would give an axiomatization as well as a rule of evidence usable in the outer theory of FOL in the first place during its definition.
Obviously, this supposed rule can not prove all valid formulas, otherwise resting on the obvious validity would make the validity of the FOL propositions decidable, but the idea is that the rule is sufficient to prove the schemes of propositions. " clearly valid ".
So my questions are: 1) Is there a known system of axioms and rules of proof of FOL that would capture the way mathematicians do mathematics, so that we can formulate their proofs formally in the initial theory in accordance with the requirements that I state? If such rules are not known, to the extent that no one has verified the proofs of mathematicians with a seemingly unusable rule of evidence like a normal Hilbert system (that's pretty difficult, in my opinion), 2) what is exactly the formalize the mathematics using the theory of sets of materials that is (I believe) generally claimed to exist?
One could postulate that there is no set of rules showing "obviously valid" proposal schemes, but I have a little difficulty believing it. In fact, it seems that mathematicians can reasonably argue their arguments, so if anything, a "manifestly valid" schema should be very quickly descriptible, so brief as a schema in formal FOL. So, my instinct is that they should be able to be described, if not at most, quite soon, or at least they can be proved together by a computer with modus ponens and a rule of quantification drawn from a very short list of axiomatic schemas (after all, the schemas should for example probably have a small upper limit in their length).
However, if no one has described these seemingly valid schemes of propositions (for example, specified them as axioms), I have a hard time understanding how mathematicians can claim that there is a formality in their arguments. It is true that theories on material sets are supposedly fundamental, that is to say that we think that the arguments of mathematicians "could be" transferred to a simple set of rules proof, 3) what exactly is the value of this type of rule. "conviction", that is, is there perhaps some kind of formal argumentation that I could formally see as having meaning, but which is distinct from the specification of "clearly valid" patterns?
I would also be interested in what follows. Even if I find a suitable way for FOL to work fairly well, 4) when will I get by with hardware issues? and 5) is a kind of theory of types already a viable solution, that is to say that someone has actually shown that it would not be necessary, for example, to have a kind of theory of types? to dynamically assume "clearly valid" claims or to use a calculation otherwise to describe the system (or is it perhaps at least likely)?
Overall, I have so many questions about the formalization ability of type theory and I have not found any source that answers satisfactorily. For example, I would like to point out 6) whether the classical demonstration in type theory is such a fundamental concept that one can be assured of not having to argue in meta-theory, that is, to say outside the theory of types itself (which seems to be a major problem weak system as theories of material sets)? I have not yet found a book that could even correctly describe any theory of expressible formal types, so that I can really approach the theory of types.