Greater Order Logics
I've reviewed around higher-order logics (i.e. those that improve first-order predicate logic) yet am not also clear on their applications. While they can sharing a better series of evidence (though never ever all, by Godel's Incompleteness theory), they are usually claimed to be much less "mannerly".
Mathematicians usually appear to remain free from such reasonings when feasible, yet they are absolutely essential for prooving some extra difficult concepts/theorems, as I recognize. (As an example, it appears the reals can just be created making use of at the very least 2nd order logic.) Why is this, what makes them much less mannerly or much less valuable relative to logic/proof theory/other areas?
Second - order logic does not please the efficiency and also density theories. Below is an evidence that the density theory falls short (which itself indicates that the efficiency theory falls short, due to the fact that efficiency indicates density). Particularly, there is a 2nd - order means (call it $F$) of sharing that a set is limited : every injective function on the set is surjective. This suggests that if $P$ is a relationship on the item set $S \times S$ such that $P$ pleases the problems to be a function, and also $P(x,y), P(z,y)$ indicate $x=z$, after that for all $w$ there is $q$ with $P(w,q)$. This declaration is evaluated over $P$ along with the variables $x,y,z,w,q$ so is 2nd - order and also plainly shares finiteness.
Nonetheless, we additionally have the declaration $T_n$ "there exist distinctive $x_0, \dots x_n$ in the set," which is also first order, and also mentions that the set has cardinality at the very least $n$. So the combination of $F$ and also any kind of limited collection of the $T_n$ is satisfiable, yet every one of them with each other can not be pleased.
Unfortunately the term is unclear : there 2 sort of semiotics of greater - order languages, and also just one is bothersome. Take into consideration the language of 2nd - order arithmetic, where there are quantifiers over both all-natural numbers and also collections of all-natural numbers.
First is what Quine called "set - concept in lamb's apparel" : this is where metrology over collections of all-natural numbers is specified to be over all the collections of number that can be assumed. It's the concept we make use of when we confirm that there can be just one full, entirely gotten area. It isn't actually a logic, there's no full idea of evidence formalisation for it. Wikipedia calls this "typical semiotics" ; I'm not exactly sure if there is an authority for this.
After that there's Henkin semiotics, which makes use of the regulation similar to the lambda - calculus to specify a semiotics for 2nd - order quantifiers. This can be viewed as still in the world of first - order logic, because feeling that the 2nd - order system can be converted right into a first - order system preserving provability. This is just how 2nd - order arithmetic is specified.
All the theories of the Henkin semiotics will certainly be "theories" in the first.
A more feeling in which Higher Order Logics with typical or saturated semiotics (HOL, hereafter) are much less mannerly than First Order Logic (FOL, hereafter) is a straight effects of the failing of Completeness (and also hence, as clarified in various other solutions, of Compactness). The set of sensible facts and also the set of proper cases of semantic effect for these reasonings are not recursively enumerable.
FOL is Complete, yet not Decidable. So, we establish of an approximate sentences and also collections of sentences of the language of FOL if those sentences are sensible facts or if a set has actually an offered sentence therefore. Yet, given that FOL is Complete and also evidence are finitely long, we can (in the mathematicians feeling of "can") identify the evidence and also evaluate individually, examining what sentence the evidence reveals as a theory or what sentence the evidence stems from what set of presumptions. This obtains us a recursive list of the facts and also the sentence/set sets that stand in the effect relationship. (This does not negate the failing of decidability as we can not end that given that we've yet ahead throughout an evidence in out list, there isn't one so we maintained looking.)
Given that HOL's are not Complete, this methods of revealing them recursively enumerable is not readily available. Without a doubt, there can be no methods ; existed such a method, maybe manipulated to generate a Complete evidence system, and also there can not be such a Complete evidence system as the HOL's are not Compact.