Logic in the metatheory

In Goldstern and Judah's The Incompleteness Phenomenon we are asked to confirm that any kind of version of the first 2 Peano Axioms : $$\forall x [Sx\neq0]$$ $$\forall x\forall y[Sx=Sy\implies x=y]$$ have to be boundless. S is a unary function planned as follower. If the version is limited there have to be $m$ and also $n$ with $S^m0=S^n0$ where the superscript stands for model. After that $|n-m|$ applications of the 2nd axiom bring about an opposition with the first.

We are plainly saying in the metatheory, as our axioms have no suggestions of boundless collections. Exists a clear definition of what axioms and also logic are permitted below? Can you recommend a publication on the topic?

2019-12-02 02:49:24
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Answers: 2

You are doing it indirectly, the straight means to do this is by specifying a one to one function from $N$ to the participants of the version.

So allow $M$ be a version of the first 2 axioms. (To able to to speak about the version you require a concept that can speak about collections to some level or a concept that can analyze that quantity of set concept.)

The next point is specifying the function inductively in the noticeable means and afterwards make use of induction and also the reality that $M$ pleases the first and also 2nd axiom to confirm that the function is one to one.

2019-12-04 09:20:23

There are 2 means to manage this.

One of the most usual means is merely to think some kind of set concept in the metatheory, as an example ZFC. After that you can make use of all the regular semantic and also model - logical approaches that you are made use of to. The benefits of this method are that it matches the manner in which mathematicians in fact think of points, which it permits us to make use of all the theories we are made use of to.

An additional alternative is to reinterpret cases made in the metatheory as syntactic cases. As an example, the case "the version have to be infinite" can be reinterpreted to suggest that the concept itself (!) confirms each of the sentences $$ \lnot (\exists c_0) (\forall x) (x = c_0) $$ $$ \lnot (\exists c_0, c_1) (\forall x) (x = c_0 \lor x = c_1) $$ $$ \lnot (\exists c_0, c_1, c_2) (\forall x) (x = c_0 \lor x = c_1 \lor c = c_2) $$ and also even more usually, for each and every $k > 0$, $$ \lnot (\exists c_0, \ldots, c_k ) (\forall x) (x = c_0 \lor \cdots \lor c = c_k) $$

The reality that each of these sentences is conclusive in the concept can be validated making use of a really weak metatheory. There is a straightforward and also consistent means, for any kind of set $k$, to create an evidence from your 2 axioms for that certain $k$.

This reinterpretation procedure is popular to logicians, and also it can be used really usually. The major constraint is whether the building concerned can be mentioned in the language of the concept itself (perhaps as a boundless set of sentences, similar to the instance over).

Gödel is efficiency theory claims that if a sentence in the language of a concept holds true (semantically) in every version of the concept, then that sentence is conclusive (syntactically) within the concept itself. So we could, if we desired, overlook the "true" component in those instances and also concentrate simply on the "provable" component. The price of doing that is that it perhaps misreads, given that we generally assume semantically concerning versions, which it does not relate to buildings of versions that can not be shared within the language of the concept itself.

Due to the fact that this reinterpretation procedure is generally really regular, it is uncommon for logicians to trouble to state it, despite the fact that we are cognizant maybe done. We simply write in the regular version - logical (set logical) means unless there is a certain factor that we require to concentrate on the syntactic analysis.

2019-12-03 04:23:27