Proof Theory: The First Step into Impredicativity (Universitext)


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Abbreviating x1 ,. Proof We induct on n.

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A cut possesses no critical formula. It is exactly the proof of Lemma 5. See Note 7. As a consequence of Theorem 7. By Theorem 7. The following results masters this situation, too. Now the Basic Elimination Lemma. In his sense, a semi-formal system consists of a decidable2 set T of axioms together with a collection R of rules. In a more strict sense, however, one requires that the premises of a rule of a semi-formal systems are primitive recursively enumerated, i.

Proof Theory, The First Step Into Impredicativity by Wolfram Pohlers | | Booktopia

Semi-formal systems in this strict sense have subtle applications in proof theory cf. It is not too difficult to see that the semiformal system of Definition 7. Also cut-elimination holds true for strict semi-formal systems. In this book, however, we will not need the properties of semi-formal systems in the strict sense.

Therefore we only introduced the liberated notion, omitting the requirement of primitive recursive enumerability of the premises of an infinite inference rule. This becomes especially clear by property 7. But observe that some of the properties of the verification calculus may be lost by turning it into a semi-formal system in the strict sense.

An example of such a property is in fact 7. This difference will become even more evident when we later introduce operator controlled derivations. In general, the verification calculus cannot be controlled by an operator. Hint: Just add the additional case of an atomic cut to the proof of Lemmas 6. There are only two cases, which need some attention. First the case of an axiom. But this is shown as in the proof of the Tautology Lemma Lemma 7.

To avoid this, we have to show that T T obvious by induction on m. The remaining cases are all simple. Formulating Mathematical Induction as a scheme is, however, inevitable. Once we have ii we also get 7. It remains to prove i. Again we work informally in NT. Assume Prog X. Theorem 7. Getting rid of free set variables could, however, be of importance for the separation of theories in bounded arithmetic cf. Corollary Confer the result with Exercise 7.

Proof Theory

Hint: Use mathematical induction to show NT0 niously. We start with some general remarks on the consistency of formal systems. For a theory T that contains some kind of infinity axiom it is impossible to prove by finitist means that T possesses a model. An infinite model cannot be constructed by finitist means.

For reasonable formal systems we can show that semantical and syntactical consistency coincide. To explain which formal systems are reasonable we need some preparations. Let us first recall some basic notions. A propositional atom is either an atomic formula or a formula whose outermost logical symbol is a quantifier. A propositional assignment is a map that assigns a truth value to every propositional atom.

Here we require that atoms that are dual in the Tait-language i. There are good reasons why this system is regarded as finitist that have been widely discussed in the literature. We do not want to add further arguments to that discussion. A formula is propositionally valid if it becomes true under all propositional assignments. We say that a rule A1 ,. A rule A1 ,. So T, R is syntactically consistent.

So T, R is syntactically inconsistent. To obtain the semantical consistency of NT it therefore suffices to show that there is a formula A that is not derivable from the axioms in NT. We will prove that via the syntactical consistency of the semi-formal system introduced in Definition 7. This, however, contradicts Lemma 7. Proof This follows from Corollary 7.


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Proof We already remarked that theories are always propositionally closed. So it remains to show that NT is syntactically consistent. Nevertheless, we want to at least touch this issue. First we observe that the infinitary proof tree constructed in the proof of Theorem 7. We will later extend the semi-formal system to a language with infinitely long formulas.

To show that these systems are also propositionally closed we will then have to reprove that all propositionally valid formulas are derivable.

Proof Theory: The First Step into Impredicativity (Universitext)

By a slight modification of the cut-elimination procedure it is even possible to define the reduction steps for non well-founded trees. So we have shown by transfinite induction along the notation system developed in Sect. On the other hand we have seen in Sect. In this sense the result is optimal. That this is malicious has been pointed out by Kreisel who presented the following counterexample.

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To sketch it let us work in PRA. But this implies F y , a contradiction.


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The well-ordering in the counterexample given here is admittedly extremely artifical. But whatever definition we choose, it is nearly impossible to avoid artifical counter examples. The true reason for that is the fact that we have to refer to presentations of ordinals by primitive recursive well-orderings or equivalently to ordinal notations instead of ordinals themselves.

Therefore these definitions become extremely sensitive to the way in which ordinals are presented. Our motivation, however, was mainly ordinal analysis and not primarily consistency. We want to discuss how tiny the extension really is. Of course we assume that our opponent is able to follow mathematical reasoning in an unbiased way.

This s he says is a bit beyond his or her finitist horizon. We avoid talking about ordinals and just want to convince him or her that the order relation we used in showing the syntactical consistency is well-founded.

Therefore we introduce the order relation as in Sect. We did not pay attention to define J X parsimoniously. But even under the most careful definition of J X we will have at least l quantifiers in the defining formulas for Jl X which means that Jl X belongs at least to the lth level in the arithmetical hierarchy. At this point our opponent will argue that in doing so we exhaust full first-order number theory and even a bit more.

Proof Theory: The First Step into Impredicativity (Universitext) Proof Theory: The First Step into Impredicativity (Universitext)
Proof Theory: The First Step into Impredicativity (Universitext) Proof Theory: The First Step into Impredicativity (Universitext)
Proof Theory: The First Step into Impredicativity (Universitext) Proof Theory: The First Step into Impredicativity (Universitext)
Proof Theory: The First Step into Impredicativity (Universitext) Proof Theory: The First Step into Impredicativity (Universitext)
Proof Theory: The First Step into Impredicativity (Universitext) Proof Theory: The First Step into Impredicativity (Universitext)
Proof Theory: The First Step into Impredicativity (Universitext) Proof Theory: The First Step into Impredicativity (Universitext)

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