In [2] John Burgess describes predicative versions of Frege's logic and poses the problem of find... more In [2] John Burgess describes predicative versions of Frege's logic and poses the problem of finding their exact arithmetical strength. I prove here that PV, the simplest such theory, is equivalent to Robinson's arithmetical theory Q.
It is shown that Robinson's arithmetic Q is in fact equivalent to the Fregean system with an open... more It is shown that Robinson's arithmetic Q is in fact equivalent to the Fregean system with an open comprehension schema (SPV), and thus can be claimed to be a strictly predicative theory.
A notion of logical computation is introduced and shown to be equivalent to Turing computation, t... more A notion of logical computation is introduced and shown to be equivalent to Turing computation, thus providing another argument for the Church-Turing thesis.
The Church-Turing thesis is given a provable interpretation based on the idea that a computation ... more The Church-Turing thesis is given a provable interpretation based on the idea that a computation by an idealized human agent must be a logically definable finite mathematical object. The argument is preserved under a large variation in the expressive power of the underlying logical language, thus providing a possible explanation of why the notion of effective computability is so robust.
Robinson’s arithmetic Q is given a simple interpretation in Hajek’s weaker relational version QH ... more Robinson’s arithmetic Q is given a simple interpretation in Hajek’s weaker relational version QH that does not use Solovay’s technique of shortening cuts. The result is placed within two research themes regarding relational arithmetics.
Relations between some theories of semigroups (also known as theories of strings or theories of c... more Relations between some theories of semigroups (also known as theories of strings or theories of concatenation) and arithmetic are surveyed. In particular Robinson's arithmetic Q is shown to be mutually interpretable with TC, a weak theory of concatenation introduced by Grzegorczyk. Furthermore, TC is shown to be interpretable in the theory F studied by Tarski and Szmielewa, thus confirming their claim that F is essentially undecidable. §0. Introduction. Prior to Gödel's work on the arithmetization of syntax Tarski developed a rigorous mathematical treatment of the syntax of formal languages in . He used a second-order version of the theory of concatenation on strings of symbols from a finite alphabet, or, from an algebraic viewpoint, the theory of free semigroups with a finite set of generators. Much later, after important work by Quine and Bennett, Corcoran, Frank, and Maloney investigated the relations between Tarski's theory, a generalized successor theory put forward by Hans Hermes in [10] and second-order Peano arithmetic 1 .
the objects of a finitistic theory are finite (given in intuition), such as finite sequences of s... more the objects of a finitistic theory are finite (given in intuition), such as finite sequences of symbols. On the other hand it is unclear what principles regarding these objects can be considered finitistic. Within the Hilbert school there was considerable indecision over this point.
It can be argued that only the equational theories of some sub-elementary function algebras are f... more It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition (Ganea 2010). The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's 'predicative arithmetic' program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly as a restricted form of logicism.
Determining the computational requirements of formal reasoning is a key task in implementing Davi... more Determining the computational requirements of formal reasoning is a key task in implementing David Hilbert's foundational program. This paper presents a number of results on the syntactical complexity of some versions of the language of fi rst-order logic (L FOL ), some of which are already known. Two versions of L FOL with ineffi cient (tally) indexing are shown to be context-free, and the sets of their sentences (the sub-language L S-FOL ) and a version of L FOL with effi cient (positional) indexing are shown to be not context-free. The latt er is not even the intersection of a fi nite number of context-free languages but is in D-LogSpace and consequently rudimentary in Smullyan's sense.
Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the ex... more Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the existence of what might be called absolutely undecidable statements. Neil Tennant attacks epistemic optimism, the view that there are no such statements. I expose what seem serious flaws in his attack, and I suggest a way of defending the use of classical logic in arithmetic that circumvents the issue of optimism.
The following is a reconstruction and evaluation of the final argument for the immortality of the... more The following is a reconstruction and evaluation of the final argument for the immortality of the soul in Plato's Phaedo. The argument can be seen as the culmination of the dialectical part of the dialogue; what comes after it is the mythical description of afterlife by Socrates (107e-114d), the preparation for the execution (115a-117a), and the death scene (117b-118). It is unclear to me though whether this circumstance is to be interpreted as a literary device whereby Plato singles out the argument in question as carrying a special weight. The criteria of length and elaboration seem to indicate however that it does not have a special status in the collection of arguments put forward in the text: for instance the argument from recollection (72e-77a) is at least as meticulously articulated.
In [2] John Burgess describes predicative versions of Frege's logic and poses the problem of find... more In [2] John Burgess describes predicative versions of Frege's logic and poses the problem of finding their exact arithmetical strength. I prove here that PV, the simplest such theory, is equivalent to Robinson's arithmetical theory Q.
It is shown that Robinson's arithmetic Q is in fact equivalent to the Fregean system with an open... more It is shown that Robinson's arithmetic Q is in fact equivalent to the Fregean system with an open comprehension schema (SPV), and thus can be claimed to be a strictly predicative theory.
A notion of logical computation is introduced and shown to be equivalent to Turing computation, t... more A notion of logical computation is introduced and shown to be equivalent to Turing computation, thus providing another argument for the Church-Turing thesis.
The Church-Turing thesis is given a provable interpretation based on the idea that a computation ... more The Church-Turing thesis is given a provable interpretation based on the idea that a computation by an idealized human agent must be a logically definable finite mathematical object. The argument is preserved under a large variation in the expressive power of the underlying logical language, thus providing a possible explanation of why the notion of effective computability is so robust.
Robinson’s arithmetic Q is given a simple interpretation in Hajek’s weaker relational version QH ... more Robinson’s arithmetic Q is given a simple interpretation in Hajek’s weaker relational version QH that does not use Solovay’s technique of shortening cuts. The result is placed within two research themes regarding relational arithmetics.
Relations between some theories of semigroups (also known as theories of strings or theories of c... more Relations between some theories of semigroups (also known as theories of strings or theories of concatenation) and arithmetic are surveyed. In particular Robinson's arithmetic Q is shown to be mutually interpretable with TC, a weak theory of concatenation introduced by Grzegorczyk. Furthermore, TC is shown to be interpretable in the theory F studied by Tarski and Szmielewa, thus confirming their claim that F is essentially undecidable. §0. Introduction. Prior to Gödel's work on the arithmetization of syntax Tarski developed a rigorous mathematical treatment of the syntax of formal languages in . He used a second-order version of the theory of concatenation on strings of symbols from a finite alphabet, or, from an algebraic viewpoint, the theory of free semigroups with a finite set of generators. Much later, after important work by Quine and Bennett, Corcoran, Frank, and Maloney investigated the relations between Tarski's theory, a generalized successor theory put forward by Hans Hermes in [10] and second-order Peano arithmetic 1 .
the objects of a finitistic theory are finite (given in intuition), such as finite sequences of s... more the objects of a finitistic theory are finite (given in intuition), such as finite sequences of symbols. On the other hand it is unclear what principles regarding these objects can be considered finitistic. Within the Hilbert school there was considerable indecision over this point.
It can be argued that only the equational theories of some sub-elementary function algebras are f... more It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition (Ganea 2010). The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's 'predicative arithmetic' program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly as a restricted form of logicism.
Determining the computational requirements of formal reasoning is a key task in implementing Davi... more Determining the computational requirements of formal reasoning is a key task in implementing David Hilbert's foundational program. This paper presents a number of results on the syntactical complexity of some versions of the language of fi rst-order logic (L FOL ), some of which are already known. Two versions of L FOL with ineffi cient (tally) indexing are shown to be context-free, and the sets of their sentences (the sub-language L S-FOL ) and a version of L FOL with effi cient (positional) indexing are shown to be not context-free. The latt er is not even the intersection of a fi nite number of context-free languages but is in D-LogSpace and consequently rudimentary in Smullyan's sense.
Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the ex... more Michael Dummett's argument for intuitionism can be criticized for the implicit reliance on the existence of what might be called absolutely undecidable statements. Neil Tennant attacks epistemic optimism, the view that there are no such statements. I expose what seem serious flaws in his attack, and I suggest a way of defending the use of classical logic in arithmetic that circumvents the issue of optimism.
The following is a reconstruction and evaluation of the final argument for the immortality of the... more The following is a reconstruction and evaluation of the final argument for the immortality of the soul in Plato's Phaedo. The argument can be seen as the culmination of the dialectical part of the dialogue; what comes after it is the mythical description of afterlife by Socrates (107e-114d), the preparation for the execution (115a-117a), and the death scene (117b-118). It is unclear to me though whether this circumstance is to be interpreted as a literary device whereby Plato singles out the argument in question as carrying a special weight. The criteria of length and elaboration seem to indicate however that it does not have a special status in the collection of arguments put forward in the text: for instance the argument from recollection (72e-77a) is at least as meticulously articulated.
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