arXiv subjects

Payam Seraji

Publications and source records attributed to Payam Seraji.

3 recordsLinked to original sources

arXiv2018

On Constructivity and the Rosser Property: a closer look at some Gödelean proofs

The proofs of Kleene, Chaitin and Boolos for Gödel's First Incompleteness Theorem are studied from the perspectives of constructivity and the Rosser property. A proof of the incompleteness theorem has the Rosser property when the independence of the true but unprovable sentence can be shown by assuming only the (simple) consistency of the theory. It is known that Gödel's own proof for his incompleteness theorem does not have the Rosser property, and we show that neither do Kleene's or Boolos' proofs. However, we show that a variant of Chaitin's proof can have the Rosser property. The proofs of Gödel, Rosser and Kleene are constructive in the sense that they explicitly construct, by algorithmic ways, the independent sentence(s) from the theory. We show that the proofs of Chaitin and Boolos are not constructive, and they prove only the mere existence of the independent sentences.

math.LO↗

arXiv2016

Godel-Rosser's Incompleteness Theorems for Non-Recursively Enumerable Theories

Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true $Π_n$-sentences or equivalently the $Σ_n$-soundness of the theory, and the other is $n$-consistency the restriction of $ω$-consistency to the $Σ_n$-formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense. Though the proof of the incompleteness theorem using the $Σ_n$-soundness assumption is constructive, it is shown that there is no constructive proof for the incompleteness theorem using the $n$-consistency assumption, for $n\!>\!2$.

math.LO↗

arXiv2016

Godel's Second Incompleteness Theorem for Definable Theories

It is proved that if $T$ is a $Σ_{n+1}$ Definable theory which is $Σ_n$-sound and extends $PA$, then $T$ can not prove the sentence $Σ_n-sound(T)$ that expresses the $Σ_n$-soundness of $T$. Optimality of this result is showed by constructing a $Σ_{n+1}$-definable and $Σ_{n-1}$-sound theory extending $PA$ such that $Σ_n-sound(T)$ is $T$-provable. It is also proved that no R.E. arithmetical theory, evevn very weak theories which are not $Σ_1$-complete, can prove $Σ_1$-soundness of itself.

math.LO↗