# Extensions of Some Theorems of Gödel and Church

@article{Rosser1936ExtensionsOS, title={Extensions of Some Theorems of G{\"o}del and Church}, author={J. Barkley Rosser}, journal={J. Symb. Log.}, year={1936}, volume={1}, pages={87-91} }

We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Godel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Godel calls “rekursiv” will be called “primitive recursive.” By an “ Entscheidungsverfahren ” will be meant a general recursive function ϕ ( n ) such that, if n is the Godel number of a provable formula, ϕ ( n ) = 0 and, if n is not the… Expand

#### Topics from this paper

#### 322 Citations

Gödel's Third Incompleteness Theorem

- Mathematics
- 2016

In a note appended to the translation of “On consistency and completeness” (1967), Godel reexamined the problem of the unprovability of consistency. Godel here focuses on an alternative means of… Expand

Arithmetical Proof and Open Sentences

- Mathematics
- 2012

If the concept of proof (including arithmetic proof) is syntactically restricted to closed sentences (or their Gödel numbers), then the standard accounts of Gödel’s Incompleteness Theorems (and Löb’s… Expand

Some properties of pseudo-complements of recursively enumerable sets

- Mathematics
- 1966

Introductory remarks. Those first order systems which exhibit some real mathematical pretensions fall into what is called in [1] the class of arithmetical logics; it is there demonstrated that that… Expand

Internal and external consistency of arithmetic

- Mathematics
- 2001

What Godel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the… Expand

A case for weakening the Church-Turing Thesis

- Mathematics
- 2011

We conclude from Godel's Theorem VII of his seminal 1931 paper that every recursive function f(x1,x2) is representable in the first-order Peano Arithmetic PA by a formula (F(x1,x2,x3)) which is… Expand

Hilberťs Programme and Gödel's Theorems

- Philosophy, Mathematics
- 2005

In this paper, we attempt to show that a weak version of Hilberťs metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates.… Expand

Character Strings and Gödel's Incompleteness Proof

- Computer Science, Mathematics
- AFL
- 2014

A rather easy yet rigorous proof of a version of Gödel’s first incompleteness theorem is presented. The version is “each recursively enumerable theory of natur al numbers with 0, 1, +, ·, =,∧, ¬,… Expand

On the Philosophical Relevance of Godel's Incompleteness Theorems

- Philosophy
- 2005

Godel began his 1951 Gibbs Lecture by stating: "Research in the founda tions of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves,… Expand

Can Gödel's Incompleteness Theorem be a Ground for Dialetheism?

- Philosophy
- 2017

【Abstract】Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of… Expand

On the Truth of G\"odelian and Rosserian Sentences

- Mathematics
- 2020

There is a longstanding debate in the logico-philosophical community as to why the Godelian sentences of a consistent and sufficiently strong theory are true. The prevalent argument seems to be… Expand

#### References

SHOWING 1-2 OF 2 REFERENCES

An Unsolvable Problem of Elementary Number Theory

- Mathematics
- 1936

Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use… Expand