Home
Hostname: page-component-5c569c448b-4wdfl Total loading time: 0.182 Render date: 2022-07-03T04:13:20.206Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

# A note on the Entscheidungsproblem

Published online by Cambridge University Press:  12 March 2014

## Extract

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.

In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.

Consider the system L of symbolic logic which arises from the engere Funktionenkalkül by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

Type
Research Article
Information
The Journal of Symbolic Logic , June 1936 , pp. 40 - 41

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

## References

1 An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936)Google Scholar.

2 Grundzüge der theoretischen Logik, Berlin 1928Google Scholar.

3 Definitions of the terms well-formed formula and convertible are given in the cited paper.

4 Cf. Ackermann, Wilhelm, Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit, Mathemaiische Annalen, vol. 93 (19241925), pp. 1136Google Scholar; Neumann, J. v., Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift, vol. 26 (1927), pp. 146Google Scholar; Herbrand, Jacques, Sur la non-contradiction de l'arithmétique, Journal für die reine und angewandte Mathematik, vol. 166 (19311932), pp. 18Google Scholar.

5 In lectures at Princeton, N. J., 1936. The methods employed are those of existing consistency proofs.

6 By the Entscheidungsproblem of a system of symbolic logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system. Hilbert and Ackermann (loc. cit.) understand the Entscheidungsproblem of the engere Funktionenkalkül in a slightly different sense. But the two senses are equivalent in view of the proof by Kurt Gödel of the completeness of the engere Funktionenkalkiil (Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349360Google Scholar).

7 From this follows further the unsolvability of the particular case of the Entscheidungsproblem of the engere Funktionenkalkül which concerns the provability of expressions of the form (Ex 1)(Ex 2)(Ex 3)(y 1)(y 2) …(y n)P, where P contains no quantifiers and no individual variables except x1, x2, x3, y1, y2, …, yn. Cf. Gödel, Kurt, Zum Entscheidungsproblem des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 433–143Google Scholar.

273
Cited by

# Save article to Kindle

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A note on the Entscheidungsproblem
Available formats
×

# Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

A note on the Entscheidungsproblem
Available formats
×

# Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

A note on the Entscheidungsproblem
Available formats
×
×

#### Conflicting interests

Do you have any conflicting interests? *