Talk:Gödel's β function

Falsche rem-Funktion
Ich habe die Defintion von $$rem(x,y)$$ mal angepasst. Es muss Rest bei $$y/x$$ sein, anstatt $$x/y$$.

Im Beweis in Mendelson werden $$u_i$$ konstruiert, sodass $$rem(u_i, b) = k_i$$ sein soll. Das $$b$$ soll es nach Chinesischem Restsatz geben. Dort sind die Moduln jedoch eben die $$u_i$$, sodass in $$rem$$ "durch sie geteilt" werden muss. —Preceding unsigned comment added by 88.74.91.131 (talk) 07:57, 6 August 2010 (UTC)


 * Danke. You're right that the variables were backwards. &mdash; Carl (CBM · talk) 13:23, 6 August 2010 (UTC)

Origin of the β function and lemma
Neither the β function nor the β lemma are mentioned in Gödel's incompleteness article:

Kurt Gödel: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsheft für Mathematik und Physik, volume 38, pages 173-198, 1931

An English transaltion is given in

Solomon Feferman, John W. Dawson, Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, editors: Kurt Gödel - Collected Works, Oxford, UK, and New York, USA, pages 144-194

I believed the following: We can therefore safely assume that the β function nor the β lemma have been devised by Elliott Mendelson for his treatment of Gödel's First Incompeteness Theorem. Indeed, that function and lemma are used in Mendelson's book for a complete proof of definability which is not given (but only sehr briefly sketched) in Gödel's incompleteness article.

This was not ccorrect: the β function has been introduiced wirthout its name in Gödel's incomleterness article of 1931. Gödel gaver the function its name in 1934 a talk.

I added the information corresponding references. In my opinion, the issue is resolved.

The remainder function definable in Q
The article currently claims "the remainder function ... is arithmetically definable", and then follows this up with mention of Robinson Arithmetic (usually denoted "Q"). It's not obvious that $$rem$$ is definable in Q! So at the very least, I think a citation would be in order, or an explanation why 220.244.237.15 (talk) 09:22, 8 May 2022 (UTC)


 * Since Robinson arithmetic admits multiplication, my first guess for a definition of rem would be $$ z = rem(x,y) \;\;\stackrel{def}{\Longleftrightarrow}\;\; (\exists d. z+d = y) \land (\exists q. q \cdot y+z = x)$$. - Jochen Burghardt (talk) 18:02, 9 May 2022 (UTC)

New section Elimination of Parameters
I made a new section "All primitive recursive functions".

I move the section Elimination of Parameters from there:

https://en.wikipedia.org/wiki/Primitive_recursive_function

To this article here. Jan Burse (talk) 00:12, 29 February 2024 (UTC)