Grundgesetze

It remains the case as I write these words, in 2004, that there is no published translation of Gottlob Frege's Grundgesetze der Arithmetik. That's partly my fault, I suppose. Some years ago, Jason Stanley and I began such a translation of Part II (which is concerned with arithmetic). It has been used in courses taught by myself and by others, but, for a variety of reasons (including the need to get tenure), we never finished it. There is now a group at Arché working on a complete translation, but it could still be some time before they finish theirs, and it will be even longer before it is published.

I am therefore posting here the very rough translation Jason and I did. Please note the words "very rough". This work was a very preliminary draft, intended largely to (i) make my life easier, since my German isn't very good, and (ii) make Grundgesetze accessible to more people. Please do not rely upon this translation for serious research purposes: Check the German before you quote anything in English.

The translation is copyrighted by Richard Heck and Jason Stanley, 1992. You are welcome to make free use of this translation for scholarly and educational purposes only. Please let us know of any mistakes you may find.

To view these files, you will need an appropriate sort of reader. Adobe Acrobat will work with the PDF files. But please consider using an open source viewer, such as Ghostview, which will handle not only the PDF files but also Postscript). Versions exist for all of the major desktop OSs. If you use MacOS or Linux, you have even more choices. But then, you probably already knew that.

Notes

1. The translations of sections Nu, Xi, and Omicron are incomplete: The "Analysis" sections are done, but the formal proofs themselves are missing. I hope to complete scanning them before too long.
2. These are big files. I have split the translation into a dozen or so smaller files, divided by the sections of Part II, to make downloading easier. I am also posting compressed versions of the whole batch, both as a zip file and a bzip2 tarball. These are 8.8MB and 8.7MB, respectively.
3. I have had some display problems with these files using XPDF and other viewers based upon it, such as Gnome PDF. I have had no problems with GhostView and viewers based upon it, such as Gnome GhostView.

Here are the files. These links will open in this window.

• Title Page
• Section Alpha
  The proof of the "safe" direction of Hume's Principle: If the Fs are in one-one correspondence with the Gs, then the number of Fs is the same as the number of Gs.
• Section Beta
  The proof of the "unsafe" direction of Hume's Principle and of the fact that predecession is many-one: ∀x∀y∀z(Pxy & Pxz → y=z).
• Section Gamma
  The proof that predecession is one-many: ∀x∀y∀z(Pxz & Pyz → x=y) and so one-one.
• Sections Delta and Epsilon
  Proofs of various facts about zero and one.
• Sections Zeta, Eta, and Theta
  Section Zeta contains the proof that there are no "loops" in the number series: ∀x(P^*=0x → ¬P*xx). Section Eta contains the proof of the crucial theorem that every number has a successor. Section Theta contains a couple of subsidiary results.
• Section Iota
  Contains proofs of various facts about the number Frege calls "Endlos", which is defined as the number of natural numbers. The most important of these are (i) that Endlos is not a finite number; (ii) that, in Frege's words, "If Endlos is the number of a concept, then the objects falling under the concept can be ordered in an unbranching series which begins with a certain object and, without coming back on itself, continues without end" (or, as Dedekind would put it, they can be ordered as a simply infinite sequence); and (iii) the converse of (ii), thus yielding a characterization of countably infinite concepts.
• Section Kappa
  Contains a proof of a similar result concerning concepts whose number is finite: "[T]he number of a concept is finite if the objects falling under it may be ordered in a simple (unbranching, not coming back on itself) series which begins with a certain object and ends with a certain object."
• Section Lambda
  The proof of the converse of the theorem from Kappa, thus yielding a characterization of finite concepts.
• Section Mu
  The main theorem here is that, if S⊂T, and T is countably infinite, then S is either finite or countably infinite. Subsection a contains a proof of a form of the least number principle.
• Section Nu
  The proof that every subset of a finite set is finite.
• Title Xi
  Proofs of some results regarding cardinal addition.
• Section Omicron
  Proofs of some further results regarding finite and infinite sets.
• Tables of Definitions and Basic Laws
• Table of Important Theorems
• The whole shebang as a zip file
• The whole shebang as a bzip2 tarball