Frege's Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction. Abstract. Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze. Gottlob Frege - Paragraphen in moderner Notation - Corrigenda nicht . im Auge gehabt und in meinen Grundlagen der Arithmetik vom.


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We will examine these derivations in the following sections. Both are biconditionals asserting the equivalence of an identity among singular terms the left-side condition with an equivalence relation on concepts the right-side condition.

Indeed, both correlate concepts with certain objects.

Grundgesetze der arithmetik

See the independent work of Geach—7Hodes, Burgess and Hazen They are included here for those who wish to have a more complete understanding grundgesetze der arithmetik frege what Frege in fact attempted grundgesetze der arithmetik frege do. Frege in fact identifies the cardinal number 2 with this extension, for it contains all and only those concepts under which two objects fall.

Similarly, Frege identifies the cardinal number 0 with the extension consisting of all those first-level concepts under which no object falls; this extension would include such concepts as unicorn, centaur, prime number between 3 and 5, etc.

We know that Basic Law V does not offer such a coherent conception. In other words, the proof relies on a kind of higher-order version of the Law of Extensions described abovethe ordinary version of which we know to be a consequence of Basic Law V.

In Gg, extensions do not contain concepts as members but rather objects. So Frege had to find another way to express the explicit definition described in the previous subsection.

Heck : Grundgesetze der Arithmetik I §§29‒32

His technique was to let extensions go proxy for their corresponding concepts. Instead, he derives both directions separately without combining them or indicating that the two directions should be conceived as a biconditional.

In the next section, we go through the proof. The first is that the following series of concepts has a rather interesting grundgesetze der arithmetik frege Frege realized that though we may identify this sequence of numbers with the natural numbers, such a sequence is simply a list: No two natural numbers have the same grundgesetze der arithmetik frege.

Frege's Theorem and Foundations for Arithmetic

Principle of Mathematical Induction Every natural number has a successor. Moreover, Frege recognized the need to employ the Principle of Mathematical Induction in the proof that every number grundgesetze der arithmetik frege a successor. One cannot prove the claim that every number has a successor simply by producing the sequence of expressions for cardinal numbers e.

All such a sequence demonstrates is that for every expression listed in the sequence, one can define an expression of the appropriate form to follow it in the sequence.

Frege's Theorem and Foundations for Arithmetic (Stanford Encyclopedia of Philosophy)

This is not grundgesetze der arithmetik frege same as proving that every natural number has a successor. Russell falls under grundgesetze der arithmetik frege concept author of Principia Mathematica, i.

In what follows, we sometimes introduce other such abbreviations. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever.


Grundgesetze is thus a work of much greater significance than has often been thought. Frege wrote a hasty, last-minute Appendix to Vol. Frege opened the Appendix with the exceptionally honest comment: This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion.

Frege's proposed remedy was subsequently shown to imply that there is but one object in the universe of discourseand hence is worthless indeed, this would make for a contradiction in Frege's system if he had axiomatized the idea, fundamental to his discussion, that the True and the False grundgesetze der arithmetik frege distinct objects; see, for example, Dummettbut recent work has shown that much of the program of the Grundgesetze might be salvaged in other ways: Basic Law V can be weakened in other ways.

The best-known way is due to philosopher and mathematical logician George Boolos —who was an expert on the work of Frege. A "concept" F is "small" if the objects falling under F cannot be put into one-to-one correspondence with the universe of discourse, that is, unless: Basic Law V can simply be grundgesetze der arithmetik frege with Hume's principlewhich says that the number of Fs is the same as the number of Gs if and only if the Fs can be put into a one-to-one correspondence with the Gs.