On the Consistency of the Δ1 1-CA Fragment of Frege's Grundgesetze

Plural Grundgesetze

“…See also [1]. 5 See [3] 119. 6 I claim that PG is at least as strong as third-order Peano arithmetic.…”

“…Given the formulae permitted on the right-hand side of PRC, extension-terms containing Σ 1 1 -formulae of the form ∃X(xηX) are allowed in PG, beside Σ 0 1 -terms of the form {x : F x}. 8 [8] at most interprets Robinson arithmetic, whereas it is not clear how much arithmetic [5] and [12] interpret. These latter, though, can hardly be expected to get to second-order Peano arithmetic.…”

“…[5], [8], [12], where comprehension is restricted: [8] employs predicative comprehension, while both [5] and [12] adopt Δ 1 1 -comprehension. Beyond Δ 1 1 -comprehension, inconsistency reappears.…”

THE CONVENIENCE OF THE TYPESETTER; NOTATION AND TYPOGRAPHY IN FREGE’SGRUNDGESETZE DER ARITHMETIK

“…by nonsense words (azig, bezig, zezig, arung, berung, poll, etc.). 18 Any influence of prior knowledge of the subject matter is thereby ruled out, and Frege argues that the sense conveyed upon these unknown words and symbols by Cantor's explanation is not sufficiently precise to serve as a proper definition of these notions. The text in Figure 8 We are thus here taken back to Frege's methodological reason for the adoption of unfamiliar symbols in his own development of arithmetic that we mentioned above ( §3).…”

“…Since the discovery of what is now called Frege's Theorem [33,58] -the proof that the axioms of arithmetic can be derived in second-order logic using Hume's Principle (a principle governing the identity of cardinal numbers and derived by Frege from Basic Law V) and Frege's definition of zero, predecession and natural number -there has been a revival of the logicist idea often under the heading neo-Fregeanism or neo-logicism, in both philosophy and mathematics. Removing valueranges (and Basic Laws V and VI, which feature them), Frege's system is essentially standard second-order logic, and it has also been shown that fragments of the Grundgesetze system using predicative versions of Basic Law V are consistent [2,5,18,32,56]. This recent revival has also lead to a closer study of Frege's original writings (e.g., [34]) and lead to the first full translation of Frege's magnum opus: Grundgesetze der Arithmetik into English [28].…”