That is, there is no reasonable list of axioms from which we can prove exactly all true statements of number theory (Gödel 1931).
On Frege's view, the statements of number which answer such questions (e.g., "There are n Fs") tell us something about the concept involved.
2.5 The Analysis of Statements of Number In what has come to be regarded as a seminal treatise, Die Grundlagen der Arithmetik (1884), Frege began work on the idea of deriving some of the basic principles of arithmetic from what he thought were more fundamental logical principles and logical concepts.
In this way, Frege analyzed a statement of number ('there are two authors of Principia Mathematica') as higher-order logical statements about concepts.
The seminal idea of Gl §46 was the observation that a statement of number (e.g., "There are eight planets") is an assertion about a concept.
But it was just the analysis of ordinary language that led Frege to his insight that a statement of number is an assertion about a concept.
The leading idea is that a statement of number, such as 'There are eight planets' and 'There are two authors of Principia Mathematica', is really a statement about a concept.
Gödel's theorem does not merely claim that such statements exist: the method of Gödel's proof explicitly produces a particular sentence that is neither provable nor refutable in F; the "undecidable" statement can be found mechanically from a specification of F. The sentence in question is a relatively simple statement of number theory, a purely universal arithmetical sentence.
The database that we have developed contains added material such as algorithms, consistent statements of the number of required terms, and verified computer programs for evaluating eigenvalues and solutions.
Furthermore, there was no statement of the number of minimal invasive FTC included in the study, and the total number of cases used for immunohistochemistry was limited to27 carcinomas and 22 adenomas.
