Such a machine can not be constructed. That's what Turing established in the 1930's. This result is intimately related to the work of Gödel.A_Seagull wrote: The theorems that are generated are then necessarily true within that system, for they have been generated following the axioms and inferential logic of the system. And only those theorems that it generates can be considered to be statements of mathematics.
It is the role of mathematicians to design the axioms from which the machine can be constructed.
This article may be of interest. https://en.wikipedia.org/wiki/Entscheidungsproblem
From the article:
There is no way to program a computer to recognize when a given statement is a theorem following from a given set of axioms. This is subtly different from what you suggested, namely generating all theorems. According to the article you linked,Wiki wrote: In mathematics and computer science, the Entscheidungsproblem (pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm], German for 'decision problem') is a challenge posed by David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.
https://en.wikipedia.org/wiki/Automated ... he_problemWiki wrote:...given unbounded resources, any valid formula can eventually be proven. However, invalid formulas (those that are not entailed by a given theory), cannot always be recognized.