Sentence examples for arithmetical statement from inspiring English sources

Moreover, Gödel's second incompleteness theorem implies that the formal (arithmetical) statement $CON ZFC $, which asserts that ZFC is consistent, while true, cannot be proved in ZFC.

And he thought that every arithmetical statement that can be proved by making a detour through higher mathematics, can also be proved directly in Peano Arithmetic.

A set of T-sentences that does not imply any false arithmetical statement may be obtained by allowing only those φ in T-sentences T ⌈φ⌉ ↔ φ that contain T only positively, that is, in the scope of an even number of negation symbols.

The history of mathematics has shown that making a "detour" through higher mathematics can sometimes lead to a proof of an arithmetical statement that is much shorter and that provides more insight than any purely arithmetical proof of the same statement.

Then Kurt Gödel proved that there exist arithmetical statements that are undecidable in Peano Arithmetic (Gödel 1931).

This implies that many consistent sets of T-sentences prove false arithmetical statements.

In Leigh (2012) it is shown that the theory proves the same Π02 arithmetical statements as classical FS.

And, fortunately, it seemed possible in principle to do this, for in the final analysis consistency statements are, again modulo coding, arithmetical statements.

Because of the finiteness of a natural number in contrast to, for example, a real number, many arithmetical statements of a finite nature that are true in classical mathematics are so in intuitionism as well.

It is not hard to see how a successor function and addition and multiplication operations can be defined on the number-candidates of I and on the number-candidates of II so that all the arithmetical statements that we take to be true come out true.

The mapping between arithmetical expressions and meta-arithmetical statements, based on Gödel's numbering, cannot go awry in any "non-standard" model, as Sloman suggested.