Sentence examples for the theory of arithmetic from inspiring English sources

According to Platonists, the theory of arithmetic says what this sequence of abstract objects is like.

But Quine does hold that the sentence, and the theory of arithmetic of which it is an inseparable part, earns its place in our body of knowledge by contributing to the overall success of that body of knowledge in dealing with experience as a whole.

These two pure forms of intuition time and space delivered, for Kant, the theories of arithmetic and Euclidean geometry respectively, and endowed them both with their a priori character.

Consider first the conception of natural number that underlies the theory of Peano Arithmetic (PA).

The background motivation is this: First, we know that in the presence of large cardinal axioms the theory of second-order arithmetic and even the entire theory of L is invariant under set forcing.

Tait's technical analysis yields that the finitistic functions are exactly the primitive recursive ones, and the finitistic number-theoretic truths are exactly those provable in the theory of primitive recursive arithmetic PRA.

More interestingly, the natural first-order theory of arithmetic of real numbers (with both addition and multiplication), the so-called theory of real closed fields (RCF), is both complete and decidable, as was shown by Tarski (1948); he also demonstrated that the first-order theory of Euclidean geometry is complete and decidable.

The informal notion of interpretability is ubiquitous in mathematics; for example, Poincaré provided an interpretation of two dimensional hyperbolic geometry in the Euclidean geometry of the unit circle; Dedekind provided an interpretation of analysis in set theory; and Gödel provided an interpretation of the theory of formal syntax in arithmetic.

From a philosophical point of view, provability logic is interesting because the concept of provability in a fixed theory of arithmetic has a unique and non-problematic meaning, other than concepts like necessity and knowledge studied in modal and epistemic logic.

Books VII IX deal with what the Greeks called "arithmetic," the theory of whole numbers.

Note the stress here on 'significant parts'.[35] We know from Gödel's second incompleteness theorem that any consistent and sufficiently strong theory of arithmetic is unable to prove or refute (the formalized statement of) its own consistency.