Heck (1997b) dealt with so-called ‘finite Frege arithmetic’.
The consistency of arithmetic can be proved only by means stronger than those provided by arithmetic itself.
We now describe the proof of the two theorems, formulating Gödel's results in Peano arithmetic.
Thus it does not contain classical (Peano) arithmetic but only intuitionistic (Heyting) arithmetic.
For instance, the classical theory of elementary arithmetic, Peano Arithmetic, can no longer be accepted.
Then the consistency of intuitionistic arithmetic would guarantee also the consistency of classical arithmetic.
The Second Incompleteness Theorem shows that the consistency of arithmetic cannot be proved in arithmetic itself.
In fact, he strongly suspected that every problem of elementary arithmetic can be decided from the axioms of Peano Arithmetic.
But this means that arithmetic truth and arithmetic provability are not co-extensive — whence the First Incompleteness Theorem.
Unfortunately, as Meyer and Friedman have shown, relevant arithmetic does not contain all of the theorems of classical Peano arithmetic.
We now consider Gödel 1933e, in which Gödel showed, in effect, that intuitionistic or Heyting arithmetic is only apparently weaker than classical first-order arithmetic.
Indeed, after the theorems on Arithmetic in §54 which concludes Part II of Basic Laws of Arithmetic, Frege jumps directly to the topic of Real Numbers for the remainder of Volume II.
In order to maximize expected utility, we would have to accept any bet we were offered on the truths of arithmetic, and reject any bet we were offered on false sentences in the language of arithmetic.
Taken together with his 1933e, which reduces classical first order arithmetic to Heyting arithmetic, a justification in these terms is also obtained for classical first order arithmetic.
Standard arguments had indicated that arithmetic progressions in the set of primes might not be very long, so the discovery that they can be arbitrarily long was a profound discovery about the building blocks of arithmetic.
Skolem's definable ultrapower construction from 1933 (see Skolem 1933) gives a direct construction of a non-standard model of True Arithmetic (which extends Peano arithmetic, being the set of arithmetic sentences true in the natural numbers).
Nonetheless, just like the classical non-standard models of arithmetic, there is a class of inconsistent models of arithmetic (or more accurately models of inconsistent arithmetic) which have an interesting and important mathematical structure.
While there is no reason to think that mental arithmetic (mental calculation in the integers and rational numbers) typically involves much visual thinking, there is strong evidence of substantial visual processing in the mental arithmetic of highly trained abacus users.
If the only ways of proving the consistency of arithmetic make essential use of notions which arguably belong to higher-order mathematics, then the consistency of arithmetic, even though it can be expressed in the language of Peano Arithmetic, is a non-arithmetical problem.
Gödel also published a number of significant papers on modal and intuitionistic logic and arithmetic during this period, principal among which is his “On intuitionistic arithmetic and number theory,” (Gödel 1933e), in which he showed that classical first order arithmetic is interpretable in Heyting arithmetic by a simple translation.
- the branch of pure mathematics dealing with the theory of numerical calculations
- relating to or involving arithmetic
Example: arithmetical computations
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Gödel also published a number of significant papers on modal and intuitionistic logic and arithmetic during this period principal among which is his On intuitionistic arithmetic and number theory Gödel 1933e in which he showed that classical first order arithmetic is interpretable in Heyting arithmetic by a simple translation