Gödel's program for new axioms: why, where, how and what?

@inproceedings{Feferman1996GdelsPF,
  title={G{\"o}del's program for new axioms: why, where, how and what?},
  author={Solomon Feferman},
  year={1996}
}
Summary. From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of infinity. In particular, he speculated (in his 1946 Princeton… Expand
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References

SHOWING 1-10 OF 66 REFERENCES
Some Problems and Results Relevant to the Foundations of Set Theory
Publisher Summary This chapter reports on some meta-mathematical results obtained by William P. Hanf (as a solution of a problem suggested by the author) and some mathematical problems involvingExpand
On the Problem of Schemata of Infinity in Axiomatic Set Theory
Publisher Summary This chapter describes Zermelo's set theory that was substantially extended in 1921 when Fraenkel added his axiom of replacement to the original axioms; for, it is this axiom thatExpand
Recent advances in ordinal analysis: pi12 - CA and related systems
  • M. Rathjen
  • Mathematics, Computer Science
  • Bull. Symb. Log.
  • 1995
TLDR
Recent success is reported in obtaining an ordinal analysis for the system of Π2 analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to Π1-formulae, giving hope for an ordinals analysis of Z2 in the foreseeable future. Expand
Believing the Axioms II
  • P. Maddy
  • Computer Science
  • J. Symb. Log.
  • 1988
Believing the Axioms I
  • P. Maddy
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1988
§0. Introduction. Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, “because we have proofs!” The more sophisticated might addExpand
AXIOM SCHEMATA OF STRONG INFINITY IN AXIOMATIC SET THEORY
l Introduction. There are, in general, two main approaches to the introduction of strong infinity assertions to the Zermelo-Fraenkel set theory. The arithmetical approach starts with the regularExpand
Principles of Proof and Ordinals Implicit in Given Concepts
TLDR
Godel's incompleteness theorem, with an absolute minimum of assumptions on the details of the distinction, shows the existence of theorems stated in elementary terms that do not have an elementary proof. Expand
Systems of Predicative Analysis, II: Representations of Ordinals
  • S. Feferman
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1968
The eventual purpose of this paper is to provide certain specific representations of ordinals and develop their basic properties as needed for the proofs of the results announced in [3]. Since theExpand
The Formal Language of Recursion
TLDR
This is the first of a sequence of papers in which this approach takes recursion to be a fundamental (primitive) process for constructing algorithms, not a derived notion which must be reduced to others—e.g. iteration or application and abstraction. Expand
Systems of Predicative Analysis
  • S. Feferman
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1964
TLDR
The authors can never speak sensibly (in the predicative conception) of the "totality" of all sets as a "completed totality" but only as a potential totality whose full content is never fully grasped but only realized in stages. Expand
...
1
2
3
4
5
...