## Internal set theory: A new approach to nonstandard analysis (1977)

Venue: | Bull. Amer. Math. Soc |

Citations: | 56 - 0 self |

### BibTeX

@ARTICLE{Nelson77internalset,

author = {Edward Nelson},

title = {Internal set theory: A new approach to nonstandard analysis},

journal = {Bull. Amer. Math. Soc},

year = {1977},

volume = {83},

pages = {1165--1198}

}

### Years of Citing Articles

### OpenURL

### Abstract

1. Internal set theory. We present here a new approach to Abraham Robinson's nonstandard analysis [10] with the aim of making these powerful methods readily available to the working mathematician. This approach to nonstandard analysis is based on a theory which we call internal set theory (1ST). We start with axiomatic set theory, say ZFC (Zermelo-Fraenkel set theory with the axiom of choice [1]). In addition to the usual undefined binary predicate E of set theory we adjoin a new undefined unary predicate standard. The axioms of 1ST are the usual axioms of ZFC plus three others, which we will state below. All theorems of conventional mathematics remain valid. No change in terminology is required. What is new in internal set theory is only an addition, not a change. We choose to call certain sets standard (and we recall that in ZFC every mathematical object-a real number, a function, etc.-is a set), but the theorems of conventional mathematics apply to all sets, nonstandard as well as standard.

### Citations

335 | Introduction to Mathematical Logic - Mendelson - 1964 |

106 |
Set Theory and the Continuum Hypothesis
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- 1969
(Show Context)
Citation Context .... This approach to nonstandard analysis is based on a theory which we call internal set theory (1ST). We start with axiomatic set theory, say ZFC (Zermelo-Fraenkel set theory with the axiom of choice =-=[1]-=-). In addition to the usual undefined binary predicate E of set theory we adjoin a new undefined unary predicate standard. The axioms of 1ST are the usual axioms of ZFC plus three others, which we wil... |

15 | Some remarks on set theory - ERDős, G - 1956 |

10 |
Axiom schemata of strong infinity in axiomatic set theory
- Lévy
- 1960
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Citation Context ... in the context of Banach space valued functions. We have followed Lamperti [5] closely in §7.s1198 EDWARD NELSON Theorem 8.7 is a reflection principle. Our proof is taken from an argument of A. Levy =-=[6]-=- who credits the method to A. Montague. Our proof uses implicitly the theorem that for all V, if Vx G V3y A{x,y) then there exists a function ƒ on F such that A(x,y( x )) f° r a ll x i n V* If the y i... |

9 |
Strong derivatives and inverse mappings
- Nijenhuis
- 1974
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Citation Context ...engthy proof which is also valid for enlargements. The theorem of Example 14, and its consequence in Example 15, are also due to Luxemburg [7]. The term "strongly differentiable" is used by Nijenhuis =-=[9]-=- for what we have called "derivable" in §5. We have avoided the use of "strongly differentiable" because this term has a different meaning in the context of Banach space valued functions. We have foll... |

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A general theory of monads, in Applications of Model Theory to Algebra, Analysis and Probability
- Luxemburg
- 1969
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Citation Context ...re given by adequate ultrapowers rather then adequate ultralimits. The use of the unrestricted idealization principle produces a natural closure, and allows the reduction algorithm to work. Luxemburg =-=[7]-=- gives examples of results which may be false for enlargements. The distinction between E and E in §3 is necessary. We cannot treat external sets on the same footing as internal sets when we are discu... |

5 |
Perturbations of the Schrödinger equation by potentials with small support
- Friedman
- 1972
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Citation Context ...iscussing a problem of interest in quantum mechanics. We omit some details of a functional analytic nature-for a full discussion in conventional terms see the original accounts by Charles N. Friedman =-=[3]-=-, [4]. Let A be the Laplace operator on L 2 (R n ) with the usual domain which makes it a selfadjoint operator and let H0 = —A. The one-parameter unitary group 11-> e~ ltH ° describes the motion of a ... |

4 |
Nonstandard Analysis, 2nd ed
- Robinson
- 1974
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Citation Context ..., Number 6, November 1977 INTERNAL SET THEORY: A NEW APPROACH TO NONSTANDARD ANALYSIS EDWARD NELSON 1 1. Internal set theory. We present here a new approach to Abraham Robinson's nonstandard analysis =-=[10]-=- with the aim of making these powerful methods readily available to the working mathematician. This approach to nonstandard analysis is based on a theory which we call internal set theory (1ST). We st... |

1 |
of the Schrödinger equation by potentials with small support
- Perturbations
- 1972
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Citation Context ...sing a problem of interest in quantum mechanics. We omit some details of a functional analytic nature-for a full discussion in conventional terms see the original accounts by Charles N. Friedman [3], =-=[4]-=-. Let A be the Laplace operator on L 2 (R n ) with the usual domain which makes it a selfadjoint operator and let H0 = —A. The one-parameter unitary group 11-> e~ ltH ° describes the motion of a free ... |