Post History
#2: Post edited
by
The Amplitwist
·
2024-02-13T07:33:50Z (9 months ago)
godel -> Gödel; i -> I
Minimal non-standard number in non-standard models of PA
- Excuse me, if the question sounds too naive.
From godel's incompleteness theorem we know that there would be non-standard models where the godel sentence would be false. These models will have an initial segment isomorphic to standard natural numbers. Will there be a minimal non-standard number in such models such that every number smaller than it is a standard natural number and every number bigger than it would be non-standard ?Since non-standard model would be a model of arithmetic then i think there should be a minimal non-standard number, but then maybe my concept is unclear about it. Any help ?
- Excuse me, if the question sounds too naive.
- From Gödel's incompleteness theorem we know that there would be non-standard models where the Gödel sentence would be false. These models will have an initial segment isomorphic to standard natural numbers. Will there be a minimal non-standard number in such models such that every number smaller than it is a standard natural number and every number bigger than it would be non-standard ?
- Since non-standard model would be a model of arithmetic then I think there should be a minimal non-standard number, but then maybe my concept is unclear about it. Any help?
#1: Initial revision
by
Aditya
·
2024-02-12T18:11:12Z (9 months ago)
Minimal non-standard number in non-standard models of PA
Excuse me, if the question sounds too naive. From godel's incompleteness theorem we know that there would be non-standard models where the godel sentence would be false. These models will have an initial segment isomorphic to standard natural numbers. Will there be a minimal non-standard number in such models such that every number smaller than it is a standard natural number and every number bigger than it would be non-standard ? Since non-standard model would be a model of arithmetic then i think there should be a minimal non-standard number, but then maybe my concept is unclear about it. Any help ?