Non-standard model of arithmetic: Difference between revisions
Algebraist (talk | contribs) refined stub type |
Zero sharp (talk | contribs) compactness theorem |
||
| Line 1: | Line 1: | ||
In [[Model Theory]], a '''nonstandard model of arithmetic''' (or, equivalently, a nonstandard model of number theory) is a model of all of number theory (i.e. the true statements in number theory are true in the new model), but with a larger underlying set (or universe). Both countable and uncountable nonstandard models of arithmetic exist. It is important to note that while the nonstandard model satisfies all of standard number theory, it also satisfies new sentences (e.g. one could construct a model of number theory in which the [[Twin Prime Conjecture]] holds). |
In [[Model Theory]], a '''nonstandard model of arithmetic''' (or, equivalently, a nonstandard model of number theory) is a model of all of number theory (i.e. the true statements in number theory are true in the new model), but with a larger underlying set (or universe). Both countable and uncountable nonstandard models of arithmetic exist. It is important to note that while the nonstandard model satisfies all of standard number theory, it also satisfies new sentences (e.g. one could construct a model of number theory in which the [[Twin Prime Conjecture]] holds). |
||
The existence of non-standard models of arithmetic can be demonstrated by an application of the [[compactness theorem]]. |
|||
A simple result is that any [[countable]] nonstandard model of arithmetic has [[order type]] <math>\omega + (\omega^{*} +\omega)\cdot\eta</math> |
A simple result is that any [[countable]] nonstandard model of arithmetic has [[order type]] <math>\omega + (\omega^{*} +\omega)\cdot\eta</math> |
||
Revision as of 20:50, 10 September 2007
In Model Theory, a nonstandard model of arithmetic (or, equivalently, a nonstandard model of number theory) is a model of all of number theory (i.e. the true statements in number theory are true in the new model), but with a larger underlying set (or universe). Both countable and uncountable nonstandard models of arithmetic exist. It is important to note that while the nonstandard model satisfies all of standard number theory, it also satisfies new sentences (e.g. one could construct a model of number theory in which the Twin Prime Conjecture holds).
The existence of non-standard models of arithmetic can be demonstrated by an application of the compactness theorem.
A simple result is that any countable nonstandard model of arithmetic has order type
References
Boolos, G., and Jeffrey, R. 1974. Computability and Logic. Cambridge: Cambridge University Press.
 | This mathematical logic-related article is a stub. You can help Wikipedia by expanding it. |