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The set N of finite naturals, is known to be the minimal model of Peano system, in the sense that the model N has no proper sub-model of Peano system (BTW, N must be a sub-model of every model of Peano system, this being a stronger aspect of the minimality of N).
Let U be a consistent union of Peano system along with another set of axioms. Does U necessarily have a minimal model (i.e. a model having no proper sub-model of U)? For example, let S be an infinite set of axioms such that - for every finite natural n - the n-th axiom states that ω is greater than n. Does the union of Peano system along with S have a minimal model M (i.e. a model having no proper sub-model of that union)? If it does, then: does that hold also if one replaces S by another set of axioms whose union with Peano system is consistent? HOTmag ( talk) 19:27, 3 September 2016 (UTC)
Mathematics desk | ||
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< September 2 | << Aug | September | Oct >> | September 4 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
The set N of finite naturals, is known to be the minimal model of Peano system, in the sense that the model N has no proper sub-model of Peano system (BTW, N must be a sub-model of every model of Peano system, this being a stronger aspect of the minimality of N).
Let U be a consistent union of Peano system along with another set of axioms. Does U necessarily have a minimal model (i.e. a model having no proper sub-model of U)? For example, let S be an infinite set of axioms such that - for every finite natural n - the n-th axiom states that ω is greater than n. Does the union of Peano system along with S have a minimal model M (i.e. a model having no proper sub-model of that union)? If it does, then: does that hold also if one replaces S by another set of axioms whose union with Peano system is consistent? HOTmag ( talk) 19:27, 3 September 2016 (UTC)