All Questions
Tagged with model-theory nonstandard-models
90
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Is this a non-standard extension?
I started reading Henson's "Foundations of Nonstandard Analysis. A Gentle Introduction to Nonstandard Extensions" a couple of days ago, and I am a bit confused about something.
Let $F$ be an ...
- 3,545
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27
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Double exponentials in weak arithmetic
In arithmetic without total exponentiation, the exponentiable numbers are closed under addition. It is thus common to think of natural numbers as binary strings $n : |n| \to 2$, whose (unary) lengths $...
- 924
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2
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85
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Showing existence of a non-standard model of arithmetic elementarily equivalent to standard model of arithmetic
Let $\mathcal{M}_A=\langle\mathbb N, 0^{\mathcal{M}_A}, s^{\mathcal{M}_A}, +^{\mathcal{M}_A}, \times^{\mathcal{M}_A}, <^{\mathcal{M}_A}\rangle$ be the standard model for the language of arithmetic $...
- 589
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Confusion about the model in Robert’s Nonstandard Analysis
I’m working through Alain M. Robert’s “Nonstandard Analysis”.
I’m intrigued by his suggestion that his axiomatic approach, rather than adding elements to ℕ to create a nonstandard model *ℕ, “discerns ...
- 5,830
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If Goldbach's conjecture G is undecidable in PA, then can we prove $\mathbb N\models G$?
Suppose Goldbach's conjecture $G$ is undecidable in first-order Peano arithmetic, $\sf{PA}$. That would mean there are models in which $G$ is true and other in which it is false. But intuitively, this ...
- 6,672
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77
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Does anyone describe Henkinization as closing under Skolem Functions?
When I'm constructing a nonstandard model by by adding in a new constant symbol and invoking compactness, it feels much more like I'm adding in $c$ and "closing under Skolem functions" (at ...
- 3,079
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If I use ZFC as a metatheory for FOL, doesn't it make it weaker?
Basically, there is a theorem of first-order logic that says that no L-structure whose domain is infinite can be axiomatized up to isomorphism, so in particular there is no set $\Phi$ of formulas that ...
- 95
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Extending a Model of $ T + \operatorname {Con} ( T ) $ to a model of $ T + \neg \operatorname {Con} ( T ) $
Let $ T $ be a recursively axiomatizable extension of $ \mathsf {PA} $ and $ \mathfrak M $ be a model of $ T + \operatorname {Con} ( T ) $. Is it true that there must exist a model $ \mathfrak N $ ...
- 6,766
1
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60
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Which theories of arithmetic have non-standard computable models?
From this answer:
In particular, while PA is still overkill, there are theories of arithmetic much stronger than arithmetic with successor which are too weak for the Tennenbaum phenomenon to hold for ...
3
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1
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146
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Every nonstandard model of arithmetic has an element which is a multiple of every $n\in \Bbb N$.
Let $\mathfrak N$ be the structure of the natural numbers on the language of arithmetic $\mathcal L=\{0,1,+,\cdot,<\}$. Let $\mathfrak M$ be any nonstandard model of arithmetic. Show that there ...
- 5,696
6
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Which models of PA can be standard in some model of ZFC?
From wikipedia:
For example, there are models of Peano arithmetic in which Goodstein's theorem fails. It can be proved in Zermelo–Fraenkel set theory that Goodstein's theorem holds in the standard ...
5
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1
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56
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Hyper-extensions of Hom space
We fix an ultrafilter $\mathcal{F}$ of $\mathbb{N}$ which contains the cofinite filter.
Let $A,B$ be sets and ${}^{*}A,{}^{*}B$ their hyper-extensions. Then is
$$
{\rm Hom}({}^{*}A,{}^{*}B)
$$
equal ...
- 161
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66
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Number of models of the naturals and reals with and without CH
The Wikipedia page on true arithmetic says that it has $2^\kappa$ models for each uncountable cardinal $\kappa$. This refers to the theory of all first-order statements of the naturals.
I'm curious ...
- 8,000
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96
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Soundness and Completeness for a single Model Only?
Question modified to hopefully answer the questions (I'm a physicist to all might not be mathematically watertight)
In Enderton "A Mathematical Introduction to Logic", logical Implication is ...
user239186
3
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187
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"Real-closed" vs "transfer principle"
The hyperreals are "real-closed," which means that any first-order statement that is true of the reals is true of the hyperreals. However, they also satisfy a "transfer principle," ...
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