A nonstandard model for an ill

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It only takes a minute to sign up. We often use sets to represent natural numbers, but we can also use natural numbers to represent sets. For example, we can use the binary expansion of a natural number to represent a set.

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Assume we have a nonstandard model of PA. But then again the definition of "non-standard" is external too. But it doesn't mean that every subset of the model is a set in the model. They have very limited powers when it comes to talking about sub-collections of the universe.

Both can only ensure that a small fragments of these have a minimal element. So we can't quite write down the statement "Every set has a minimal element", and this fact exactly is what allows non-standard elements to exist. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Ask Question. Asked 6 years, 3 months ago. Active 6 years, 3 months ago. Viewed times.

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This seems to be an infinite descending chain of nested sets. Russell Easterly Russell Easterly 1, 7 7 silver badges 14 14 bronze badges. Active Oldest Votes. It is easy to show that, too.

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N could be the standard natural numbers. I'm sorry. M could be a nonstandard model and N the standard model. You only assume N is a subset of M, contains 0, and is closed under successor. The standard natural numbers satisfy this definition.

Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Responding to the Lavender Letter and commitments moving forward. Related Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled.The axiomatic method, as conceived for most of the two millennia leading up to the critical developments in formal mathematics in the 20th century, was the view that all properties of a mathematical structure, say the natural numbers, can be proved from a correctly chosen collection of axioms.

The axioms themselves were truths so self-evident that they did not require a proof. The intention was also to have the axioms determine the structure in question up to isomorphism.

In this simple universe, the natural numbers would be the only structure satisfying the Peano axioms and every true arithmetical statement would be provable from them. The 20th century saw a rapid succession of fundamental developments in formal mathematics.

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Many of them could be interpreted as demonstrating the limitations of the formal approach, but rather they revealed its hidden complexity.

Cantor developed a theory of infinite collections and Zermelo codified its properties in his axiomatization of set theory. Led by Skolem, first-order logic was developed as a powerful and robust language of formal mathematics. The Lowenheim-Skolem theorems showed that a collection of axioms cannot determine the size of a model: every collection of axioms having an infinite model, also has models of every infinite cardinality.

A model for a collection of statements is simply a mathematical structure in which the statements hold true.

A powerful consequence of this, the compactness theoremshowed that if every finite fragment of a collection of statements has a model, then so does the entire no matter how infinitely large collection. It already follows from the Lowenheim-Skolem theorems that there are uncountable models of the Peano axioms.

Yes, there are uncountable structures in which induction holds! Let's call the natural numbers the standard model of the Peano axioms and all other models nonstandard. An easy application of the compactness theorem shows that there are countable nonstandard models of the Peano axioms, or indeed of any collection of true arithmetic statements.

Can we compute inside a nonstandard model of arithmetic?

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This question can be made precise in the following way. But we cannot!

a nonstandard model for an ill

Tennenbaum famously showed in that there is no countable nonstandard model for which there is an algorithm to compute any of the above see this post. So finally we have that the natural numbers are unique in one sense, it is the only model of the Peano axioms in which we can compute.In model theorya discipline within mathematical logica non-standard model is a model of a theory that is not isomorphic to the intended model or standard model.

The non-standard models can be chosen as elementary extensions or elementary substructures of the intended model. Non-standard models are studied in set theorynon-standard analysis and non-standard models of arithmetic.

From Wikipedia, the free encyclopedia. Mathematical logic.

a nonstandard model for an ill

Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus. Propositional calculus and Boolean logic. Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic. First-order Quantifiers Predicate Second-order Monadic predicate calculus. Recursion Recursive set Recursively enumerable set Decision problem Church—Turing thesis Computable function Primitive recursive function.

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It only takes a minute to sign up. This set is called the minimal model of ZFC. Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets which are models of ZF assuming ZF is consistent. However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded.

I don't get it. OK, so in this case, one assumes that there exists some standard model that is not a class, but a set. Then one constructs a constructible universe inside such set which is also a set.

Then one uses Lowenheim-Skolem to show that the smallest such model is countable. My question is, it seems that the model is externally not standard viewed from a bigger model - so why call it standard first place? Or am I being confused, and is the model really standard? And the quote in the later part seems to say that the model is not standard, which confuses me much. And I do not get the second paragraph - why would any consistent theory having at least one model relates to the minimal model containing models of ZF?

I am not sure how Godel's completeness theorem is being matched here. Within the minimal model, however, we can't do this sort of trick anymore. There is no set which is a standard model that we can begin with. But this model cannot be standard, it has to be ill-founded.

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The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Standard models being non-standard?

a nonstandard model for an ill

Ask Question. Asked 7 years, 3 months ago.In mathematical logica non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment.

The construction of such models is due to Thoralf Skolem There are several methods that can be used to prove the existence of non-standard models of arithmetic.

The existence of non-standard models of arithmetic can be demonstrated by an application of the compactness theorem. The element of this model corresponding to x cannot be a standard number, because as indicated it is larger than any standard number. Using more complex methods, it is possible to build non-standard models that possess more complicated properties.

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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 model, so a model where Goodstein's theorem fails must be non-standard.

By the completeness theoremthis means that G is false in some model of Peano arithmetic. However, G is true in the standard model of arithmetic, and therefore any model in which G is false must be a non-standard model.

Another method for constructing a non-standard model of arithmetic is via an ultraproduct. Identify two sequences if they agree almost everywhere. The resulting semiring is a non-standard model of arithmetic. It can be identified with the hypernatural numbers. The ultraproduct models are uncountable.

One way to see this is to construct an injection of the infinite product of N into the ultraproduct. One way to define such a model is to use Henkin semantics. In other words, a countable non-standard model begins with an infinite increasing sequence the standard elements of the model.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. How is that? Any reference will be welcome. This is a standard application of the Compactness Theorem, and works basically the same in producing nonstandard models of ZFC as it does for producing nonstandard models of PA or real-closed fields. This follows from two facts:.

Viewed externally, any nonstandard model of Peano arithmetic is non-well-founded. So if the natural numbers within a model of set theory are not well founded, then the model's set inclusion relation is also not well founded.

These are, formally, different meanings of "well founded". It's interesting to ask how the model can think that its natural numbers are well founded, given that the descending sequence above was completely concrete. Sign up to join this community. The best answers are voted up and rise to the top.

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Non-standard model of arithmetic

Carl Mummert 9, 1 1 gold badge 40 40 silver badges 62 62 bronze badges. In case it is not clear enough, I am asking for some sketch of a proof for that result, not for an explanation of why such models exist or for what they look like. Active Oldest Votes. Joel David Hamkins Joel David Hamkins k 28 28 gold badges silver badges bronze badges.

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This follows from two facts: Viewed externally, any nonstandard model of Peano arithmetic is non-well-founded. Carl Mummert Carl Mummert 9, 1 1 gold badge 40 40 silver badges 62 62 bronze badges.Aston Villa have been very good at home this season while Millwall are yet to win a game away. Aston Villa to win looks like a great pick. On top of this page, you can filter to see only free football predictions. StakeHunters will strive hard to be the best football prediction site in the world.

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a nonstandard model for an ill

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