science

Inner model

Inner model is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Inner model rather than just read about it. In short: In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M. Definition Let L = ⟨∈⟩ be the language of set theory.

Key takeaways

Inner model belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Inner model to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Inner model from memory before moving on to harder problems.

Reference excerpt

In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M.

Definition Let L = ⟨∈⟩ be the language of set theory. Let S be a particular set theory, for example the ZFC axioms and let T (possibly the same as S) also be a theory in L. If M is a model for S, and N is an L-structure such that

N is a substructure of M, i.e. the interpretation ∈N of ∈ in N is ∈M ∩ N2 N is a model of T the domain of N is a transitive class of M N contains all ordinals in M then we say that N is an inner model of T (in M). Usually T will equal (or subsume) S, so that N is a model for S 'inside' the model M of S. If only conditions 1 and 2 hold, N is called a standard model of T (in M), a standard submodel of T if (S = T and) N is a set in M. A model N of T in M is called transitive when it is standard and condition 3 holds. If the axiom of foundation is not assumed (that is, is not in S) all three of these concepts are given the additional condition that N be well-founded. Hence inner models are transitive, transitive models are standard, and standard models are well-founded. The assumption that there exists a standard submodel of ZFC (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard submodel, then there is a smallest standard submodel called the minimal model contained in all standard submodels. The minimal submodel contains no standard submodel (as it is minimal) but (assuming the consistency of ZFC) it contains some model of ZFC by the Gödel completeness theorem. This model is necessarily not well-founded otherwise its Mostowski collapse would be a standard submodel. (It is not well-founded as a relation in the universe, though it satisfies the axiom of foundation so is "internally" well-founded. Being well-founded is not an absolute property.) In particular in the minimal submodel there is a model of ZFC but there is no standard submodel of ZFC.

Use Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension of ZFC (like ZFC + "a measurable cardinal exists"). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of subtheories of ZFC (like ZF or KP) as well.

Related ideas Kurt Gödel proved that any model of ZF has a least inner model of ZF, the constructible universe, which is also an inner model of ZFC + GCH. There is a branch of set theory called inner model theory that studies ways of constructing least inner models of theories extending ZF. Inner model theory has led to the discovery of the exact consistency strength of many important set theoretical properties.

References

Worked examples

Example 1 — a first encounter with Inner model

Start with the simplest possible case. Write down what Inner model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Inner model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Inner model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Inner model

In research: Inner model appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Inner model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Inner model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Inner model theory, so understanding it makes those chapters shorter.
In everyday life: Look for Inner model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Inner model in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Inner model means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Inner model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Inner model in simple terms?

In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M. Definition Let L = ⟨∈⟩ be the language of set theory.

Why does Inner model matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Inner model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Inner model.