science

L(R)

L(R) 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 L(R) rather than just read about it. In short: In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals. Construction L(R) can be constructed in a manner analogous to the construction of Gödel's constructible universe, L, by adding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.

Key takeaways

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

Reference excerpt

In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals.

Construction L(R) can be constructed in a manner analogous to the construction of Gödel's constructible universe, L, by adding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.

Assumptions In general, the study of L(R) assumes a wide array of large cardinal axioms, since without these axioms one cannot show even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the axiom of choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, given only that the von Neumann universe, V, also satisfies that axiom.

Results Under the assumption of sufficiently strong large cardinal axioms, some additional results of the theory are:

L(R) satisfies ZF + AD+. (In particular, it satisfies the axiom of determinacy.) Every projective set of reals – and therefore every analytic set and every Borel set of reals – is an element of L(R). Every set of reals in L(R) is Lebesgue measurable (in fact, universally measurable) and has the property of Baire and the perfect set property. L(R) does not satisfy the axiom of uniformization or the axiom of real determinacy. R#, the sharp of the set of all reals, has the smallest Wadge degree of any set of reals not contained in L(R). While not every relation on the reals in L(R) has a uniformization in L(R), every such relation does have a uniformization in L(R#). Given any (set-size) generic extension V[G] of V, L(R) is an elementary submodel of L(R) as calculated in V[G]. Thus the theory of L(R) cannot be changed by forcing.

References Woodin, W. Hugh (1988). "Supercompact cardinals, sets of reals, and weakly homogeneous trees". Proceedings of the National Academy of Sciences of the United States of America. 85 (18): 6587–6591. doi:10.1073/pnas.85.18.6587. PMC 282022. PMID 16593979.

Worked examples

Example 1 — a first encounter with L(R)

Start with the simplest possible case. Write down what L(R) 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 L(R) 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 L(R) 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 L(R)

In research: L(R) 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 L(R) 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: L(R) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Descriptive set theory, Determinacy, Inner model theory, so understanding it makes those chapters shorter.
In everyday life: Look for L(R) 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “L(R)” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study L(R) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what L(R) 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 L(R) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is L(R) in simple terms?

In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals. Construction L(R) can be constructed in a manner analogous to the construction of Gödel's constructible universe, L, by adding in all the reals at the start, and then…

Why does L(R) 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 L(R)?

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 L(R).