science

Internal set theory

Internal set theory 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 Internal set theory rather than just read about it. In short: Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment.

Key takeaways

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

Reference excerpt

Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, Nelson's approach modifies the axiomatic foundations through syntactic enrichment. Thus, the axioms introduce a new term, "standard", which can be used to make discriminations not possible under the conventional ZFC axioms for sets. Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate "standard" satisfies three additional axioms I, S, and T. In particular, suitable nonstandard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements. Nelson's formulation is made more accessible for the lay-mathematician by leaving out many of the complexities of meta-mathematical logic that were initially required to justify rigorously the consistency of number systems containing infinitesimal elements.

Intuitive justification

Whilst IST has a perfectly formal axiomatic scheme, described below, an intuitive justification of the meaning of the term standard is desirable. This is not part of the formal theory, but is a pedagogical device that might help the student interpret the formalism. The essential distinction, similar to the concept of definable numbers, contrasts the finiteness of the domain of concepts that we can specify and discuss, with the unbounded infinity of the set of numbers; compare finitism.

The number of symbols one writes with is finite. The number of mathematical symbols on any given page is finite. The number of pages of mathematics a single mathematician can produce in a lifetime is finite. Any workable mathematical definition is necessarily finite. There are only a finite number of distinct objects a mathematician can define in a lifetime. There will only be a finite number of mathematicians in the course of our (presumably finite) civilization. Hence there is only a finite set of whole numbers our civilization can discuss in its allotted lifespan. What that limit actually is, is unknowable to us, being contingent on many accidental cultural factors. This limitation is not in itself susceptible to mathematical scrutiny, but that there is such a limit, whilst the set of whole numbers continues forever without bound, is a mathematical truth. The term standard is therefore intuitively taken to correspond to some necessarily finite portion of "accessible" whole numbers. The argument can be applied to any infinite set of objects whatsoever – there are only so many elements that one can specify in finite time using a finite set of symbols and there are always those that lie beyond the limits of our patience and endurance, no matter how we persevere. We must admit to a profusion of nonstandard elements—too large or too anonymous to grasp—within any infinite set.

Principles of the standard predicate The following principles follow from the above intuitive motivation and so should be deducible from the formal axioms. For the moment we take the domain of discussion as being the familiar set of whole numbers.

Any mathematical expression that does not use the new predicate standard explicitly or implicitly is an internal formula. Any definition that does so is an external formula. Any number uniquely specified by an internal formula is standard (by definition). Nonstandard numbers are precisely those that cannot be uniquely specified (due to limitations of time and space) by an internal formula. Nonstandard numbers are elusive: each one is too enormous to be manageable in decimal notation or any other representation, explicit or implicit, no matter how ingenious your notation. Whatever you succeed in producing is by definition merely another standard number. Nevertheless, there are (many) nonstandard whole numbers in any infinite subset of N. Nonstandard numbers are completely ordinary numbers, having decimal representations, prime factorizations, etc. Every classical theorem that applies to the natural numbers applies to the nonstandard natural numbers. We have created, not new numbers, but a new method of discriminating between existing numbers. Moreover, any classical theorem that is true for all standard numbers is necessarily true for all natural numbers. Otherwise the formulation "the smallest number that fails to satisfy the theorem" would be an internal formula that uniquely defined a nonstandard number. The predicate "nonstandard" is a logically consistent method for distinguishing large numbers—the usual term will be illimited. Reciprocals of these illimited numbers will necessarily be extremely small real numbers—infinitesimals. To avoid confusion with other interpretations of these words, in newer articles on IST those words are replaced with the constructs "i-large" and "i-small". There are necessarily only finitely many standard numbers—but caution is required: we cannot gather them together and hold that the result is a well-defined mathematical set. This will not be supported by the formalism (the intuitive justification being that the precise bounds of this set vary with time and history). In particular we will not be able to talk about the largest standard number, or the smallest nonstandard number. It will be valid to talk about some finite set that contains all standard numbers—but this non-classical formulation could only apply to a nonstandard set.

Formal axioms IST is an axiomatic theory in the first-order logic with equality in a language containing a binary predicate symbol ∈ and a unary predicate symbol st(x). Formulas not involving st (i.e., formulas of the usual language of set theory) are called internal, other formulas are called external. We use the abbreviations

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Internal set theory

Start with the simplest possible case. Write down what Internal set theory 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 Internal set theory 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 Internal set theory 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 Internal set theory

In research: Internal set theory 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 Internal set theory 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: Internal set theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Nonstandard analysis, Systems of set theory, so understanding it makes those chapters shorter.
In everyday life: Look for Internal set theory 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 Internal set theory in 20 minutes

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

Frequently asked questions

What is Internal set theory in simple terms?

Why does Internal set theory 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 Internal set theory?

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 Internal set theory.