Ω-logic

Key takeaways

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

Reference excerpt

In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure H ℵ 2 {\displaystyle H_{\aleph _{2}}} . Just as the axiom of projective determinacy yields a canonical theory of H ℵ 1 {\displaystyle H_{\aleph _{1}}} , he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.

Analysis Woodin's Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most results in the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completeness theorem. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive over H ℵ 2 {\displaystyle H_{\aleph _{2}}} (in Ω-logic), it must imply that the continuum is not ℵ 1 {\displaystyle \aleph _{1}} . Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent Π 2 {\displaystyle \Pi _{2}} (over H ℵ 2 {\displaystyle H_{\aleph _{2}}} ) sentence is true; this axiom implies that the continuum is ℵ 2 {\displaystyle \aleph _{2}} . Woodin also related his Ω-conjecture to a proposed abstract definition of large cardinals: he took a "large cardinal property" to be a Σ 2 {\displaystyle \Sigma _{2}} property P ( α ) {\displaystyle P(\alpha )} of ordinals which implies that α is a strong inaccessible, and which is invariant under forcing by sets of cardinal less than α. Then the Ω-conjecture implies that if there are arbitrarily large models containing a large cardinal, this fact will be provable in Ω-logic. The theory involves a definition of Ω-validity: a statement is an Ω-valid consequence of a set theory T if it holds in every model of T having the form V α B {\displaystyle V_{\alpha }^{\mathbb {B} }} for some ordinal α {\displaystyle \alpha } and some forcing notion B {\displaystyle \mathbb {B} } . This notion is clearly preserved under forcing, and in the presence of a proper class of Woodin cardinals it will also be invariant under forcing (in other words, Ω-satisfiability is preserved under forcing as well). There is also a notion of Ω-provability; here the "proofs" consist of universally Baire sets and are checked by verifying that for every countable transitive model of the theory, and every forcing notion in the model, the generic extension of the model (as calculated in V) contains the "proof", restricted its own reals. For a proof-set A the condition to be checked here is called "A-closed". A complexity measure can be given on the proofs by their ranks in the Wadge hierarchy. Woodin showed that this notion of "provability" implies Ω-validity for sentences which are Π 2 {\displaystyle \Pi _{2}} over V. The Ω-conjecture states that the converse of this result also holds. In all currently known core models, it is known to be true; moreover the consistency strength of the large cardinals corresponds to the least proof-rank required to "prove" the existence of the cardinals.

Notes

References Bagaria, Joan; Castells, Neus; Larson, Paul (2006), "An Ω-logic primer", Set theory (PDF), Trends Math., Basel, Boston, Berlin: Birkhäuser, pp. 1–28, doi:10.1007/3-7643-7692-9_1, ISBN 978-3-7643-7691-8, MR 2267144 Koellner, Peter (2013), "The Continuum Hypothesis", The Stanford Encyclopedia of Philosophy, Edward N. Zalta (Ed.) Woodin, W. Hugh (1999), The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, Walter de Gruyter, doi:10.1515/9783110804737, ISBN 3-11-015708-X, MR 1713438 Woodin, W. Hugh (2001), "The continuum hypothesis. I" (PDF), Notices of the American Mathematical Society, 48 (6): 567–576, ISSN 0002-9920, MR 1834351 Woodin, W. Hugh (2001b), "The Continuum Hypothesis, Part II" (PDF), Notices of the AMS, 48 (7): 681–690 Woodin, W. Hugh (2005), "The continuum hypothesis", in Cori, Rene; Razborov, Alexander; Todorčević, Stevo; et al. (eds.), Logic Colloquium 2000, Lect. Notes Log., vol. 19, Urbana, IL: Assoc. Symbol. Logic, pp. 143–197, MR 2143878

External links W. H. Woodin, Slides for 3 talks

Worked examples

Example 1 — a first encounter with Ω-logic

Start with the simplest possible case. Write down what Ω-logic 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 Ω-logic 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 Ω-logic 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 Ω-logic

In research: Ω-logic 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 Ω-logic 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: Ω-logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Set theory, Systems of formal logic, so understanding it makes those chapters shorter.
In everyday life: Look for Ω-logic 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.

How to study Ω-logic in 20 minutes

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

Frequently asked questions

What is Ω-logic in simple terms?

In set theory, Ω-logic is an infinitary logic and deductive system proposed by W. Hugh Woodin (1999) as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure H ℵ 2 {\displaystyle H_{\aleph _{2}}} .

Why does Ω-logic 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 Ω-logic?

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 Ω-logic.