November 5, 2003
New Axioms for Set Theory
Note: This paper is a condensed draft.
The base theory is ZFC.
Add predicate Tr for infinitary statements such that Tr(S) ↔ S is a set that codes a true infinitary statement.
Axiom schema: If ZFC proves that for all measurable cardinals κ, φ holds in the structure (V(κ), membership, Tr), then φ.
Justification: Take a mental image of the universe. Various theorems strongly suggest that such images can be made with arbitrary accuracy. Try extending it with new cardinals. There must be some reason why the new cardinals are not realized in the universe. A good reason is non-rigidity, that is the existence of a nontrivial elementary embedding of V into M if the cardinals existed: Such embedding raises the question as to why the universe is V rather than M; metaphorically speaking, such embedding implies that V is not held together by its structure. (Still, there is a substantial likelihood that measurable cardinals do exist in the universe.) Detailed studies have shown that no breakdown of structure occurs at lower cardinal levels, and that the hierarchy of cardinals below measurable is canonical. The theory of sets is canonical and the axiom schema appears to be simply a canonical schema stating that the universe is endless and that an extension would be nonrigid, and providing powerful reflection principles for set theory. Also, the axiom schema with measurable replaced by inaccessible can be approximated by allowing greater expressiveness in the formulas in the replacement axiom schema and be viewed as the natural limit of such approximations.
Generalized Continuum Hypothesis: For every infinite cardinal κ, 2κ = κ+.
Justification: The hypothesis holds in all canonical inner models in fact, a J-structure is considered acceptable only if it satisfies GCH. It can be easily forced by collapsing cardinals; in fact, ground models for forcing usually satisfy GCH. GCH is a natural strengthening of the Axiom of Choice: GCH implies the Axiom of Choice over ZF and the “unintuitive” consequences of GCH can be viewed as the natural strengthening of the consequences of the axiom of choice. For example, the axiom of choice produces a well-ordering of real numbers, while the GCH produces a well-ordering of real numbers with each member having only countably many predecessors. GCH is the only canonical theory of cardinal arithmetic. The theory of the generalized Cantor spaces is much more natural with the GCH. It is possible in models of ZFC to cause ω1 to behave like a countable ordinal and to make the theory of H(ω2) not very expressive, but such behavior is a restriction, as the development of the theory will clearly illustrate. Finally, no definable set of real numbers negates the Continuum Hypothesis, so the negation of the CH postulates entire cardinalities of sets of real numbers, none of which are definable. Minimization of unexplainable requires their nonexistence.
Determinacy Maximum: Every perfect information game of length ω1
with ordinal definable payoff set and with the set of allowed moves
being an ordinal is determined.
(See my paper "Determinacy Maximum" for analysis.)
Principle of Undefinability of Well-Orderings: For every infinite cardinal κ and for every cardinal λ > κ, a sequence of length λ of distinct subsets of κ (if it exists) is not ordinal definable from any sequence of ordinals of length less than λ.
Justification: The principle captures our intuition that
there is no natural way to
pick which elements go first in a well-ordering of a power set, so one has to use the axiom of choice to choose the elements. For example, given two arbitrary real numbers, it appears that there is no way to decide which of them should be first in a well-ordering. It is a maximization principle as it suggests that arbitrary subsets exist rather than only those that have enough regularity to fit in a definable well-ordering.
In addition, there are interesting hypotheses that appear to be true but not yet sufficiently analyzed to be proposed axioms. To be effective, set theory must have sufficient set existence axioms. While ZFC suffices for many purposes, the large number of incompletenesses shows that more axioms are needed. However, the new existence axioms—determinacy hypotheses being the most prominent examples—are subtle and require careful analysis to establish their truthfulness.
An interesting strengthening of the principle of undefinability of well-orderings for regular cardinals is that for every regular infinite cardinal κ, every set of subsets of κ that is ordinal definable from κ ordinal parameters either has cardinality at most κ or contains a perfect subset. (This implies the GCH.) Specifically, for a limit ordinal κ, a tree T of height κ is said to be perfect if every totally ordered subset of T is contained in a path, and every path has κ points at which the tree branches. A set of subsets of κ is a perfect subset if it is the set of paths through a perfect tree of height κ.
On a similar line, one can ask whether for every regular cardinal κ, every Baire game of length κ with the payoff set ordinal definable from κ ordinal parameters is determined.
An interesting forcing hypothesis is that collapsing all uncountable cardinals below an inaccessible to ω1 does not change the theory of H(ω2). The hypothesis should resolve in a canonical way many incompletenesses of third order arithmetic. More generally, for every (?) infinite regular cardinal κ, collapsing all cardinals strictly between κ and a cardinal above κ, yields with respect to H(κ+) an elementary extension. κ+ should be so inaccesibly large that making it larger by collapsing some cardinals above κ should not be noticeable in H(κ+). To produce stronger results, one can ask for invariance under collapsing of all cardinals strictly between κ (such as ω1) and a cardinal above κ for the theory L(H(κ+)).
The defining property of regular cardinals is that for every
regular cardinal κ, every partition of κ into less than κ
sets contains a set of cardinality κ. A natural strengthening
of this is the Generalized Suslin Hypothesis: For every infinite
regular cardinal κ, a tree of height κ either has a path
(of length κ) or contains an anti-chain of cardinality κ.
Note: The Generalized Suslin Hypothesis is certainly an interesting statement, but it may well be false (it contradicts ◊). There is an uncountable partial order that includes neither an uncountable set of pairwise comparable elements, nor an uncountable set of pairwise incomparable elements.
Absoluteness under Forcing
One way to capture the maximality of the parts of the universe is through absoluteness under forcing: Extending an image of the parts with certain generic sets should not alter the theory (the generic extension should be an elementary extension of the theory) because the parts are already the largest possible, so it should not be possible to make them larger. The statement that L(R) is absolute under forcing provides true canonical theory of L(R), which maximizes which sets are constructible above the reals.
As the level of expressiveness becomes higher, the ability to change the truth of statements between models increases, so the amount of absoluteness decreases: Δ01 statements are absolute under all models, Σ0n statements are absolute for all ω-models, Π11 statements are absolute for well-founded models, and Σ12 statements are absolute for wellfounded models (containing basic set theory) that include ω1 . At the level of third order arithmetic, the expressiveness is so great that there are plenty of opportunities for forcing and hence no absoluteness under all forcing.
Absoluteness requires that the models are sufficiently
well-behaved. Adding real numbers can dramatically upset the
structure of higher set theory: V[G] would satisfy that it is a
generic extension by a single real number. Accordingly, a reasonable
restriction is to require the forcing not to add real numbers. Under
the restriction, we do have an absoluteness theorem:
Theorem: If H(ω2) is Σ21 absolute under forcing that does not add reals, then the Continuum Hypothesis holds. Under suitable large cardinal axioms, the converse also holds.
The restriction is too weak for Σ22 statements as one can force the Diamond Principle (◊) and the Suslin Hypothesis by forcing that does not add reals. Instead, one can require models to be well-behaved by being countably closed. However, forcing ordinarily works on countable transitive models, so to work with countably closed models, an appropriate restriction is for the partial ordering to be countably closed (for every countable descending sequence, there is an element smaller than every member of the sequence).
Theorem: If H(ω2) is Σ22 absolute under countably closed forcing, then ◊ holds.
In fact, whenever countably closed forcing adds subsets of ω1, ◊ is forced.
Conjecture (based on a conjecture of Woodin): Under suitable large cardinal axioms the generic version of ◊ (absoluteness under Coll(ω1, R)) implies that H(ω2) is Σ22 absolute under countably closed forcing.
Unfortunately, both the
Kurepa Hypothesis (a Σ23
statement) and its negation can be forced by countably closed
forcing. An interesting addtional restriction is that the cardinality
of the generic set is ω1
Question: How much absoluteness is possible under countably closed forcing for which the generic set has cardinality ω1 in the generic extension or under products of such forcings?
For example, I do not see any way to force the Kurepa Hypothesis with such forcing.
Forcing and Definability: In M[G], every set is definable from parameters in M and G. One could argue that after adding a generic subset of ω1 G, for some properties of G the witnesses are not definable from M and G, so one has to add sets that witness the specific properties G. However, any addition of witnesses by countably closed forcing preserves ◊. No well-ordering of real numbers is definable because one has to pick continuum many parameters and there is no way to choose which of the real numbers go before which. A generic subset of ω1 already picks which the parameters go first, so there is no reason to expect that no matter how sufficiently closed the ground model M is, none of the witnesses for a property of G expressible in the first order language of H(ω2) are available in M[G]. For example, if the ground model for second order arithmetic is sufficiently closed, then adding a generic real does not change its properties as the witnesses for the properties are definable from the real number.