New Foundations - Strong Axioms of Infinity

Strong Axioms of Infinity

In this section we mainly discuss the effect of adding various "strong axioms of infinity" to our usual base theory, NFU + Infinity + Choice. This base theory, known consistent, has the same strength as TST + Infinity, or Zermelo set theory with Separation restricted to bounded formulas (Mac Lane set theory).

One can add to this base theory strong axioms of infinity familiar from the ZFC context, such as "there exists an inaccessible cardinal," but it is more natural to consider assertions about cantorian and strongly cantorian sets. Such assertions not only bring into being large cardinals of the usual sorts, but strengthen the theory on its own terms.

The weakest of the usual strong principles is:

• Rosser's Axiom of Counting. The set of natural numbers is a strongly cantorian set.

To see how natural numbers are defined in NFU, see set-theoretic definition of natural numbers. The original form of this axiom given by Rosser was "the set {m|1≤m≤n} has n members", for each natural number n". This intuitively obvious assertion is unstratified: what is provable in NFU is "the set {m|1≤m≤n} has members" (where the T operation on cardinals is defined by ; this raises the type of a cardinal by one). For any cardinal number (including natural numbers) to assert is equivalent to asserting that the sets A of that cardinality are cantorian (by a usual abuse of language, we refer to such cardinals as "cantorian cardinals"). It is straightforward to show that the assertion that each natural number is cantorian is equivalent to the assertion that the set of all natural numbers is strongly cantorian.

Counting is consistent with NFU, but increases its consistency strength noticeably; not, as one would expect, in the area of arithmetic, but in higher set theory. NFU + Infinity proves that each exists, but not that exists; NFU + Counting (easily) proves Infinity, and further proves the existence of for each n, but not the existence of . (See beth numbers).

Counting implies immediately that one does not need to assign types to variables restricted to the set of natural numbers for purposes of stratification; it is a theorem that the power set of a strongly cantorian set is strongly cantorian, so it is further not necessary to assign types to variables restricted to any iterated power set of the natural numbers, or to such familiar sets as the set of real numbers, the set of functions from reals to reals, and so forth. The set-theoretical strength of Counting is less important in practice than the convenience of not having to annotate variables known to have natural number values (or related kinds of values) with singleton brackets, or to apply the T operation in order to get stratified set definitions.

Counting implies Infinity; each of the axioms below needs to be adjoined to NFU + Infinity to get the effect of strong variants of Infinity; Ali Enayat has investigated the strength of some of these axioms in models of NFU + "the universe is finite".

A model of the kind constructed above satisfies Counting just in case the automorphism j fixes all natural numbers in the underlying nonstandard model of Zermelo set theory.

The next strong axiom we consider is the

• Axiom of strongly cantorian separation: For any strongly cantorian set A and any formula (not necessarily stratified!) the set {x∈A|φ} exists.

Immediate consequences include Mathematical Induction for unstratified conditions (which is not a consequence of Counting; many but not all unstratified instances of induction on the natural numbers follow from Counting).

This axiom is surprisingly strong. Unpublished work of Robert Solovay shows that the consistency strength of the theory NFU* = NFU + Counting + Strongly Cantorian Separation is the same as that of Zermelo set theory + Replacement.

This axiom holds in a model of the kind constructed above (with Choice) if the ordinals which are fixed by j and dominate only ordinals fixed by j in the underlying nonstandard model of Zermelo set theory are standard, and the power set of any such ordinal in the model is also standard. This condition is sufficient but not necessary.

Next is

• Axiom of Cantorian Sets: Every cantorian set is strongly cantorian.

This very simple and appealing assertion is extremely strong. Solovay has shown the precise equivalence of the consistency strength of the theory NFUA = NFU + Infinity + Cantorian Sets with that of ZFC + a schema asserting the existence of an n-Mahlo cardinal for each concrete natural number n. Ali Enayat has shown that the theory of cantorian equivalence classes of well-founded extensional relations (which gives a natural picture of an initial segment of the cumulative hierarchy of ZFC) interprets the extension of ZFC with n-Mahlo cardinals directly. A permutation technique can be applied to a model of this theory to give a model in which the hereditarily strongly cantorian sets with the usual membership relation model the strong extension of ZFC.

This axiom holds in a model of the kind constructed above (with Choice) just in case the ordinals fixed by j in the underlying nonstandard model of ZFC are an initial (proper class) segment of the ordinals of the model.

Next consider the

• Axiom of Cantorian Separation: For any cantorian set A and any formula (not necessarily stratified!) the set {x∈A|φ} exists.

This combines the effect of the two preceding axioms and is actually even stronger (precisely how is not known). Unstratified mathematical induction enables proving that there are n-Mahlo cardinals for every n, given Cantorian Sets, which gives an extension of ZFC that is even stronger than the previous one, which only asserts that there are n-Mahlos for each concrete natural number (leaving open the possibility of nonstandard counterexamples).

This axiom will hold in a model of the kind described above if every ordinal fixed by j is standard, and every power set of an ordinal fixed by j is also standard in the underlying model of ZFC. Again, this condition is sufficient but not necessary.

An ordinal is said to be cantorian if it is fixed by T, and strongly cantorian if it dominates only cantorian ordinals (this implies that it is itself cantorian). In models of the kind constructed above, cantorian ordinals of NFU correspond to ordinals fixed by j (they are not the same objects because different definitions of ordinal numbers are used in the two theories).

Equal in strength to Cantorian Sets is the

• Axiom of Large Ordinals: For each noncantorian ordinal, there is a natural number n such that .

Recall that is the order type of the natural order on all ordinals. This only implies Cantorian Sets if we have Choice (but is at that level of consistency strength in any case). It is remarkable that one can even define : this is the nth term of any finite sequence of ordinals s of length n such that, for each appropriate i. This definition is completely unstratified. The uniqueness of can be proved (for those n for which it exists) and a certain amount of common-sense reasoning about this notion can be carried out, enough to show that Large Ordinals implies Cantorian Sets in the presence of Choice. In spite of the knotty formal statement of this axiom, it is a very natural assumption, amounting to making the action of T on the ordinals as simple as possible.

A model of the kind constructed above will satisfy Large Ordinals, if the ordinals moved by j are exactly the ordinals which dominate some in the underlying nonstandard model of ZFC.

• Axiom of Small Ordinals: For any formula φ, there is a set A such that the elements of A which are strongly Cantorian ordinals are exactly the strongly cantorian ordinals such that φ.

Solovay has shown the precise equivalence in consistency strength of NFUB = NFU + Infinity + Cantorian Sets + Small Ordinals with Morse–Kelley set theory plus the assertion that the proper class ordinal (the class of all ordinals) is a weakly compact cardinal. This is very strong indeed! Moreover, NFUB-, which is NFUB with Cantorian Sets omitted, is easily seen to have the same strength as NFUB.

A model of the kind constructed above will satisfy this axiom if every collection of ordinals fixed by j is the intersection of some set of ordinals with the ordinals fixed by j, in the underlying nonstandard model of ZFC.

Even stronger is the theory NFUM = NFU + Infinity + Large Ordinals + Small Ordinals. This is equivalent to Morse–Kelley set theory with a predicate on the classes which is a κ-complete nonprincipal ultrafilter on the proper class ordinal κ; in effect, this is Morse–Kelley set theory + "the proper class ordinal is a measurable cardinal"!

The technical details here are not the main point, which is that reasonable and natural (in the context of NFU) assertions turn out to be equivalent in power to very strong axioms of infinity in the ZFC context. This fact is related to the correlation between the existence of models of NFU, described above and satisfying these axioms, and the existence of models of ZFC with automorphisms having special properties.