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Zermelo-Fraenkel Axioms, Internal Classes, External Sets
Leonid A. Levin∗ (https://www.cs.bu.edu/fac/Lnd)
Abstract
Usual math sets have special types: countable, compact, open, occasionally Borel, rarely
projective, etc. They are described by a single Set Theory formula with parameters unrelated to
other formulas. Exotic expressions involving sets related to formulas of unlimited complexity or
to Powerset axiom appear mostly in esoteric or foundational studies.
Recognizing internal to math (formula-specified) and external (based on parameters in those
formulas) aspects of math objects greatly simplifies foundations. I postulate external sets
(not internally specified, treated as the domain of variables) to be hereditarily countable and
independent of formula-defined classes, i.e. with finite Kolmogorov Information about them.
This allows elimination of all non-integer quantifiers in Set Theory statements.
arXiv:2209.07497v16 [cs.LO] 29 May 2025
I always wondered why math foundations as taken by logicians are so distant from the actual
math practice. For instance, the cardinality theory - the heart of the Set Theory (ST) - is almost
never used beyond figuring out which sets are countable and which are not.
I see the culprit in the blurred distinction between two types of collections different in nature.
One are pure classes defined by math formulas. The other are externals that math handles but
generally does not specify internally. Random sequences are an example.1 Externals are proper sets
and are the domain of ST variables. I postulate them to be hereditarily countable and independent
of pure classes, i.e. have only finite Kolmogorov Information about them. Math objects (only
informally called sets) are classes {q: Fp (q)} of sets q satisfying formulas F with external parameters p.
Collections of objects are treated as ones of those parameters. Excluded are “Logics” objects involving
classes of unlimited complexity, such as e.g., the collection of all true statements of Arithmetic.
Cantor’s axioms asserted that all Set Theory formulas define (quantifiable) sets. In effect, this
allows formulas with quantifiers over formulas. This self-referential aspect turned out fatal.
Zermelo and Fraenkel reduced this aspect by (somewhat ad-hoc) restrictions on cardinality
treatment by Cantor’s Axioms: Replacement must preserve it; only a separate Power Set increases it.
This cardinality focus has questionable relevance. Distinctions between uncountable cardinalities
almost never looked at in math papers. Besides, unspecific subsets from Power Set classes,
with no other descriptions, find little use in math and greatly complicate its foundations.
And as all axiom systems have countable models, cardinalities feel like artifacts, designed to hide
self-referential aspects. Many papers in Logic (e.g., [2, 3, 7]) aimed at isolating math segments where
more ingenious proofs can replace the use of Power Set Axiom and its uncountable sets. But this
breaks the unity of math: a very unfortunate effect. And complicating proofs is unattractive, too.
Expanding Set Theory with fancier formula types, axioms, etc. has no natural end. Benefits are
little, and eventual consistency loss inevitable. Let us see how to drop any such excesses.
Some Formalities. In effect, math objects would have ranks, each quantifier binding only classes below
one rank. Rank fk classes Fpk are specified by a hereditarily countable (“proper” set) parameter p via
the universal Σ1k formula Fpk (q). In p, a v.Neumann ordinal o(p) is included. Fpk (q) defines membership
′ ′
Fqk ∈Fpk conditioned on o(q)2 Zermelo-Fraenkel Axioms, Internal Classes, External Sets. Leonid A. Levin.
1 Purging the Tricksome Objects
Making explicit the internal to math (defined by formulas) and external (parameters based) aspects
of math objects clarifies their nature, allows a better focus on issues brought by these different
(albeit with many similarities) sources. “Pure” classes defined by parameter-free ST formulas F (n)
are specific but tricksome. Treated carelessly they easily bring paradoxes. They will not be quantified
or put in unlimited definable collections: that would only extend the language of allowed formulas.
Proper sets are external data which math handles without specifying internally. They constitute
the domain of ST variables. They can be put in collections based on properties and relations with
other sets. This forms “mixed” classes, specified by ST formulas with some free variables taken for
data parameters. Data may be chaotic, but “simpleminded,” playing no self-referential tricks.
General math objects – mixed classes – carry both tricks and chaos. But their tricks are limited
to those of a single formula. Distinguishing sets from pure classes allows another radical insight:
Even with infinite complexity, external sequences α ∈ Ω = df
{0, 1}N can have only finite dependence
on pure sequences Ψ. Complexity theory (see sec. 2 for some background) allows to formalize that
as having finite (small, really) mutual Kolmogorov Information I(α: Ψ) 0 where F ′ has all 2nd order variables αi in the closed statement F
replaced by Ai (γ (ki ) ), with ki and algorithms Ai quantified as integers. Cf. [4].
2 Kolmogorov Information: A Brief Background
I freely use concepts involving quantifiers only over hereditarily finite sets (I often call “integers”).
Presuming reader knows how to develop such basics as, e.g., algorithms, I do not try to work
out elegant ways to define them directly in membership terms. Below ∥x∥ = df
n for x∈{0, 1}n and
N k df −∥x∥
∥t∥ = +log2 t,−1 for t∈R+ . Uniform measure on Ω = {0, 1} (or on Ω ≃ Ω) is λ(x Ω) =
df
2 .
2
Some statements can be a meta-theorem: a family with formula parameter F , as done now by Category theorists.Leonid A. Levin. Zermelo-Fraenkel Axioms, Internal Classes, External Sets. 3
Partial continuous transforms (PCT) on Ω may fail to narrow-down the output to
a single sequence, leaving a compact set of eligible results. So, their graphs are compact sets
A ¢ Ω×Ω with A(α) = df
{β : (α, β)∈A} = ̸ ∅. Singleton outputs {β} are interpreted as β∈ Ω.
−1
Preimages A (s) = df
{α : A(α) ¢ s} of open s ¢ Ω are always open.
Closed A also have closed preimages of all closed s.
Computable PCTs have algorithms enumerating the clopen subsets of Ω2 \A. U (pα) is a
universal PCT. It computes n=∥x∥ bits of Ap (α)¢xΩ in tpα (n) steps (and U (p)∈N in tp steps).
A computably enumerable (c.e.) function to R+ is sup of a c.e. set of basic continuous ones.
Dominant
P in a Banach space C of functions is its c.e. f ∈ C if all c.e. g in C are O(f ).
Such is i gi /(i2 +i) if (gi ) is a c.e. family of all c.e. functions in the unit ball of C.
λ-test is d(α) =df
∥+T(α),∥ for a c.e. T: Ω→R+ , λ(T)f1, dominant in L1 (Ω, λ).
Martin-Lof λ-random are α ∈ Rλ∞ , where Rλc = df
{α : d(α)i Fj ; F = ∪i Fi ¢F = ∩i Fi . Any F ∈∆∗ has such (Fi )∈∆∗ with F ¢F ¢F +.
− df + df 0 0 −
Note that F + \F − contains no generic γ. If F ¢ Rλ , λ(F ) = 0 then also M(Fi ) = O(2−i ), implying
I(γ : (Fi )i ) = ∞ for all γ∈F . Thus, any arithmetic class F includes no generic γ ∈ Rλ , if λ(F )=0,
and includes all such γ or none if F is invariant under single digit flips.
2.1 Weak Truth-table (Closed PCT) Reductions to Generic Sequences
Pχ Turing-reduces each α ∈ Ω to a β ∈ Rλ : α = U (β). PCT U can use unlimited segments βm ,
discarding nearly all their information, to produce small segments αn . But IP makes Pχ equivalent
to its stronger form, requiring a closed PCT u, with m ∼ n. Yet, by [9], one cannot require a total
u, nor β Turing equivalent to α: some information loss cannot be avoided.
Let snt =
df
λ({α: tα (n)r}, r∈Q, is computable.
Let I(α: s∞ )4 Zermelo-Fraenkel Axioms, Internal Classes, External Sets. Leonid A. Levin.
For some fixed c, let RλΨ =
df
{α : (Ψ, α) ∈ Rλc }. Combine the first i of sets the (countable) M lists
into αi ∈ RλΨ , so K(αi | αi+1 ) < ∞. We need to merge all αi into (so reduce to) a single γ ∈ RλΨ .
This takes constructing a sequence of γi = u(γi−1 , βi ) ∈ RλΨ , each Turing-equivalent to αi .
3 Some Discussion
Cantor Axioms license on formula-defined sets led to fatal consistency problems. Zermelo-Fraenkel’s
(somewhat ad-hoc) cardinality-based restrictions diffused those but left intact their self-referential
root. The result was a Babel Tower of cardinalities, other hierarchies finding little relevance in math.
A clean way out may be recognizing the distinction between internal to math collections specified
by its formulas and external ones that math handles as values of variables, without specifying.
Internal collections have a limited hierarchy: the type of allowed formulas is clear-cut. Any
extension would make a new theory, with its own clear limits. External objects would be fully
independent of internal ones: having finite information about them. Complexity theory allows to
formalize that, justify the validity for “external data,” and use that for simplifying math foundations.
General math objects are collections specified by formulas with external sets as parameters.
Collections of them are treated as collections of those parameters. So, any such object would rely
on a single formula, not on all of them, thus excluding back-door extensions of formula language.
The uniform set concept of all math objects with no explicit types hierarchy is luring but illusive.
The hierarchy of ever more powerful axioms, models, cardinals remains, if swiped under the rug.
Making it explicit and matching the math relevance may be a path to simpler foundations.
What is left out? – “Logical” sets, related to infinite hierarchies of formulas, such as “the set of
all true sentences of Arithmetic.” Those should be subject of math foundations. Theories cannot
include their own foundations. Math Logic then could focus on math rather than on itself.
Acknowledgments. I am grateful to Albert Meyer for helpful criticism.
References
[1] George Barmpalias, Xiaoyan Zhang. 2025. Collision-resistant hash-shuffles on the reals.
https://arxiv.org/abs/2501.02604
[2] H.Friedman. 1974. Some systems of second-order arithmetic and their use. ICM proc. 1:235-242.
[3] Randall Holmes. 2006. Alternative Set Theories. Stanford Encyclopedia of Philosophy.
https://plato.stanford.edu/entries/settheory-alternative/
[4] Leonid A. Levin. 1976. On the Principle of Conservation of Information in Intuitionistic
Mathematics. Soviet Math. Dokl. 17/2:601-605 = DAN SSSR 227/6:1293-1296.
[5] Leonid A. Levin. 2013. Forbidden Information. JACM, 60/2. Sec. “The Independence Postulate.”
https://arxiv.org/abs/cs/0203029
[6] Leonid A. Levin. 2022. Gacs-Kucera Theorem. TCS, 929:172-173. https://arxiv.org/abs/2207.12489
[7] Stephen G. Simpson. 2009. Subsystems of Second Order Arithmetic. Cambridge U. Press.
[8] E.Zermelo. 1908. Untersuchungen über die Grundlagen der Mengenlehre. Math.Ann. 65:261-281.
[9] Vladimir V. V’yugin. The Algebra of Invariant Properties of Binary Sequences.
Probl.Inf.Transm. 18/2:147-161, 1982.Leonid A. Levin. Zermelo-Fraenkel Axioms, Internal Classes, External Sets. 5
A Appendix. Some Other IP Applications
Foundations of Probability. Paradoxes in its application led to the K-ML randomness concept
Rλ . IP clarifies its use: For any S¢ Ω: λ(S)=0 if and only if ∃σ S∩Rλ ¢ {γ : I(γ : σ) = ∞}.
Goedel Theorem Loophole. Goedel writes:
“It is not at all excluded by the negative results mentioned earlier that nevertheless
every clearly posed mathematical yes-or-no question is solvable in this way.
For it is just this becoming evident of more and more new axioms on the basis
of the meaning of the primitive notions that a machine cannot imitate.”
No way ! Let a predicate P on {0, 1}n extend “proven/refuted” partial predicate of Peano
Arithmetic. Let rn be the n-bit prefix of a c.e. real r = min Rλ0 . Then I(P : rn ) = n ± O(log n).
B Appendix. ZFC Axioms
ZFC axioms are sometimes given to undergraduates in an unintuitive,
hard to remember list. Setting them in three pairs seems to help intuition.
Sets with a given ST property F (possibly with parameters c) are said to form a class {x : Fc (x)}.
They may or may not form a set, but only sets are the domain of ZFC variables.
1. Membership chains: sources, sinks. (1b anti-dual to 1a):
1a. Infinity (a set with no membership source): ∃S̸=∅ ∀x∈S∃y∈S(x∈y)
1b. Foundation (each set has sinks: members disjoint with it): ¬ ∃S̸=∅ ∀x∈S∃y∈S(y∈x)
2. Sets with formula-defined membership:
2a. Extensionality: (content identifies sets uniquely): x£y£x∈t ⇒ y∈t
2b. Replacement:3 (∀x ∃Y £ Rc ({x})) ⇒ ∀X ∃Y £Rc (X)£Y
df
3. Functions Inverses. f −1 = {g: f (g(x))=x, Dom(g)=Im(f )}:
3a. Powerset (f −1 ¢ G is a set, as g ¢ h = f T ): ∀h∃G ∀g (g¢h ⇒ g∈G)
3b. Choice4 (f −1 is not empty): ∀f ∃ g ∈ f −1
The modifications discussed above include:
1. Restrict Replacement to computable R, drop Power Set;
2. add IP, Pχ, strengthened;
3. extend Foundation to classes of sets (as a family of axioms, like Peano Induction);
4. some (unclear yet) replacement for Choice.5
(May be dropping it, or adding to the language a postulated (not described) class
mapping countable v.Neumann ordinals onto reals, implying continuum hypothesis, too.)
Math objects are classes {q: Fp (q)} of sets q satisfying formulas F with external parameters p. Collections
of objects are treated as collections of those parameters, with conditions for Foundation and Extensionality.
Quantifiers bind parameters, not properties F .
3 df
An axiom for each ST-defined relation Rc (x, y). Rc (X) = {y : ∃x∈X Rc (x, y)}.
4
The feasibility of computing inverses is the most dramatic open Computer Theory problem.
5
I am not clear yet on how to handle some math theorems that depend on Axiom of Choice, as it provides classes
not specified by ZF formulas. One option may be extending the language with a postulated (not otherwise described)
class that maps countable v.Neumann ordinals onto reals (implying Continuum Hypothesis, too).
But any such options would require careful analysis.(More details in https://www.cs.bu.edu/fac/lnd/.m/sta.pdf)
Set Theory in the Foundation of Math;
Internal Classes, External Sets
Leonid A. Levin
www.cs.bu.edu/fac/Lnd
Boston University
BU, July 10, 2024
1 / 15Set Theory in the Foundation of Math;
Internal Classes, External Sets.
Outline
Set Theory: Some History, Self-Referentials
Dealing with the Concerns; Cardinalities
Going at the Self-Referential Root
Radical Computer Theorist Hits Back
Some Complexity Background
Independence Postulate
Reducing All Quantifiers to those on Integers
A Problem: One-Way Functions
Takeout: the Issues
Takeout: a Way to Handle
Some More IP Applications
Appendix: ZFC Axioms
To Modify ZFC
2 / 15Set Theory: Some History, Self-Referentials
Cantor’s Axioms: All Set Th. formulas define (quantifiable) sets.
In effect: formulas with quantifiers over formulas.
This self-referential aspect turned out fatal.
Zermelo, Fraenkel: Restrict cardinality in Cantor’s Axioms:
Replacement preserves it. A separate Power Set increases it.
Somewhat ad-hoc as foundations for math. And cardinality focus
has questionable relevance. Distinctions between uncountable
cardinalities almost never looked at in math papers.
Usual math sets have special types: countable, compact,
open, occasionally Borel, rarely projective, etc.
Generic subsets from Power Set classes, with no other descriptions,
find little use in math and greatly complicate its foundations.
All consistent axiom systems have countable models. Cardinalities
look like an artifact, designed to hide some self-referential aspects.
3 / 15Dealing with the Concerns; Cardinalities
Logicians: Isolate math segments where more ingenious proofs can
replace the use of Power Set Axiom and its uncountable sets.
Math folk: Bad to mess with math unity. Must keep whole its
monumental structure ! And better not to complicate proofs.
Computer Theorist breaks in: Are there really any infinite objects?
Computer T(errorist): Timidity never works! Reject infinite sets.
Dear C.T.: Agreed about timidity, but drop your errorist aspect!
Infinities are neat: R is compact, “less infinite” in that than Q.
And handling (often ambiguous) termination points of objects is
awkward. And 0, ∞ are great simplifying approximations to ϵ, 1ϵ .
4 / 15Going at the Self-Referential Root
Expanding Set Th. with more formula types, axioms, etc. has no
natural end. Benefits little, eventual consistency loss inevitable.
ZF-restricted self-referentials, such as implicit quantifiers over
formulas, brought no trouble so far, but find little math use either.
Let us try to drop any such excesses.
Externals: sets math handles (as values of variables, e.g., random
strings), but does not internally specify. Mark them apart from
classes: collections defined by math properties.
Math objects (only informally called sets) are classes {q: Fp (q)}
of sets q satisfying formulas F with external parameters p.
Collections of objects are treated as collections of those
parameters. Quantifiers bind parameters, not properties F .
5 / 15Radical Computer Theorist Hits Back
Independence Postulate
Even with infinite complexities, external objects have finite
information (small, really) about formula-defined classes.
Besides, it would be redundant for math objects Fp to duplicate
in the external parameter p their formula-defined information.
Complexity theory allows to formalize that, justify the validity for
“external data”, and use that for simplifying math foundations.
This gives a way to handle infinitely complex sets, but reduce their
quantifiers to those on integers. All with no seeming need to
change anything in math papers, only reinterpret some formalities.
(In some cases one can state meta-theorems: a family with
formula parameter F , as done now by Category Theorists.)
6 / 15Some Complexity Background
df
Length ∥t∥ = n for t∈{0, 1}n ; ∥t∥ =df
+log2 t,−1 for t∈R+ .
The uniform on Ω =df
{0, 1}N distribution ¼(t Ω) = df
2−∥t∥ .
C.e. (computably enumerable) sets are ranges of algorithms.
C.e. function to R+ is sup of a c.e. set of (continuous) basic ones.
C.e. f ∈D dominates Banach space D if all c.e. g ∈D are O(f ).
The ¼-test is d(³) =df
∥+T(³),∥ for c.e. T: Ω→R+ , ¼(T)f1 that
dominates L1 (Ω, ¼). M(x) =df
¼(u −1 (x Ω)) (u - universal alg.) is
dominant among semimeasures {µ : ∀x µ(x) g µ(x0)+µ(x1)}.
Kolmogorov–Martin-Lof ¼-randomness: R¼c =
df
{³ : d(³)Independence Postulate
IP: ∀³ I(³: F ) < ∞
(A family of axioms, one for each property F ∈ ∆0∗ =
df
∪n ∆0n ¢ Ω.)
(By IP, classes ³∈∆0∗ double as sets only if computable.)
Justifications and Applications
Conservation laws: no processing of ³, algorithmic, or random, or
mixed, increases I(³:F )+O(1). Arguably, no physical process can.
Little expressive power loss: Any object F³ is also F ´ , I(´: G )Reducing All Quantifiers to those on Integers
IP opens a way: excludes ³ ∈ F´ unless such ³ reduce to a positive
fraction of u −1 (´). (Note: ¼(A|B)>t has only integer quantifiers.)
But what about the reverse?
Primal Chaos axiom (Pχ): “Each ³ reduces to
even-indexed digits of some K-ML random ´ ∈ R¼ .”
(For classes, i.e. in ZF, it is the famous Gacs-Kucera theorem.)
A model (countable, in ZFC): Take a µ∈ Ω, outside all X ∈∆0∗ of
¼(X )=0. Let µ (k) : i 7→ (i mod k) µi . µ-model includes all sets with
“∈” on transitive closures enumerable from µ (k) for some k.
The model eliminates 2nd order quantifiers, obeying a c.e. family of
axioms: F ô ¼({µ: F ∗ (µ)}) > 0, where F ∗ has all real variables ³i
in a sentence F replaced by Ai (µ (ki ) ); Ai , ki quantified as integers.
9 / 15A Problem: One-Way Functions
Extending IP with such a c.e. family of axioms does not strike me
as really elegant and intuitive. I hoped, adding a single Gacs-Kucera
Theorem as a fundamental Set Theory axiom would suffice.
(Hint: by IP, ∃³P(³, ´) ⇒ ¼({µ: ∃f P(f (µ), ´)} | u(µ)=´) > 0).
But deriving “⇐” via Pχ meets an obstacle:
Recursively One-Way Functions on S, ¼(S)>0.
Let ¼(f −1 (x Ω) ∩ S) = O(¼(x Ω)). f is OW on S if no g inverts it
(f , g computable): ¼2 ({(´, µ) : ³=
df
g (´, µ)∈S, f (³)=´}) = 0.
They do exist: [Barmpalias, Gacs, Zhang: 2024].
Handling OWFs demands more tools. A single axiom would be
more elegant and intuitive than the whole c.e. family from
the above model. I have some ideas but the problem is still open.
10 / 15Takeout: the Issues
1. Cardinality-based ZF restrictions of Cantor’s Axiom defuse
self-referential problems but do not eliminate their source.
A bit ad-hoc, and result in a Babel Tower of cardinalities,
other hierarchies, that find little relevance in math.
2. Replacing Power Set in segments of math with more elaborate
proofs (as Reverse Math, some others do) breaks the unity
of math, so does not seem to be the right solution.
3. I blame the blurred distinction between internal (math-defined)
and external (the domain of variables) aspects of math objects.
4. Extending Set Theory reach has no limits. Including in
quantifiable domains formulas or classes they define just climbs
higher in that direction. Little relevance to mainstream math.
11 / 15Takeout: a Way to Handle
5. Separating formulas F from external variables values p in math
objects Fp allows restricting formula-related information from p.
6. Complexity theory allows to formalize that, justify the validity
for “external data”, and use that for simplifying math foundations.
7. What is left out? – “Logical” sets, related to infinite hierarchies
of formulas, such as “The set of all true sentences of Arithmetic”.
Those should be subject of math foundations.
Theories cannot include their own foundations.
IP has a number of impressive (to myself at least) applications.
But I am against holding hostages for long.
So I will mention a couple and let you be free.
12 / 15Some More IP Applications
Foundations of Probability Theory. Paradoxes in its application
led to the concept of K-ML Randomness R¼ . IP clarifies its use:
For any S¢ Ω: ¼(S)=0 if and only if ∃Ã S ∩ R¼ ¢ {µ : I(µ:Ã)=∞}.
Goedel Theorem Loophole. Goedel writes:
“It is not at all excluded by the negative results
mentioned earlier that nevertheless every clearly posed
mathematical yes-or-no question is solvable in this way.
For it is just this becoming evident of more and more new
axioms on the basis of the meaning of the primitive notions
that a machine cannot imitate.”
No way ! Let a predicate P on {0, 1}n extend “proven/refuted”
partial predicate of Peano Arithmetic. Let rn be the n-bit prefix
of a c.e. real r = min R¼0 . Then I(P : rn ) = n ± O(log n).
13 / 15Appendix: ZFC Axioms
1. Membership chains: sources, sinks. (1b anti-dual to 1a) :
1a. Infinity (a set with no source): ∃S,s∈S ∀x∈S∃y ∈S(x∈y )
1b. Foundation (sink in any set): ¬ ∃S,s∈S ∀x∈S∃y ∈S(y ∈x)
df
2. Formula-defined Sets ( Rc (X ) = {y : ∃x∈X Rc (x, y )} ) :
2a. Extensionality (content identifies sets): x£y £x∈t ⇒ y ∈t
2b. Replacement: (∀x ∃Y £ Rc ({x})) ⇒ ∀X ∃Y £Rc (X )£Y
3. Function Inverses f −1 =
df
{g : f (g (x))=x, Dom(g )=Im(f )}:
3a. Powerset (f −1 is a set: h=f T ): ∀h∃G ∀g (h£g ⇒ g ∈G )
3b. Choice (f −1 is not empty): ∀f ∃ g ∈ f −1
14 / 15To Modify ZFC
1. Restrict Replacement to computable R, drop Power Set;
2. add IP, Pχ, strengthened;
3. extend Foundation to classes of sets (as a family of axioms);
4. some (unclear yet) replacement for Choice.
(May be dropping it, or adding to the language a postulated
(not described) class mapping countable v.Neumann ordinals
onto reals, implying continuum hypothesis, too).
Math objects are classes {q: Fp (q)} of sets q satisfying formulas F
with external parameters p. Collections of objects are treated as
collections of those parameters, with conditions for Foundation and
Extensionality. Quantifiers bind parameters, not properties F .
15 / 15You can also read
