
List of Statements Independent of ZFC - killlameme99
https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC
======
triska
A very interesting discussion about this topic that also includes many
examples and references is available on MathOverflow:

 _What are some reasonable-sounding statements that are independent of ZFC?_

[https://mathoverflow.net/questions/1924/what-are-some-
reason...](https://mathoverflow.net/questions/1924/what-are-some-reasonable-
sounding-statements-that-are-independent-of-zfc/6594)

With the top voted result currently being:

 _" If a set X is smaller in cardinality than another set Y, then X has fewer
subsets than Y."_

As is also mentioned in the linked entry, this statement is independent of
ZFC.

~~~
ur-whale
So, does that mean that one can assume this to be true and build a perfectly
consistent theory, or conversely assume it to be false (with - say - at least
one counter-example) and build another perfectly consistent theory?

~~~
deepsun
Well, it's not possible to prove the consistency, thanks to Godel. Maybe one
of your new theories would contain a statement, inconsistent with the rest of
ZFC.

~~~
CrazyStat
This is incorrect. It absolutely is possible to prove consistency, what Gödel
tells us is that in any consistent logic system there are true but unprovable
(in that system) statements.

For this particular list, the statements have been proven to both be
consistent with ZFC and for their negations to be consistent with ZFC.

~~~
ithinkso
This is first incompleteness theorem. What deepsun was referring to is second
incompleteness theorem - in a consistent system F the statement 'F is
consistent' is in fact unprovable (in F).

------
tomp
I'm tempted to exclude the Axiom of Choice (AC) from any math I do, and
instead include the Axiom of Determinacy (AD) [1] (which contradicts AC), so
that all subsets of _R^n_ are _measurable_ [2] (thus precluding the
Banach–Tarski paradox), and the Axiom of Dependent Choice (DC), which is
weaker than AC but sufficient to develop most of real analysis. Like, I don't
really care if not _all_ vector spaces have a basis; it's enough for me that
all _interesting_ vector spaces do (I think).

But then we have this (from [3]):

 _> For each of the following statements, there is some model of ZF¬C where it
is true:_

 _> \- In some model, there is a set that can be partitioned into strictly
more equivalence classes than the original set has elements, and a function
whose domain is strictly smaller than its range. In fact, this is the case in
all known models._

So it's really, pick your own poison - _either_ one of these:

\- you can take apart a ball and put the pieces back together into two balls

\- there exists a function whose range is larger than its domain

Math is weird.

[1]
[https://en.wikipedia.org/wiki/Axiom_of_determinacy](https://en.wikipedia.org/wiki/Axiom_of_determinacy)

[2]
[https://en.wikipedia.org/wiki/Solovay_model](https://en.wikipedia.org/wiki/Solovay_model)

[3]
[https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_con...](https://en.wikipedia.org/wiki/Axiom_of_choice#Statements_consistent_with_the_negation_of_AC)

~~~
ginnungagap
In fact it's an open problem (very likely the oldest open problem in set
theory) whether in every model of not AC there is a set with such a
paradoxical partition! Here's a nice introduction to the problem by Asaf
Karagila [http://karagila.org/2014/on-the-partition-
principle/](http://karagila.org/2014/on-the-partition-principle/)

------
mturmon
Some of this was familiar, but I also saw this:

> In 1973, Saharon Shelah showed that the Whitehead problem ("is every abelian
> group A with Ext^1(A, Z) = 0 a free abelian group?") is independent of ZFC.

(This is the same Shelah who proved what is usually called "Sauer's Lemma" on
set separability, upon which the "VC dimension" and the resulting VC learning
theory are based.)

Anyway, this was surprising because I didn't know there were any theorems
_outside_ of set theory that had been known and studied, and then later turned
out to be independent of ZFC.

Apparently, the surprise I felt is just because I wasn't paying attention --
there are a couple of other problems in the page that seem to have a similar
flavor. I'm not enough of a mathematician to appreciate to what extent they
are interesting problems that arise independently of counterexamples.

[edited to add: the comment of @triska nearby addresses exactly this point!]

~~~
ginnungagap
Another famous example of that kind is the existence of outer automorphisms of
the Calkin algebra, which is a simple question about a naturally occuring
object which turned out to be independent of ZFC by work of Farah and Weaver.

Topology and set theoretic topology is full of those statements, are there
Suslin lines? Are there S-spaces? Is the product of two ccc spaces also ccc?
The list goes on

Another one from analysis is a very strong form of Fubini's theorem that was
shown independent by Friedman.

There's surely more but those are the ones I could remember right now!

------
ArchReaper
Is there any ELI5-type explanation for us non-mathers? Whenever I see stuff
like this, I start trying to actually understand it, then fail miserably just
trying to google terms I'm not familiar with. Is advanced knowledge of these
math principles required to understand the significance of this page, or why
it is interesting?

~~~
gjm11
[The following isn't really "like you're five", but given how long it is
already that's probably just as well.]

 _Proofs and formal systems, and why we 're kinda screwed_

Mathematicians like to prove things. What we would _really_ like would be to
be able to find, for every mathematical statement, either a proof that it's
true or a proof that it's false.

It wasn't until the early 20th century that mathematicians got a clear enough
idea of what proof _is_ to figure out whether that could be done. And it turns
out it can't! That was proved in 1931 by Kurt Goedel, who did something that
these days is very familiar to anyone in computing but was novel then: he
found a way to encode mathematical statements, and sequences of mathematical
statements, as _numbers_ , in such a way that properties of those statements
turn into properties of the numbers, so that you can use ordinary mathematics
to reason about them.

This enabled him to show that if you have any system for doing mathematics
that (1) is powerful enough to describe ordinary arithmetical statements and
(2) is simple enough that you can check mechanically what is and isn't a
proof, then there are statements that that system can neither prove nor
disprove. (With a proviso I'll come to in a couple of paragraphs.)

So much for the hopes of mathematicians.

Anyway, despite that setback, mathematicians didn't abandon the idea of
building up all of mathematics in some nice simple formalized system in which
we can prove and disprove things. We still like to do that, and have come to
terms with the fact that there will be things we can neither prove nor
disprove.

... Unless the system we're working in happens to be _inconsistent_ , meaning
that there's some proposition that it can both prove _and_ disprove. In that
case, what actually happens is that it can prove _everything_ , which is of
course completely useless. So far as we know, the systems we like to use
aren't inconsistent. It would be nice to prove they aren't -- but there's a
nice variant of Goedel's theorem that says not just "for any system that isn't
inconsistent, there are propositions it can't decide" but "for any system that
isn't inconsistent, a formalized statement of its own consistency is a
proposition it can't decide".

Once, again, so much for the hopes of mathematicians.

 _The particular formal system called ZFC_

One particular nice simple formalized system is called ZFC, which is short for
"Zermelo-Fraenkel set theory with the Axiom of Choice". That's a bit of a
mouthful. So, "set theory" means that the basic idea we're starting with is
that of a _set of things_. (In order to do mathematics, we certainly need some
idea like that. It turns out that taking it as the _foundational_ idea works
fairly well.) Zermelo was a mathematician who came up with one fairly good way
to do that. Fraenkel was another mathematician who improved Zermelo's system.

The "Axiom of Choice" \-- the C in ZFC -- is a rather technical statement in
set theory that turns out to be (1) "obviously true" according to some
mathematicians' intuition, (2) useful for proving things we care about, and
(3) something that plain old ZF, without the C, can neither prove nor
disprove. (Unless, as usual, ZF is actually inconsistent.) In particular, this
means that if ZF is a consistent system -- if it isn't able to prove 1+1=3 --
then ZFC is as well. (Because if you could get a contradiction out of ZFC, you
could use it to prove not-C within ZF.) So adding the Axiom of Choice to ZF is
"safe"; it can't create contradictions that weren't already there. And, since
it's useful, we tend to keep it around.

[You can skip the next two paragraphs if you like. They describe the sort of
thing you do if you want to build up all of mathematics on top of set theory.]

So, how do you build mathematics out of set theory? The usual sort of game you
play is this. We want to be able to talk about _numbers_. So first of all we
find an implementation of numbers in terms of sets. (If you were doing
mathematics in a computer you'd want to do roughly the reverse, and build your
_sets_ out of _numbers_.) So we might, e.g., say that 0 "is" the empty set,
and then 1 "is" the set containing just 0, and then 2 "is" the set containing
just 0 and 1, and so forth. That gets you a load of sets that can (as well as
being sets) do double duty as the non-negative integers. Then you construct
the integers (negative as well as positive) out of those, and then the
rational numbers (ratios of integers) out of those, and then you need some
fancier footwork to get the "real numbers" (including things like pi and the
square root of 2).

And then you're off to the races, because once you have the real numbers you
can do _geometry_ : e.g., three-dimensional space "is" the set of triples
(x,y,z) of real numbers, and things like spheres and dodecahedra are just sets
of points in space. And now you have numbers and geometrical things, and you
can build all the other weirder more abstract things mathematicians like to
study in a similar way.

OK, so if we have sets then we have everything, so a formalized set theory is
a reasonable thing to try to use as a basis for mathematics. But, ever since
Goedel, we know that any formalized system will be unable to decide (i.e.,
either prove or disprove) some statements we might be interested in.

 _What that page is about_

The Wikipedia page linked here is a list of statements we might be interested
in that ZFC _is_ , as it turns out, unable to decide. "Independent of" means
"neither provable nor disprovable from".

(There are other systems you can use instead of ZFC. Some of them are other
varieties of set theory; some work entirely differently. It usually turns out
that you can translate most statements of ordinary mathematics to and fro
between them, and that what's provable and what isn't doesn't depend on
"implementation details" (e.g., I described one particular way to build non-
negative integers out of sets; what if we do it a different way? The answer is
usually that it doesn't matter). But some systems are "stronger" than others
and can prove or refute more statements. To be usable for mathematics, a
system can't be "too much" weaker than ZFC. And since in some sense ZFC
encodes most of the things that mathematicians are pretty sure "ought" to be
true, most systems people want to use aren't "too much" stronger than ZFC
either. A lot of the statements on that page would also be on similar pages
about systems other than ZFC.)

A lot of those statements are weird technical things that only a mathematician
could care about, and indeed that only a mathematician specifically interested
in what's provable from ZFC and what isn't could care about. Some of them are
at least potentially interesting to some mathematicians for their own sake,
but understanding them generally requires a pile of background that (1) I
don't really have myself in most cases and (2) you really don't want me to
make this long enough to deal with.

But here's one of them, one of the oldest of them all. Some sets are bigger
than others. For instance, {1,2,3} is bigger than {1,2}. What about infinite
sets? It's not obvious a priori what "bigger" should even _mean_ for infinite
sets, but Georg Cantor (a pioneer of this stuff) found a good answer, whose
details we don't need right now. It turns out that the integers are "the same
size" as the rational numbers, even though the integers seem like a tiny
subset of the rational numbers, but that the real numbers are "bigger". So
here's the question: are there any sets _bigger_ than the rational numbers but
_smaller_ than the real numbers? This is called the "continuum hypothesis",
which is a bit of a stupid name but never mind, and it is neither provable nor
disprovable in ZFC.

~~~
xigency
Well done.

Has anyone named a set in between the rationals and the reals?

~~~
daxfohl
The canonical example is
[https://en.wikipedia.org/wiki/Vitali_set](https://en.wikipedia.org/wiki/Vitali_set)

You can kinda think of it as "the set of real numbers MOD the set of
rationals". Kinda. And they're dorked up because the length is infinitesimal,
but a countably infinite number of them add up to length 1.

For a more precise explanation see here:
[https://math.stackexchange.com/a/137959/287133](https://math.stackexchange.com/a/137959/287133)

~~~
daxfohl
Also note that the construction of the above set requires the axiom of choice.
And, as we all know, the axiom of choice is equivalent to the continuum
hypothesis in ZFC. So that's how it all fits.

------
reggieband
I recall once I read the short book "The Philosophy of Set Theory" [1] since I
like philosophy and have an interest in Math. It contains much of the history
that lead up to the decision to base significant portions of the soundness of
mathematics on top of set theory (and by proxy: Cantor's work on infinities).
My recollection is fuzzy since it was years ago but I recall it starts at
Zeno's paradox and follows along to calculus and beyond.

The book suggests there was a lot of displeasure and argumentation within the
philosophy and math communities because it was felt that there was no real
basis for infinitesimals. Some mathematicians (I believe Hilbert and Frege
among them?) became determined to shut-up the pesky philosophers by proving
the soundness of math based on some logical axiomatic fundamentals. Of course,
this was later proven to be impossible by Gödel but at the time they
considered it a win that philosophers and mathematicians could at least agree
on logic (and more broadly "logical empiricism" which is a basis of
"analytical philosophy").

I recall being completely dissatisfied at the arguments presented in favour of
ZFC (not mathematically, but philosophically). I remember there was a single
paragraph somewhere in the final third of the book that I head to re-read
several times before I finally gave up in frustration. My impression of this
history is that the mathematicians "won" in some sense by railroading their
ideas. Calculus works, right? It is extremely effective and leads to correct
results ... so ignore the seeming paradox of summing an infinite quantity of
infinitesimally small values and move on already! Further, ignore the actual
paradoxes inherent in infinite sets. And this was all done not because there
was some problem to be solved but rather to shut-down debate that seemed to
undermine the philosophical position of logical empiricism.

Another interesting (if historically questionable) exploration of this topic
is the graphic novel Logicomix [2]. This work follows Bertrand Russel and
Wittgenstein through this period in our history.

1\. [https://www.amazon.com/Philosophy-Set-Theory-Introduction-
Ma...](https://www.amazon.com/Philosophy-Set-Theory-Introduction-
Mathematics/dp/0486435202)

2\.
[https://en.wikipedia.org/wiki/Logicomix](https://en.wikipedia.org/wiki/Logicomix)

~~~
empath75
I’m not sure that any side won. ZFC is merely a game with clearly defined
rules that lots of people have agreed to play, but it is not the only game, by
a long stretch.

~~~
reggieband
I think it feels easier to say that now that we are long past the point where
the debates occurred. But this happened during a time when universities were
still trying to figure out how to divide up sciences.

Nowadays, the idea that math has a role to play in pretty much every science
isn't really questioned at all. I mean, imagine I suggested that something
_other_ than math should be brought to bear on physics. I doubt a single
person in here would support such an approach on any level. I think that
represents a clear win. Answering objections about the fundamentals of math
using formalisms like ZFC was a component of that.

~~~
empath75
ZFC has very little to do with why math is used in universities or the
sciences, and it would still be used even without it, because as you said, it
works. It worked for 3000 years before we had ZFC after all.

ZFC wasn’t even the end of the debate on mathematical foundations even in
math. There are a lot of people trying to redo everything with types and
category theory today.

~~~
reggieband
ZFC follows along a path including (but not started by) Russell/Whiteheads
Principia Mathematica, which famously (infamously?) takes several hundred
pages to prove 1+1=2. I doubt very few have thought ZFC (or it's variants)
would be the last word.

Almost no scientists cared about formalizing or proving the soundness of the
mathematical tools they used. In the same way the majority of programmers do
not care about proving the soundness of their programming languages. In
general, people seem to be interested in the practical aspects of their work.

But the general idea that symbolic logic is the primary basis for
understanding the world is something a bit different and something we rarely
question now. I think people assume that this is some obvious thing but it is
actually an idea that was coordinated and forwarded. It appears to me that the
debate at the beginning of the 20th century around using set theory to
establish the foundations of math by way of logic is when the scale seems to
have heavily tipped towards that particular idea.

~~~
aratakareigen
I can't speak for scientists, but programmers usually care _very much_ about
the soundness (not in the Curry-Howard sense) of their type systems.

~~~
reggieband
I disagree and I'm sure we'd only be able to trade anecdotes and no real
evidence. However, my experience working in industry for 20+ years is that
almost no working programmers pay any attention to such things. For example,
when my team recently decided to switch from Javascript to Typescript there
was zero consideration about the fundamental soundness of either language. I
had the same experience when a team I recently worked with was debating a
switch from Java to Kotlin. Nothing approaching the topic of soundness even
came up.

I think hacker news can be a bubble since these deeper issues can sometimes
appear here. I recall a recent post about a soundness bug found in Rust and it
generated quite a lot of discussion. However, I see that as analogous to the
intense scrutiny of a small cabal of scientists/philosophers, the Vienna
Circle for example, who did take these fundamentals seriously in
math/sciences. I just do not believe and have not experienced that sentiment
to be prevalent outside of this bubble.

------
daxfohl
Why is it always zfc plus optionally something else? Is there anything other
than zfc that creates an interesting starting point?

~~~
triska
These are excellent questions! And yes, there is!

For example, I highly recommend the paper _Rethinking set theory_ by Tom
Leinster:

[https://arxiv.org/abs/1212.6543](https://arxiv.org/abs/1212.6543)

It highlights Lawvere set theory. The paper won this year's Chauvenet Prize:

[https://www.maa.org/programs-and-communities/member-
communit...](https://www.maa.org/programs-and-communities/member-
communities/maa-awards/writing-awards/chauvenet-prizes)

~~~
QuesnayJr
Isn't Lawvere set theory equiconsistent with bounded Zermelo set theory?

ZFC, its subtheories, and its extensions, serve as a universal yardstick for
consistency results. It's not really clear why that is the case, but it has so
far proven to be so.

------
est31
In other news, there's recently been a mechanized proof for one of the ZFC
independent problems:
[https://flypitch.github.io/papers/](https://flypitch.github.io/papers/)

[https://news.ycombinator.com/item?id=21823618](https://news.ycombinator.com/item?id=21823618)

------
etaioinshrdlu
I love seeing pure math on HN. I think math is the purest expression of what
it means to be curious. It sometimes gives us fun toys to build into whatever
we are coding.

------
Rerarom
Hopefully this will clear at least some of the Gödelian misconceptions that
pop up from time to time here

~~~
JadeNB
How? Misunderstandings about Gödel's theorem are unlikely to be due to a
misunderstanding of the axioms of set theory (more likely of logical issues
such as the technical meaning of 'incompleteness'), let alone to a lack of
understanding of what statements are independent of a common set of axioms.

------
joe_the_user
_One can write down a concrete polynomial p ∈ Z[x1,...x9] such that the
statement "there are integers m1,...,m9 with p(m1,...,m9)=0" can neither be
proven nor disproven in ZFC (assuming ZFC is consistent)._

So you were to specify such a thing and the thing was small enough it's value
could be determined by a command line program and you found integers m1.... m9
such that when you typed them at the command line, the value returned was 0,
would "reality" have determined a "truth" that was not deducible?

Still, I can't see each of the steps wouldn't be easily determined by existing
axioms.

Anyway, my head hurts.

~~~
joe_the_user
OK, the way I'd figure it out is: for such a polynomial, you definitely can't
find those integers. There isn't any concrete m1...m9 satisfying the
condition. The stumbling block is you can't find a proof for this fact in ZFC.

But this seems to go against the idea that for any proposition independent
from an axiom system, there is a model of the axiom system where that
proposition is true and another where it is false.

Someone enlighten me (not sarcastic).

~~~
dodobirdlord
> OK, the way I'd figure it out is: for such a polynomial, you definitely
> can't find those integers. There isn't any concrete m1...m9 satisfying the
> condition. The stumbling block is you can't find a proof for this fact in
> ZFC.

If this were formalizable it would be a proof.

I encourage you to read through the excellent piece on the busy beaver
function and computability. It's not entirely related, but it's fun! And it
touches on the same theme as your question here, that human intuition about
mathematics is weak. The writeup describes and proves the non-computability of
a particular function of positive integers with definite (but non-computable!)
value.

[https://www.scottaaronson.com/writings/bignumbers.html](https://www.scottaaronson.com/writings/bignumbers.html)

~~~
Dylan16807
> If this were formalizable it would be a proof.

A proof, but not a proof that can be expressed inside ZFC.

~~~
dodobirdlord
Obviously, the original statement has been proved to be independent from ZFC.

------
maest
What's a reasonable strategy of proving a statement like that is undecidable
in ZFC?

I think it must use some tools I'm unfamiliar with.

~~~
ginnungagap
Either you show that your statement P is equivalent to another known
independent one or you need to produce a model of P and a model of not P.

The two main techniques to produce new models of ZFC are inner models, that is
submodels of a model, for example studying the inner model L (Gödel's
constructible universe) is how the consistency of the continuum hypothesis,
the generalized continuum hypothesis, the existence of Suslin trees etc. was
shown. Looking at L also allows one to conclude that the axiom of choice is
consistent with ZF. Another commonly study inner model is the so called HOD,
and there's a whole area, "inner model theory" that essentially tries to
construct canonical inner models for some statements.

The second big way to produce a model of ZFC is by forcing. This is a very
versatile tool that allows to extend a given model of ZFC by adding a new set
to it (for example by adding a lot of new reals numbers to a model of CH you
can make CH false). Forcing is how the consistency of the negation of CH and
GCH was proved.

An interisting example is the negation of AC and its consistency with ZF. If
you happen to live in a model of AC every forcing extension will still be a
model of AC so it seems that our previous techniques are powerless. But
actually what can be done is to look at a carefully chosen submodel of a
forcing extension that still models ZF but in which AC fails.

------
chrisweekly
_" The mathematical statements discussed below are provably independent of ZFC
(the canonical axiomatic set theory of contemporary mathematics, consisting of
the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is
consistent. A statement is independent of ZFC (sometimes phrased "undecidable
in ZFC") if it can neither be proven nor disproven from the axioms of ZFC."_

\- TFA

~~~
chrisweekly
"ZFC" is an esoteric acronym; I posted the quote from the article to help
others because the title was entirely opaque to me.

------
mirimir
Might something like this be fscking unification in physics?

~~~
drdeca
I doubt it. Physics doesn’t seem to have much which is clearly connected to
proof systems.

And, besides, from any (consistent) axiom system for which a given statement
is undecidable, there is another axiom system which is the same except it adds
that statement as an additional axiom, and the statement is therefore
(trivially) provable in that system.

And, it doesn’t seem like physics is constrained to use only some specific
axiom system.

It seems to me that the relevant thing to physics would be, rather than an
axiom system, instead, a model (in the math sense, not the physics sense).

~~~
mirimir
Thanks.

But doesn't that lead to systems with infinite numbers of axioms?

~~~
drdeca
If you want to add axioms in order to be able to show each of an infinite
number of statements which are all independent of the initial axiom system,
and also none of them follow from the rest of them, that could involve an
infinite set of axioms, yes,

but any statement we could make about stuff in physics, if it could be
expressed in the language of the formal system, would, as a single statement,
be something that could be added as a single axiom.

