
To Settle Infinity Question, a New Law of Mathematics (2013) - okket
https://www.quantamagazine.org/to-settle-infinity-question-a-new-law-of-mathematics-20131126/
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pavpanchekha
The article is fine as long as it goes but thinking of set theory as the
"foundational laws" of mathematics is sort of a mid-20th century understanding
of math. We have a better understanding of what axioms are for and how
foundations should be structured. I am confident the author understands the
more subtle framing I'd like to discuss, but I think it's worth describing
here for readers who were confused.

The main difference: this is not a new "law of mathematics" so much as a new
law of set theory. No one but set theorists will be affected, and no other
mathematicians care much.

Foundations are not nearly as set as undergrad math teaches you. I don't mean
the Axiom of Choice (which pretty much no one has any problems with anymore).
However, various large cardinal axioms are pretty standard in advanced
Algebraic Geometry (for example, the existence of Grothendiek Universes). Set
theorists work with all sorts of large cardinal axioms all the time, and for a
while it was fashionable to assume various generalized continuum hypotheses.
Graph theorists, for some applications, assume "V=L", because that makes
things tidy. It's fine, and no one really has a problem with it, nor do these
disagreements really cause problems, even though some of these axioms are
contradictory. (And even when various large cardinal axioms are assumed, often
people work with an intuitive ill-founded set theory with the assumption that
it can be made well-founded by ramifying using the large cardinals instead.)

The 21-century understanding is something like: there's no need for there to
be "ultimate" foundations. As mentioned, there is little impact on "ordinary"
mathematics from different axioms of set theory, so maybe the set theorists
should choose axioms of set theory, but that needn't affect other
mathematicians. Obviously someone should study set theory, it's cool and there
are interesting theorems and so on; and if they find that "V = ultimate L" is
a useful axiom for their field of study, nothing wrong with that, much like
commutative algebra is a fine field of study though of course non-commutative
algebra is interesting too. But there's no realistic sense in which group
theory is "defined in terms of" set theory—you can tell, because group
theorists tend not to care about set theoretic axioms, and have no dog in
choosing between "V=ultimate L" and forcing.

The argument between "V = ultimate L" and the forcing axioms are a debate over
what "set theory" should study, and each mathematician involved thinks that it
should study the sort of things that they study—whether that means forcing or
large cardinal axioms. Large cardinal axiom people have a well-developed kind
of work they want to do: defining various forms of large cardinals, proving
implications between them, and then discovering that all large cardinal axioms
are totally ordered by proof strength. It's good, interesting work, and "V =
ultimate-L" will really complete the research program they've been pursuing;
or, proving independence results for various theorems, and a forcing axiom
would really put the machinery they work with deep into the foundations of set
theory, making that work much simpler and at the same time deeper. That, too,
is a valuable field of research, in fact it has led to the only Fields medal
in logic.

I assure you that _your personal_ understanding of sets will not be affected
much by which axiom which mathematicians choose to adopt.

~~~
fmap
Great post, let me just add one point:

> The 21-century understanding is something like: there's no need for there to
> be "ultimate" foundations.

The modern view is that there simply is no ultimate foundation. Large cardinal
axioms in set theory can be seen as adding more and more inner models of set
theory (with fewer large cardinals) into an ambient theory. This is basically
the same thing as asserting that "the previous theory is consistent". You can
play this game forever and it will not converge - the result hinted at in the
article states something different.

\---

In the end we use set theory as an alibi. We tell people that ultimately all
of mathematics can be encoded in set theory, but it is neither a natural
encoding, nor is it really true since we are talking about different flavors
of "set theory".

Algebraic geometry is a good example. To begin with, you are not using ZFC to
encode categories - since you need to be able to manipulate proper classes as
if they were sets - so we need to use some extension such as NBG set theory
instead. Then, in order to construct the localisation of a large category you
suddenly need large cardinals, even though intuitively the localisation is in
some sense no larger than the category you start with.

And this is the point where we forget this construction again, because it
doesn't give any useful insights.

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DonbunEf7
Here's Woodin talking about his work in this area on a more technical level:
[https://www.youtube.com/watch?v=nVF4N1Ix5WI](https://www.youtube.com/watch?v=nVF4N1Ix5WI)

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Chris2048
I don't really understand this. "Infinity" just means without finite quantity,
doesn't it? I can, theoretically, count for infinity - there are no end to the
_theoretical_ numbers.

If I can prove a value can be reduced arbitrarily close to a limit, we can say
something is equal to that limit for any _finite_ resolution (for any
resolution of "closeness to the limit" there is a precision in the calculation
capable of delivering within that resolution)

Hence, I can say "F(x) = 3 for any finite resolution", so if no infinities
exist, then none are needed for mathematics. They are a calculation artifact
that represents a single function over arbitrary levels of resolution.

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zaptheimpaler
This is crazy. How about just doing what maths does anyways - state your
assumptions?

Prove stuff with V=ultimate L? Great, slap a "V = ultimate L => ..." on it.

Prove stuff with forcing axiom? Cool, "forcing axiom implies ..."

The entire premise of asking "whats the one true axiom?" makes no sense. Its
not like the real world places any restrictions on math - its a self-contained
universe. So obviously you are free to pick whatever imaginary universe you
would like to study. If we ever get to a point where we can directly test
which of the 2 axioms leads to more useful theories in the real world, great,
but that seems pretty unlikely given they come up when talking about infinity.
The obvious truth seems to be that infinity could have a few subtly different
meanings depending on how you define it - and thats totally fine because the
real world does not contain infinities, its a purely mental construct.

~~~
placebo
Amen. Not a mathematician, but this perfectly articulates my thoughts on the
subject.

> _the real world does not contain infinities_

I have no idea what "real world" even means, but I'd say that infinity does
not exist within anything that can be conceptualized.

~~~
thomasahle
> infinity does not exist within anything that can be conceptualized.

Yet math is consistently being conceptualized?

Enabling clear thinking about concepts outside of the real world is the while
point of math.

~~~
placebo
> _Yet math is consistently being conceptualized?_

Sure, how else could we capture it or convey it to others? The thing is that I
don't see it as _that_ different than conceptualizing things in the "real
world". In the latter, conceptualization is done on input of the senses, while
in math conceptualization is done on inputs from a higher level of
abstraction. In both cases, something is defined and then manipulated for
something useful. It's just that I believe that whatever is defined
automatically becomes a finite concept otherwise it could not be grasped and
manipulated, and a finite infinity seems problematic...

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enturn
If infinity is being boundless then to make anything infinity have the bounds
as unknown. A practical use is in Haskell's lazy evaluation. Although maybe
I'm missing the point completely.

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xelxebar
“What truly infinite objects exist in the real world?”

I'm not sure what "infinite object" means, but I do find it interesting that
we can do the math anyway using only a finite number of symbols.

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m3kw9
My dad would answer when I was a kid, at the end, there is just a wall, but
then I’d ask what what’s beyond the wall and that was the end of the
conversation

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excalibur
Title needs a (2013) at the end.

In my opinion, trying to settle the matter by adopting a new axiom is a bit
like proving that blue is the best color for cars by requiring all cars to be
painted blue.

~~~
xamuel
A different viewpoint: "adding a new axiom" is synonymous with "further
restricting the things under discussion".

Imagine biologists decide that "DAG" is the correct abstraction for parent-
child relationships. Then they spend years trying to prove the obviously-true
fact that parents can't have infinitely many children. Of course, they never
can, because DAG nodes can have infinitely many children. Finally they realize
they chose the wrong abstraction, they should've chosen "DAGs where all nodes
have finitely many children".

To suggest adding a new axiom to ZFC is to suggest that ZFC was an over-broad
abstraction for capturing what things are supposed to be "sets".

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Sniffnoy
This sort of thing just strikes me as so pointless. How could one possibly
decide? Obviously, if we're talking about new axioms that aren't provable from
the old ones, you can't use those old ones to settle the matter. The axioms
under discussion don't really affect normal math so you can't appeal to their
usefulness there (unlike with, say, the axiom of choice, which famously was
controversial for a while). And I don't think people can really have much of a
claim to have any good intuition for such things, or else we'd have had such
an axiom a long time ago. My bet is neither of these will become widely
accepted.

That said, if I had to bet on one of the two becoming accepted, I'd go with "V
= ultimate L". Why? Because mathematics already has an axiom roughly
analogous, that doesn't actually affect normal mathematics but prunes away
sets we don't need so people don't waste their time with them. I'm talking, of
course, about the axiom of foundation. It's like, why is that even in ZFC?
Ordinary math doesn't actually care about the material content of sets; it
really affects nothing. The only thing it does, is that, well, without it,
non-well-founded sets are something of a problem, in that they could exist, or
they could not, and you can't really prove much about them either way. There
are non-well-founded set theories that replace the axiom of foundation with an
axiom of antifoundation, allowing you to actually prove things about non-well-
founded sets -- Peter Aczel wrote a good book on the subject -- but if you
just take it out and don't put anything in to replace it, then, well, you just
can't really say anything about the matter. The axiom of foundation prunes
these irrelevant sets away and tells you to focus on actual math rather than
this mystery that's not really much of a mystery at all. So I'd bet on another
axiom along those lines becoming accepted, rather than one that does the
opposite. After all, if we want to take the opposite perspective that anything
that can exist ought to... well, then why not non-well-founded sets, too?

(Note: Not a logician, my understanding of the two axioms being discussed is
based pretty much entirely on this article, so the analogy I claim may not
actually work as well as I think it does.)

~~~
KGIII
They speak of the Philsophy of Mathematics. I typed a lot out and chose to
delete it. In short, it is considered the highest order among Mathematicians.
However, us Applied Mathematicians just get on with the real work and have
pretty much figured out that they are insane.

It's a bit like true random and infinity. It's best not to think about it to
much and only useful for post doc bar conversations or impressing stoned Econ
undergrads of your preferred gender. Even then, it's best not to think about
it too much.

~~~
Sniffnoy
> They speak of the Philsophy of Mathematics. In short, it is considered the
> highest order among Mathematicians.

Speaking as a pure mathematician: It's really not.

When I say this stuff is pointless, that it doesn't affect ordinary
mathematics, I'm not just talking applied math; I'm talking, well, all of
ordinary mathematics, pretty much. Number theory, combinatorics... every now
and then a question comes up whose answer depends on the continuum hypothesis
admittedly but it's _seriously_ rare (and on those few occasions when it has
happened, I think often someone comes along and says that this means that you
were really asking the wrong question in the first palce).

I mean, OK, there are people such as Harvey Friedman who work out arithmetic
consequences of such assumptions. But the statements they come up with are
_seriously_ contrived, not the sort of thing ordinary mathematicians care
about, even if the statements no longer involve infinities.

Hell I work with infinities sometimes and I don't care about this stuff! Of
course, I'm not a proper logician or set theorist. But doing computations with
ordinals and well partial orders is fun... and the answers you get never end
up depending on this sort of stuff. :P

~~~
KGIII
Well, yes. They are insane.

Infinity is a rubbish idea, by the way. I've given it much thought and
concluded it is best to just not think about it. Cantor was a lunatic, and I'm
not entirely certain that infinity isn't a causal factor.

Don't get me started. It doesn't end well. ;-)

Eventually I go off on the absurdity of 'true random' and our presumption to
know the differences between it and unpredictability. It's hubris and madness,
all the way down.

They really should stop teaching mathematics by rote. If you're familiar with
my posting history, I very seldom say 'should.'

------
NotSammyHagar
This feels like the mathematical equivalent of string theory, an untestable
and apparently unprovable 'set' of ideas has shown up and captured the minds
of mathematicians. The impact of these questions is that it seems to block
progress?

~~~
rocqua
The reason to pay attention to this stuff is that it might (just might) affect
other conjectures in mathematics.

A very clear example here is the continuum hypothesis. It depends on which
axiom you choose whether or not the hypothesis holds. If other hypotheses
(e.g. Goldbach, Riemann) turn out to have different results in differing axiom
systems, that tells us something about those conjectures.

Specifically, it tells us that maybe the conjectures might not be that
important.

~~~
openasocket
> If other hypotheses (e.g. Goldbach, Riemann) turn out to have different
> results in differing axiom systems, that tells us something about those
> conjectures.

To be clear, you shouldn't (can't ?) have something like that Riemann is true
under ZFC and false under ZFC + Continuity, because that would imply that ZFC
+ Continuity is inconsistent. You could have that Riemann is true under ZFC +
Continuity but unprovable under ZFC, though.

~~~
rocqua
Indeed, if we presume ZFC + Continuity to be consistent, then your situation
cannot happen. However, it could be that ZFC + Continuity has Goldbach true,
and ZFC + Forcing has goldbach false.

I should say I recall reading that if the Riemand hypothesis is false, then it
is provably so in ZFC, hence my inclusion of the Goldbach conjecture. [1, 2]

[1] [https://mathoverflow.net/questions/79685/can-the-riemann-
hyp...](https://mathoverflow.net/questions/79685/can-the-riemann-hypothesis-
be-undecidable)

[2]
[https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numbe...](https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numberphile)

