# Why do finitists reject the axiom of infinity? [closed]

The axiom of infinity implies that there exist infinite sets. We can construct the natural numbers without this axiom, but we cannot put them together in a set, as this would violate this axiom.

The only reason I can think off why you would reject the axiom is that it looks like we can count everything: the amount of atoms on our planet, the age of the universe, ...

So my question:

What is (are) the motivation(s) of someone to reject the axiom of infinity?

• They could hardly be finitists if they believed in infinity! – Angina Seng Apr 28 '18 at 16:47
• Have you read MSE question 1989695 "Why isn't finitism nonsense?" – Somos Apr 28 '18 at 18:24
• Can you count the number of atoms on our planet? How many are there? – cody Jun 7 '18 at 21:30
• I can't count them, because it takes too long. But I can estimate how many there would be. – user370967 Jun 7 '18 at 21:32
• I'm curious as to the deletion vote. This question is too broad, but it's not (in my opinion) much too broad and it has attracted decent answers; I don't think it should be deleted. (Admittedly I'm biased - one of those "decent answers" is mine - but I do think that this question shouldn't be deleted.) – Noah Schweber Dec 11 '18 at 15:56

I occasionally am in a finitistic mood; while I have no reason to believe that the following is applicable to all, or even most, finitists, it might be worth writing down.

Note that in everything below, I'm perfectly comfortable with classical logic. Personally I actually have much less foundational interest in nonclassical logic than in finitistic frameworks, and the two are for me largely unrelated. Of course, $(i)$ this is not to say that I have no mathematical interest in nonclassical logic, and $(ii)$ this is just my opinion.

Let's start at the far opposite end from infinitism: ultrafinitism. There are various reasons one might find ultrafinitism attractive, but I think the most common is the belief that numbers must have physical meaning to be meaningful. Let's call this the "physicalist" view. Of course, pinning down what "physical meaning" means precisely is hard, but there are some things I think we can say with reasonable certainty. For example:

Based on our current understanding of physics, Graham's "number" does not in fact have any phyiscal meaning.

However, this picture has certain features which at least to me make it unsatisfactory for capturing mathematical meaning. At an immediate level, there's no reason to believe that "number" today means the same thing as "number" tomorrow: the universe does seem to be expanding, after all, and arguably this means that more "numbers" are becoming actual numbers since tomorrow the state of the universe will take more bits to describe naively than it does today. Please ignore my complete lack of actual understanding of physics here; I'm just trying to make the broader point that the fact that the universe changes over time poses a problem for the idea that "number" is a well-defined concept in the physicalist view.

A more serious problem to my mind is that this definition ironically means that we have no actual direct access to numbers! This is because I can never be absolutely certain that my understanding of the physical world is accurate, and I'm not just a brain in a vat. Maybe the true physical world is much much bigger than, and behaves quite differently from, my guess at reality. Conversely, am I really certain that $1000000$ is meaningful? The universe might be very very small . . .

The version of physicalism that has some appeal to me is a sort of "relative physicalism" (my reverse math bias might be showing here):

Except in a small handful of cases, it's outright impossible for me to say with certainty that a mathematical concept has physical meaning (even if I'm confident what "physical meaning" is!). However, there are meaningful dividing lines between mathematical concepts based on what would have to be true of the universe in order for those concepts to have physical meaning.

For example, it's completely plausible to me that Graham's number has physical meaning. All that would take would be for our understanding of physics to be wildly off-base, or for me to be the victim of the most boring conspiracy in history - and relatively speaking, these are totally plausible. By contrast, something would have to be meaningfully weird in order for $\aleph_0$ to have direct physical meaning (personally, my interpretation of "physical meaning" is strict enough that an "infinite, locally finite" universe wouldn't necessarily cut it). Intuitively, if we start from a position of severe skepticism about the universe (which I wouldn't take in day-to-day life, but I think is reasonable in foundations of mathematics), then some dividing lines are "a priori significant" while others aren't (I don't see any particularly compelling reason why Graham's number should be substantially more implausible than one million, but I definitely do see a fundamental qualitative difference between "finite" and "infinite").

At this point I might allow a drop of dogma into my skepticism, and declare that some things are just too implausible to have physical meaning at all. E.g. I might claim:

I wouldn't be too surprised if Graham's number had physical meaning, but I truly can't imagine a universe in which $\aleph_0$ has physical meaning.

(Incidentally, for me a similar-feeling claim is: "I wouldn't be too surprised if ZFC were inconsistent, but I truly can't imagine an inconsistency in PA.")

I personally would never go that far, but I can definitely get behind:

I would be fundamentally more surprised by $\aleph_0$ having physical meaning than I would by Graham's number having such physical meaning.

I actually find this "degree-of-surprise/shock/distress" line of thought quite compelling. For me, finitism often captures "Mathematics which could be physically meaningful without spiking my blood pressure." I asked my doctor, and after rolling his eyes he told me I shouldn't worry about this, so I almost never do; but then I can't be completely certain that he's not part of The Conspiracy, so I do worry from time to time.

One feature of this perspective is that finitism isn't particularly privileged: "countable" is less surprising to me than "uncountable," so "countabilism" is a meaningful-to-me stance; ditto "not-supercompact-ism," and so on. On the other hand, finitism is the most restrictive position of this type, so it is distinguished in that sense.

• The simplest view in which Graham's number can be "physically meaningless" while $1000000$ is perfectly physically meaningful is where you want to be able to perform arithmetic operations on your numbers. Certainly you can do so on all naturals less than $2^{64}$ (your computer is doing it for you every day), but you cannot even store (in a uniform encoding), much less perform arithmetic on, naturals above $2^{2^{10000}}$, as I mentioned here. =) – user21820 Nov 17 '18 at 8:19

As with many questions of belief there can be several reasons to reject the axiom of infinity and there are several variations of Finitism. There is some connection with Constructivism in mathematics as mentioned in this Mathoverflow question (Is finitism an extreme form of constructivism?). Ultrafinitism or ultra-intuitionism of A.S. Yessenin-Volpin is just one extreme case.

Ulimately, it is a choice of what to believe and, more importantly, why. Mathematical proof, no matter how rigorous, is not a sufficient reason to believe something because it is always based on certain given assumptions (axioms) which are accepted as true implictly and logical rules using them. There can be alternate axioms and logical rules. The issues here have a long complicated history.

In this forum the most fundamental question is "What is Mathematics, Really?"

From the point of view of an 'infinitist', I believe the infinitist take on their rationale is basically:

• They are interested in synthetically studying a universe of finite sets
• They want to take a non-classical approach to (higher order) logic

It's worth noting that a form of finite set theory is basically the same thing as peano arithmetic (PA). It's well known how to get natural numbers from sets; the reverse can be done with bitsets — that is, we say $m \in n$ if and only if the binary numeral for $n$ has a $1$ in the $m$-th place.

Finitists usually have a fairly computational bent on philosophy, so the fact that finite set theory and peano arithmetic are equivalent is probably very influential in finitism's preference for finite sets.

I emphasize the non-classical approach to higher order logic, because set theory is higher order logic. IIRC, bounded zermelo set theory is exactly equivalent to what you'd get by developing a classical higher order theory of peano's axioms.

Nonetheless, finitists still reject it. In my observations, they seem to be aiming for a higher order logic based on computability theory; e.g. on what "computation" can be done with Turing machines.

So to summarize, from an infinitist point of view, my current belief is:

Finitists are working internally in a universe of Turing machines.