I studied math for a long time.
I’m convinced math would be better without infinity. It doesn’t exist. I also think we don’t need numbers too big . But we can leave those
I haven't studied math beyond what was needed for my engineering courses.
However, I also am starting to believe that infinity doesn't exist.
Or more specifically, I want to argue that infinity is not a number, it is a process. When you say {1, 2, 3, ... } the "..." represents a process of extending the set without a halting condition.
There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further.
There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist.
The statement and proof of the theorem are quite accessible and eye-opening. I think the number line with ordinals is way cooler than the one without them.
I went down the rabbithole, and as far as I can tell, you have to axiomatically assume infinities are real in order to prove Goodstein’s theorem.
I challenge the existence of ordinal numbers in the first place. I’m calling into question the axioms that conjure up these ordinal numbers out of (what I consider sketchy) logic.
But it was a really fun rabbithole to get into, and I do appreciate the elegance of the Goodstein’s theorem proof. It was a little mind bending.
yes, if you want ordinal numbers in ZFC you need to take the axiom of infinity. Other than that it's a pretty straightforward construction. If you reject the axiom of infinity you also essentially reject all of standard analysis (using limits to study reals often implicitly invokes the axiom of infinity).
Whether you think infinity exists or not is up to you, but transfinite mathematics is very useful, it's used to prove theorems like Goodstein's sequence in a surprisingly elegant way. This sequence doesn't really have anything to do with infinity as first glance.
Actually, all numbers are functions in Peano arithmetic. :)
For example, S(0) is 1, S(S(0)) is 2, S(S(S(0))) is 3, and so on.
There is no end of a number line. There are lines, and line segments. Only line segments are finite.
> There is no infinity'th prime number. There is a process by which you can show that a bigger primer number must always exist.
You misunderstand the concept of infinity. Cantor's diagonal argument proves that such a bigger number must always exist. "Infinity'th" is not a place in a number line; Infinity is a set that may be countable or uncountable, depending on what kind of infinity you're working with.
There are infinities with higher cardinality than others. Infinity relates to set theory, and if you try to simply imagine it as a "position" in a line of real numbers, you'll understandably have an inconsistent mental model.
I highly recommend checking out Cantor's diagonal argument. Mathematicians didn't invent infinity as a curiosity; it solves real problems and implies real constraints. https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
>You misunderstand the concept of infinity. Cantor's diagonal argument proves that such a bigger number must always exist. "Infinity'th" is not a place in a number line; Infinity is a set that may be countable or uncountable, depending on what kind of infinity you're working with.
Diagonal argument doesn’t work in a constructive ground. It’s not a matter of whether the conclusion is valid, but if we have blind faith in the premises and are fine about speaking of something we can’t build.
They are things that humans will never be able to construct, no matter how far their control over the universe surrounding them might go. To start with, humans can create the universe, — whether it’s infinite or not.
We can construct a function G, which, given a function
f : N -> N -> {0,1}
returns a function
G(f) = h : N -> {0,1}
defined by h(i):=not(f(i)(i))
This h=G(f) has the property that, for all i, there exists a j such that f(i)(j)≠h(j) . In particular, j=i will work for this.
It seems to me that this is all constructive.
The only out that I see is to not consider the class of “functions from N to {0,1}” to be something that exists (as a set, or type, or whatever).
Like, you can fairly reasonably hold the position that there is no powerset of the natural numbers, but you can’t reasonably hold the position that it exists and that there is a surjection from the natural numbers to it. (Likewise with any other set N. This isn’t specific to the natural numbers.)
We have a constructive refutation of that claim, in the sense that we have a construction of a function which, given such a surjection (as in, a function along with a promise that the provided function is such a surjection), produces a contradiction.
You're right, I was way too imprecise in my language. Thanks for the correction. I'd rather say that all numbers can be interpreted as being constructed via a process, just as the parent comment described their interpretation of Infinity.
Let’s keep it simple. What physics or engineering is easier? Let us ignore mathematics for its own sake . If you can’t show use there … id argue it’s our math aesthetics that are wrong
Why have you picked that comment of mine to reply with that reply? I was just correcting an error. Your reply seems irrelevant to my comment.
I was just saying that “one more than 0” isn’t a function just because in Peano arithmetic, the successor function, along with the constant 0, is used to denote natural numbers.
Numbers don't exist, either. Not really. Any 'one' thing you can point to, it's not just one thing in reality. There's always something leaking, something fuzzy, something not accounted for in one. One universe, even? We can't be sure other universes aren't leaking into our one universe. One planet? What about all the meteorites banging into the one, so it's not one. So, numbers don't exist in the real world, any more than infinity does. Mathematics is thus proved to be nothing but a figment of the human imagination. And no, the frequency a supercooled isolated atom vibrates at in an atomic clock isn't a number either, there's always more bits to add to that number, always an error bar on the measurement, no matter how small. Numbers aren't real.
Why is it that when I have a stack of business cards, each with a picture of a different finger on my left hand, then when I arrange them in a grid, there’s only one way to do it,
but when I instead have each have a picture of either a different finger from either of my left or right hand, there is now two different arrangements of the cards in a grid?
I claim the reason is that 5 is prime, while 10 is composite (10 = 5 times 2).
You’re abstracting to connect the math: 2, 5, 10, multiplication, and primality are all abstract concepts that don’t exist.
What you’ve pointed out is that the interactions of your cards, when confined to a particular set of manipulations and placements, is equivalent to a certain abstract model.
You've already assumed 5 exists in order to assert that it's prime.
In any case existence of mathematical objects is a different meaning of existence to physical objects. We can say a mathematical object exists just by defining it, as long as it doesn't lead to contradiction.
I think your closing paragraph holds the key. 5 doesn't really exist, it's a constructor that parameterizes over something that does exist, eg. you never have "5", you have "5(something)". Saying 5 is prime is then saying that "for all x, 5(x) has the same structural properties as all other primes".
With numbers, I can give an explanation for the phenomenon I described above. If such reasoning cannot be done without reference to numbers, then, if such reasoning is correct, numbers must exist. If there is no other reasoning can be given that provides a good explanation, and as the explanation I gave for the phenomenon is compelling, then I think that a good reason to conclude that the reasoning is correct, and that therefore those particular numbers exist.
In particular, I would expect that if numbers don’t exist, the explanation I gave of the phenomenon I described, couldn’t be correct.
You could say they exist as concepts, that are necessary to use for some reasoning processes, without having any kind of independent existence.
It's similar for the case of programs or algorithms. We can say that a sorting algorithm exists, or a chess-playing program or whatever, which means we know how to implement the logical process in some physical system, but it doesn't mean that they have some kind of existence which is independent of the physical systems. It's just a way of talking about patterns that can be common to many physical systems
My view is that something exists iff there is any statement that is true of it.
I of course don’t mean that mathematical objects (such as the number 2, or some sorting algorithm) have the same kind of existence as my bed. To make the distinction, I would say that my bed “physically exists”.
That sounds like the same circularity, since you'll have to assume numbers exist before proving any statements about them.
Physical objects aren't like that because you can discover that they exist by empirical investigation.
In mathematics the discoveries are about the logical implications of sets of axioms. Some of those axioms contain assertions of existence, like a number 0 in Peano arithmetic or the empty set in set theory, and then you can prove statements about these objects based on the axioms. It's circular to infer from these conclusions that the axioms are true.
What's interesting is why certain axiom systems are so useful and fruitful. Personally I think it's because they evolved that way from our investigations of the physical world, but that's another matter
Wouldn't it follow that those "things" we're pointing to aren't really "things" because they're all leaking and fuzzy? Begging the question, what ends up on a list of things that do exist?
Reality does, stuff happens, etc. But physics is an abstract model we use to make predictions about reality — and triangles are part of that abstract model, not things that actually exist.
You can’t, for instance, show me a triangle. Just objects that are approximated by the abstract concept in physics.
The hierarchy doesn't seem to work. The extremely weak conjecture is the Archimedean property, but I don't see how you prove it from "every number above 7 is the sum of two other numbers". That can be true even if there are only five numbers above 7.
Or in the other direction, if numbers stop at 3, that certainly won't falsify "every number above 7 is the sum of two other numbers". It will prove that it's true. And the extremely strong conjecture immediately proves that the extremely weak one is false.
It turns out rather ok without actual infinity, by limiting oneself to potential infinity. Think a Turing Machine where every time it reaches the end of its tape, an operator ("tape ape") will come and put in another reel.
> I’m convinced math would be better without infinity. It doesn’t exist.
I agree finitism and ultrafinitism are worth more development. Infinity is a hack in the sense that it effectively represents a program that doesn't halt, and when you start thinking of it this way, then all sorts of classical mathematical arguments start looking a little fishy, at least when applying math to the real world. For instance, I do think this is at the heart of some confusions and difficulties in physics.
Math isn't concerned with what exists or not, circles doesn't exist in reality either. And infinity is useful, if we didn't have it we would need to reinvent it in a clumsier form, like "the limit if x goes to a very, very big number, bigger than anything you could name".
Infinity is simply the name for a process that does not end. Some of these processes increase faster than others (hence larger infinities). Personally, I also like the Riemann sphere model of infinity.
I don't think this is a good description of mathematicians using infinite objects. It typically doesn't involving something being given the label "infinity". That's used in like, taking limits and such, but there it is just a notation.
When mathematicians are working with infinite objects, it is not by plugging in "infinity" somewhere a number should go, in order to imagine that the rule that would construct an object if that were a number, constructs an object. No. Rather, (in a ZF-like foundations) the axiom of infinity assumes that there is a set whose elements are exactly the finite ordinals. (Or, assumes something equivalent to that.) From this, various sets are constructed such that the set is e.g. in bijection with a proper subset of itself (there are a handful of different definitions of a set being "infinite", which under the axiom of choice, are equivalent, but without AoC there are a few different senses of a set being "infinite", which is why I say "e.g.").
In various contexts in mathematics, there are properties relevant to that specific context which correspond to this notion, and which are therefore also given the name "infinite". For example, in the context of von Neumann algebras, a projection is called a "finite projection" if there is no strict subprojection that is Morray-von Neumann equivalent to it. Or, in the context of ordered fields, an element may be called "infinite" if it is greater than every natural number.
Usually, the thing that is said is that some object is "infinite", with "infinite" an adjective, not saying that some object "is infinity" with "infinity" a noun. One exception I can think of is in the context of the Surreal Numbers, in which the Gap between finite Surreal Numbers and (positive) infinite Surreal Numbers, is given the name "infinity". But usually objects are not given the name "infinity", except as like, a label for an index, but this is just a label.
I suppose that in the Riemann sphere, and other one-point compactifications, one calls the added point "the point at infinity". But, this kind of construction isn't more "make believe" than other things; one can do the same "add in a point at infinity" for finite fields, as in, one can take the projective line for a finite field, which adds a point that is doing the same thing as the "point at infinity" in the complex projective line (i.e. the Riemann sphere).
Math is a tool for structured reasoning. We often want to reason about things that physically exist. Why should we use a tool that assumes the existence of something that literally cannot physically exist? That introduces the danger that of admitting all sorts of unphysical possibilities into our theories.
I think Baez's paper, Struggles with the Continuum, shows a lot of past difficulties we've had that resulted from this:
Is this a joke or are you deeply interested in some ZFC variant that im unaware of? We absolutely need infinity to make a ton of everyday tools work, its like saying we dont need negative numbers because those dont exist either.
A ZFC variant without infinity is basically just PA. (Because you can encode finite sets as natural numbers.) Which in practice is plenty enough to do a whole lot of interesting mathematics. OTOH by the same token, the axiom of infinity is genuinely of interest even in pure finitary terms, because it may provide much simpler proofs of at least some statements that can then be asserted to also be valid in a finitary context due to known conservation results.
In a way, the axiom of infinity seems to behave much like other axioms that assert the existence of even larger mathematical "universes": it's worth being aware of what parts of a mathematical development are inherently dependent on it as an assumption, which is ultimately a question of so-called reverse mathematics.
There’s tons of variants of ZFC without the “infinity”. Constructivism has a long and deep history in mathematics and it’s probably going to become dominant in the future.
