I have trouble with the mathematical notion of infinity.
Example: Consider all of the natural numbers. It has a beginning, therefore it is bordered, therefore it cannot be infinity.
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If your definition of an "infinity" is "an ordered set that has neither a maximum or a minimum", then the natural numbers would indeed not be an "infinity".
However, it is also true that the natural numbers are an "infinite set" (or stated in more detail, "a set whose cardinality is infinite").
There is no contradiction here, because "Klimuk infinity" has very little to do with "infinite set".
My general advice regarding matters of the infinite is to ignore any conclusions on the topic that do not come from mathematical contexts, or otherwise heavily draw from the mathematical notion.
It took thousands of years for people to realize that there were a whole multitude of different ideas, notions, and objects that had previously all been called "infinity", so most history on the topic consists of all of these different things jumbled together into a confused mess. And even when someone could coherently discuss one specific idea they were calling "infinity", there is no guarantee it has any bearing on what someone else was calling "infinity".
Without really knowing much about what you're thinking, the idea you specifically have in mind is probably best captured in the notion of a "compact topological space".
Consider all of the natural numbers. It has a beginning, therefore it is bordered, therefore it cannot be infinity.
Nope. It's not really (for some definition of "really"), as you say, "bordered". Perhaps you're already familiar with the trivial demonstration that the even numbers are equinumerous with all numbers: just take 2,4,6,8,... and divide each number by 2, whereby you get 1,2,3,4,... So, you already see where I'm going with this???...
...It's pretty much equally easy to map 1,2,3,4,... into ...-4,-3,-2,-1,0,1,2,3,4,... as it was to map 2,4,6,8,... into 1,2,3,4,... For any natural number n, just map it to n/2 for even n, as above, and to -(n-1)/2 for odd n.
So, 1,2,3,4,... are fundamentally just syntactic symbols. Interpreted one way, the usual way, they're the natural numbers. But interpreted our way above, 1,2,3,4... are just a different set of labels/symbols for the entire set of integers, which aren't "bordered".
Re @FrankHubeny's comment, implicit in the above answer is a slightly more mathematical elaboration that I circumvented, thinking the details aren't of much interest here. (But I'd also have thought it would've been immediately obvious to Frank, with an ms in math and professional programming background, as per his profile.)
The underlying issue is the syntax-vs-semantics distinction I briefly alluded to. It's not 1,2,3,4,... which is bounded/"bordered", per se, but rather the poset ordering, in this case a total order, 1<2<3<4<... that lets you arrange them in a way that imposes a (more-or-less artificial) "beginning" on an otherwise unordered set. And, what's important here is that the cardinality of that set is independent of any imposed (partial or total) order.
Without the natural-number ordering, 1,2,3,4,... are just a collection of strings, "1","2","3","4",...,"998","999",... Programmers frequently generate such "collections" using Backus-Naur form (bnf), a notation for context-free grammars. Indeed, without the natural-number semantics/order imposed on them, our "1","2","3","4",... strings are just a context-free grammar typically called "numerals", and generated by bnf as illustrated at https://courses.cs.vt.edu/~cs1104/Compilers/Compilers.100.html (see the example towards the bottom of that page). Actually, the lecturer there fails to account for strings like "0123","000123",etc with any number of leading zeros. The usual textbook bnf exercise asks for a grammar that excludes them (which I'll also leave as an "exercise for the reader":).
So the upshot is that, as a collection of elements/objects, the natural numbers 1,2,3,4,... are identical/isomorphic to the strings "1","2","3","4",... And then "1" can't really be called the "first string" (although there's yet another "artificial" ordering called the lexicographic order, essentially alphabetic, where it would be first). What you've ultimately got is a very,very,very... big bucket of strings, with no beginning,middle,end; just a big jumbled-up collection. And you know that collection is countably infinite by an analysis of the underlying grammar that generates it. Then, if you want to impose a semantics and order on that collection, it's your business. But it won't affect the cardinality.
Well, let us look at the concept of ‘infinity’ from a physicist point of view.
There exist two separate assumptions: “infinitely big” and “infinitely small.” By infinitely big, one means that space can have infinite volume, that time can continue forever, and that there can be infinitely many physical objects.
By infinitely small, it represents the continuum—the idea that even a liter of space contains an infinite number of points, that space can be stretched out indefinitely without anything bad happening, and that there are quantities in nature that can vary continuously.
The two assumptions are closely related, because inflation, the most popular explanation of our Big Bang, can create an infinite volume by stretching continuous space indefinitely.
The theory of inflation explains how a subatomic speck of matter transformed into a massive Big Bang, creating a huge, flat, uniform universe, with tiny density fluctuations that eventually grew into today’s galaxies and cosmic large-scale structure—all in beautiful agreement with precision measurements from experiments such as the Planck and the BICEP2 experiments.
But by predicting that space isn’t just big but truly infinite, inflation has also brought about the so-called measure problem, which I view as the greatest crisis facing modern physics.
The problem is that whatever experiment you make, inflation predicts there will be infinitely many copies of you, far away in our infinite space, obtaining each physically possible outcome; and despite years of teeth-grinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So, strictly speaking, we physicists can no longer predict anything at all!
This means that today’s best theories need a major shakeup by retiring an incorrect assumption. Which one? Here’s my prime suspect: ∞. Infinity Doesn’t Exist
A rubber band can’t be stretched indefinitely, because although it seems smooth and continuous, that’s merely a convenient approximation. It’s really made of atoms, and if you stretch it too far, it snaps. If we similarly retire the idea that space itself is an infinitely stretchy continuum, then a big snap of sorts stops inflation from producing an infinitely big space and the measure problem goes away.
Without the infinitely small, inflation can’t make the infinitely big, so you get rid of both infinities in one fell swoop—together with many other problems plaguing modern physics, such as infinitely dense black-hole singularities and infinities popping up when we try to quantize gravity.
In the past, many venerable mathematicians were skeptical of infinity and the continuum. The legendary Carl Friedrich Gauss denied that anything infinite really exists, saying “Infinity is merely a way of speaking” and “I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.”
In the past century, however, infinity has become mathematically mainstream, and most physicists and mathematicians have become so enamored with infinity that they rarely question it. Why? Basically, because infinity is an extremely convenient approximation for which we haven’t discovered convenient alternatives.
Consider, for example, the air in front of you. Keeping track of the positions and speeds of octillions of atoms would be hopelessly complicated. But if you ignore the fact that air is made of atoms and instead approximate it as a continuum—a smooth substance that has a density, pressure, and velocity at each point—you’ll find that this idealized air obeys a beautifully simple equation explaining almost everything we care about: how to build airplanes, how we hear them with sound waves, how to make weather forecasts, and so forth. Yet despite all that convenience, the air, of course, isn’t truly continuous. I think it’s the same way for space, time, and all the other building blocks of our physical world. We Don’t Need the Infinite
Let’s face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small. We speak of infinite volumes with infinitely many planets, but our observable universe contains only about 1089 objects (mostly photons).
If space is a true continuum, then to describe even something as simple as the distance between two points requires an infinite amount of information, specified by a number with infinitely many decimal places. In practice, we physicists have never managed to measure anything to more than about seventeen decimal places. Yet real numbers, with their infinitely many decimals, have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics. We describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.
Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.
Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.
Infinity cannot have a beginning because infinity cannot exist in the material world. A common argument for the existence of infinity is the fact that infinitely many points can exist on a line segment, thus infinity has an existence. However, say there was a line segment like this:
Infinitely many points can exist on this line segment. Additionally, imagine another line segment of equal length, with an additional point marked on it.
From the second line, segment CB is removed. The segment retains the same number of points, because an infinite number of points can exist on any line segment. However, the same goes for segment CB, thus an infinite number of points are removed. The result is this:
AB=∞ points AC=∞ points CB=∞ points
We can then substitute into the first equation:
∞points = ∞points + ∞points
Divide both sides by the unit:
∞ = ∞+∞
Divide both side by infinity:
1 = 1+1 1 = 2
A clear paradox is developed.
Similar logical progressions can be used to disprove all other types of infinity as well, including infinite sets. For your example, the set of natural numbers, a one-to-one mapping can be established with this set and another of the same cardinality, such as the negative numbers. In this mapping, 1 is mapped to -1, 2 is mapped to -2, 3 is mapped to -3, and so on.
Now imagine that ten elements are removed from the set of negative numbers, specifically the numbers -1, -2, -3, -4, -5, -6, -7, -8, -9, and -10. However, because the sets are infinite, a mapping should still be possible. Thus we begin our mapping again by mapping 1 to -11, 2 to -12, 3 to -13, and so on. By the time we reach infinity, ∞ should be mapped to -∞-10, which is clearly impossible because infinity cannot be added too without reaching further paradoxes.
Both of those examples deal with infinities bounded in certain ways, but are the same issues still extent in unbounded infinities, such as a line? They are in fact. Imagine a line, stretching endlessly in opposite directions.
Now imagine a 1 foot long segment is removed from somewhere in the middle of it, and the points from which the segment was cut are drawn together. The line is still infinite, but it lost some of its length.
∞feet-1foot = ∞feet
Divide both sides by the unit and subtract infinity from both sides and you get:
-1 = 0
For these reasons, infinity cannot exist in the world governed by math. I do want to point out that this is not the view shared by the majority of mathematicians, and if someone sees an error in my thinking, please point it out in the comments.