Does infinity exist?

Something has always existed, but I wouldn’t call that something “existence” since I define existence as a relationship between subject and object. How can something be said to exist if it doesn’t exist: in something, as a function of something, in relation to something, etc? If we talk about objective existence where the object exists only in relation to actuality, then the standpoint of actuality becomes the subject.

Like James said, that which has no affect, does not exist. So if the grand totality of everything has no affect on anything else (because there is nothing else), then we can’t talk about it as existing.

Where does the absolute exist? There is no “where”. Location is only defined inside the absolute.
What does it exist in? There is no “what” because all “what” is inside the absolute.

There is no way to talk about the absolute and any concept we think we have of it simply has to be wrong.

I think it means that time, like location, is only defined inside the thing we’re trying to measure. Time is an emergent property and a consequence of relationships between moving bodies. For instance, I could drive to the next town in 1/24 revolution of the earth… or so many billion vibrations of a certain atom. Time itself does not exist and eternity is not infinite time, but absence of time.

Well, let’s talk about squared circles. Or maybe we can discuss what the universe looks like from the outside even though there is no such thing as “look” outside the universe. This goes to show that we can talk about things without conceptualizing them. We can discuss nonsense without having a concept of the nonsense. I once had a dream where a cat had its head in its mouth. Likewise with infinity: there has never been a person who could properly conceptualize what he fools himself into believing he has. All we can do is imagine the biggest thing we can and we call that “good enough, close enough” and pretend we’ve conceptualized infinity, but we’ve only approached it and our approach is still infinitely far away.

I don’t understand. Can you unpack that a little? The way I understand it is physicists generally regard time before the big bang as north of the north pole: there is no such thing as before the big bang. Time cannot exist before there is something in existence that is moving in relation to something else that is either not moving or moving with a different velocity.

Time is internal to the universe, subjective to it, and not an objective thing existing independent of the universe which could preside over the formation of the universe and record what happened before. Whatever happened before, has no affect on anything, so it doesn’t exist :wink:

Nothing north of the north pole is only true in relation to the Earth as an isolated body but not so in relation to its position within the Universe
So this is where it fails as an analogy in trying to explain why time did not exist before the Big Bang as it assumes nothing existed before it did

It is not known what did or did not exist before the Big Bang because that is only as far as back as physics can currently go
This is demonstrably not the same as saying it cannot go any further back because the BB is the absolute beginning of time

If the singularity was a space of zero volume and infinite density then nothing could have existed before it including time

This definition is wrong because both zero volume and infinite density cannot exist in actuality
As the former would have no dimension or property and the latter can only exist in finite form

A singularity less absolute in physicality however would allow for time to exist before it as it would not be the totality of all that existed

Also if time did begin at the Big Bang it would mean absolute nothing existed before it but this is actually invalidated by quantum mechanics
As absolute nothing can only exist infinitesimally not infinitely because of the existence of quantum fluctuations which disturb vacuum states

So quantum mechanics absolutely forbids the existence of a singularity as traditionally defined

Also the Big Bang was not the beginning of the Universe as such but only local cosmic expansion

I totally agree. I totally agree. I hope saying it twice will convince you that I mean it.

When infinities arise in physics equations, it doesn’t mean there’s a physical infinity. It means that our physics has broken down. Our equations don’t apply. I totally get that. In fact even our friend Max gets that.

blogs.discovermagazine.com/crux/ … g-physics/

The point I am making is something different. I am pointing out that:

All of our modern theories of physics rely ultimately on highly abstract infinitary mathematics

That doesn’t mean that they necessarily do; only that so far, that’s how the history has worked out. There is at the moment no credible alternative. There are attempts to build physics on constructive foundations (there are infinite objects but they can be constructed by algorithms). But not finitary principles, because to do physics you need the real numbers; and to construct the real numbers we need infinite sets.

I collected some examples of the infinitary math underlying physics. I tried to be brief. Each example could be expanded to a book or the work of a lifetime. I’ll do my best to answer specific questions. As with Fubini I regret that it’s beyond me to explain any of these examples fully and in detail with perfect clarity and without requiring effort on the part of the reader. That’s what TED talks are for. /s

  1. The rigorization of Newton’s calculus culminated with infinitary set theory.

Newton discovered his theory of gravity using calculus, which he invented for that purpose. However, it’s well-known that Newton’s formulation of calculus made no logical sense at all. If (\Delta y) and (\Delta x) are nonzero, then (\frac{\Delta y}{\Delta x}) isn’t the derivative. And if they’re both zero, then the expression makes no mathematical sense! But if we pretend that it does, then we can write down a simple law that explains apples falling to earth and the planets endlessly falling around the sun.

It took another 200 years for mathematicians to develop a rigorous account of calculus from first principles; and those first principles are infinitary set theory. No set theory, no real numbers, no calculus, no gravity.

encyclopediaofmath.org/inde … f_analysis

  1. Einstein’s gneral relativity uses Riemann’s differential geometry.

In the 1840’s Bernhard Riemann developed a general theory of surfaces that could be Euclidean or very far from Euclidean. As long as they were “locally” Euclidean. Like spheres, and torii, and far weirder non-visualizable shapes. Riemann showed how to do calculus on those surfaces. 60 years later, Einstein had these crazy ideas about the nature of the universe, and the mathematician Minkowski saw that Einstein’s ideas made the most mathematical sense in Riemann’s framework. This is all abstract infinitary mathematics.

en.wikipedia.org/wiki/Differential_geometry

en.wikipedia.org/wiki/Introduct … relativity

  1. Fourier series link the physics of heat to the physics of the Internet; via infinite trigonometric series.

In 1807 Joseph Fourier analyzed the mathematics of the distribution of heat through an iron bar. He discovered that any continuous function can be expressed as an infinite trigonometric series, which looks like this:

$$f(x) = \sum_{n=0}^\infty a_n \cos(nx) + \sum_{n=1}^\infty b_n \sin(nx)$$

I only posted that because if you managed to survive high school trigonometry, it’s not that hard to unpack. You’re composing any motion into a sum of periodic sine and cosine waves, one wave for each whole number frequency. And this is an infinite series of real numbers, which we cannot make sense of without using infinitary math.

Fast forward to present time. Fourier series underlie the propagation of digital signals over the Internet. They allow us to converse in this very moment.

en.wikipedia.org/wiki/Fourier_series

  1. Quantum theory is functional analysis.

If you took linear algebra, then functional analysis can be thought of as infinite-dimensional linear algebra combined with calculus. Functional analysis studies spaces whose points are actually functions; so you can apply geometric ideas like length and angle to wild collections of functions. In that sense functional analysis actually generalizes Fourier series.

Quantum mechanics is expressed in the mathematical framework of functional analysis. QM takes place in an infinite-dimensional Hilbert space. To explain Hilbert space requires a deep dive into modern infinitary math. In particular, Hilbert space is complete, meaning that it has no holes in it. It’s like the real numbers and not like the rational numbers.

QM rests on the mathematics of uncountable sets, in an essential way.

ps – There’s our buddy Hilbert again. He did many great things. William Lane Craig misuses and abuses Hilbert’s popularized example of the infinite hotel to make disingenuous points about theology and in particular to argue for the existence of God. That’s what I’ve got against Craig.

  1. Cantor was led to set theory from Fourier series.

In every online overview of Georg Cantor’s magnificent creation of set theory, nobody ever mentions how he came upon his ideas. It’s as if he woke up one day and decided to revolutionize the foundations of math and piss off his teacher and mentor Kronecker. Nothing could be further from the truth.

Cantor was in fact studing Fourier’s trigonometric series! One of the questions of that era was whether a given function could have more than one distinct Fourier series. To investigate this problem, Cantor had to consider the various types of sets of points on which two series could agree; or equivalently, the various sets of points on which a trigonometric series could be zero. He was thereby led to the problem of classifying various infinite sets of real numbers; and that led him to the discovery of transfinite ordinal and cardinal numbers. (Ordinals are about order in the same way that cardinals are about quantity).

In other words, and this is a fact that you probably will not find stated as clearly as I’m stating it here:

If you begin by studying the flow of heat through an iron rod; you will inexorably discover transfinite set theory.

I do not know what that means in the ultimate scheme of things. But I submit that even the most ardent finitist must at least give consideration to this historical reality.

ias.ac.in/article/fulltext/ … /0977-0999

Conclusion

I hope I’ve been able to explain why I completely agree with your point that infinities in physical equations don’t imply the actual existence of infinities. Yet at the same time, I am pointing out that our best THEORIES of physics are invariably founded on highly infinitary math. As to what that means … for my own part, I can’t help but feel that mathematical infinity is telling us something about the world. We just don’t know yet what that is.

Most definitely, and one that I would never personally use. Nothing is “near infinite,” I agree with you about that. Physicists and others use it to mean “really big.”

When physicists talk about infinity they often have NO IDEA what they’re saying in terms of math. Physicists misuse the word infinity terribly; and of all the physicists who do that, the celebrity physicists do it the worst.

You’re reading way too much into words people are using very informally.

I almost don’t feel like this needs saying…

To be accurate about infinite sets, it’s proper to say, “the sequence approaches 2”. Rather than, “the sequence is 2”

If there is no time, then what does “before” mean?

If time had a beginning, then there is no before. If there were a before, then after the before would not be the beginning of time.

Oh I see. But in order to have time, we need things in motion through a spacial construct. How can there be things in motion if the universe is so small?

But time doesn’t exist relative to light, yet there is not nothing.

Yes that makes sense.

That could be true.

That was an excellent post and qualifies as a treasure to be found on this site! :obscene-drinkingcheers:

Thanks for the link and I would have showcased it all on its own had I seen it first :slight_smile:

I see what you mean, but as Max pointed out when describing air as seeming continuous while actually being discrete, it’s easier to model a continuum than a bazillion molecules, each with functional probabilistic movements of their own. Essentially, it’s taking an average and it turns out that it’s pretty accurate.

But what I was saying previously is that we work with the presumed ramifications of infinity, “as if” this or that were infinite, without actually ever using infinity itself. For instance, y = 1/x as x approaches infinity, then y approaches 0, but we don’t actually USE infinity in any calculations, but we extrapolate.

Hilbert pointed out there is a difference between boundless and infinite. For instance space is boundless as far as we can tell, but it isn’t infinite in size and never will be until eternity arrives. Why can’t we use the boundless assumption instead of full-blown infinity?

I didn’t know he developed calculus specifically to investigate gravity. Cool! It does make sense now that you mention it.

I’m going to need some help with this one. If dx = 0, then it contains no information about the change in x, so how can anything result from it? I’ve always taken dx to mean a differential that is smaller than can be discerned, but still able to convey information. It seems to me that calculus couldn’t work if it were based on division by zero, and that if it works, it must not be. What is it I am failing to see? I mean, it’s not an issue of 0/0 making no mathematical sense, it’s a philosophical issue of the nonexistence of significance because there is nothing in zero to be significant.

Isn’t this the same problem as previous? dx=0?

I can’t make sense of it WITH infinitary math lol! What’s the cosine of infinity? What’s the infnite-th ‘a’?

Well, thanks to Hilbert, I’ve already conceded that the boundless is not the same as the infinite and if it were true that QM required infinity, then no machine nor human mind could model it. It simply must be true that open-ended finites are actually employed and underpin QM rather than true infinite spaces.

Like Max said, “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.”

We can claim physics is based on infinity, but I think it’s more accurate to say pretend or fool ourselves into thinking such.

Max continued with, “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.”

He said, “let go of it” like we’re clinging to it for some reason external to what is true. I think the reason is to be rid of god, but that’s my personal opinion. Because if we can’t have infinite time, then there must be a creator and yada yada. So if we cling to infinity, then we don’t need the creator. Hence why Craig quotes Hilbert because his first order of business is to dispel infinity and substitute god.

I applaud your effort, I really do, and I’ve learned a lot of history because of it, but I still cannot concede that infinity underpins anything and I’d be lying if I said I could see it. I’m not being stubborn and feel like I’m walking on eggshells being as amicable and conciliatory as possible in trying not to offend and I’m certainly ready to say “Ooooohhh… I see now”, but I just don’t see it.

Craig is no friend of mine and I was simply listening to a debate on youtube (I often let youtube autoplay like a radio) when I heard him quote Hilbert, so I dug into it and posted what I found. I’m not endorsing Craig lol

I still can’t understand how one infinity can be bigger than another since, to be so, the smaller infinity would need to have limits which would then make it not infinity.

Right, because of what Max said about the continuum model vs the actual discrete. Heat flow is actually IR light flow which is radiation from one molecule to another: a charged particle vibrates and vibrations include accelerations which cause EM radiation that emanates out in all directions; then the EM wave encounters another charged particle which causes vibration and the cycle continues until all the energy is radiated out. It’s a discrete process from molecule to molecule, but is modeled as continuous for simplicity’s sake.

I’ve long taken issue with the 3 modes of heat transmission (conduction, convention, radiation) because there is only radiation. Atoms do not touch, so they can’t conduct, but the van der waals force simply transfers the vibrations more quickly when atoms are sufficiently close. Convection is simply vibrating atoms in linear motion that are radiating IR light. I have many issues with physics and have often described it as more of an art than a science (hence why it’s so difficult). I mean, there are pages and pages on the internet devoted to simply trying to define heat.

quora.com/What-is-heat-1
quora.com/What-is-meant-by-heat
quora.com/What-is-heat-in-physics
quora.com/What-is-the-definition-of-heat
quora.com/What-distinguishes-work-and-heat

Physics is a mess. What gamma rays are, depends who you ask. They could be high-frequency light or any radiation of any frequency that originated from a nucleus. But I’m digressing…

It just means we’re using averages rather than discrete actualities and it’s close enough.

I think it means there are really no separate things and when an aspect of the universe attempts to inspect itself in order to find its fundamentals or universal truths, it will find infinity like a camera looking at its own monitor. Infinity is evidence of the continuity of the singular universe rather than an existing truly boundless thing. Infinity simply means you’re looking at yourself.

Anyway, great post! Please don’t be mad. Everyone here values your presence and are intimidated by your obvious mathematical prowess :sunglasses: Don’t take my pushback too seriously :slight_smile: I’d prefer if we could collaborate as colleagues rather than competing.

Small is a relative term and from a classical perspective the quantum is incredibly so but this does not mean motion is not actually possible at this level
Three of the four fundamental forces operate at the quantum level inside of an atom in relation to the motion of its protons and neutrons and electrons
Also motion is a universal constant so any space no matter how small will feature motion even if it is just quantum fluctuations disturbing vacuum states

This statement violates the Law Of Non Contradiction as it is impossible for an infinity to also be not infinity
The problem here is that in relation to small infinity you are focusing on the small rather than the infinity

All infinite sets contain an infinite number of members but some of them will have more members than others and some will have less
For example the infinite set of primes is smaller than the infinite set of reals because primes occur less frequently on the number line

But primes are no less infinite than the reals despite this [ and yes it is incredibly counter intuitive but more importantly it is also true ]

I think we first need to answer the question of if there is a smallest point in the universe.

If there is a smallest point, then the extents of the universe would have to be bigger than that smallest point in order to have concepts of moving from point to point.

If there is not a smallest point, then the universe has no size relative to its smallest point, and regardless how small the universe became, it would still be infinitely larger than its own points.

The evidence so far suggests that the smaller we go, the less certain we can be of location and the more probabilistic location becomes. I am under the impression that space is not empty space with things moving about independent of the space-substance, but a collection of fields analogous to a computer screen where things cannot be smaller than the pixel (a quantum) that generates them. So if the universe were smaller than a pixel, then how could movement exist?

Also, neither time nor space exists relative to a photon. Every photon emission and reception from the beginning of time to the end of time, took no time and covered no space from its own perspective, which seems to imply there is no objective time standard by which to measure spacetime itself, but that time is an artifact of spacetime and contained only therein. Time is subject to and emergent from spacetime and not objective to it.

It seems obvious that something existed before, but I can’t understand what “before” means. From the photon’s perspective, where was it yesterday? “Yesterday” only applies to me and not the photon which itself caused a sense of “yesterday” to manifest. Time cannot apply to the thing that causes time. It’s like trying to use a knife to cut the same knife.

Primes are infinite only because reals are infinite. If reals were not, then primes could not be, so the cardinality of primes is dependent upon the cardinality of reals. This is like saying the age of a son will always be less than the age of his father, except if they live forever, in which case they will be the same age on the day that forever arrives (which can never happen). It’s the same with primes and reals: there will always be more reals than primes until the day eternity arrives.

If a completed infinity of reals are ALL numbers in actual existence and potential existence and nonexistence and any other existence you can conceive, then from where will you find more numbers to have a larger infinity? And if you do find more numbers to make a larger infinity, then it wasn’t ALL the numbers to start with.

Ordinals assume an infinity of reals can be completed and then some new number comes in order right after that completed infinity. Like Gauss, I can’t make sense of a completed infinity. Further, the first infinity should include all of the second or else it wasn’t infinity.

[youtube]https://www.youtube.com/watch?v=SrU9YDoXE88[/youtube]

This is the point that cannot be made any smaller without it becoming non existent / non physical
However this is not a real point that can actually be measured either physically or mathematically
In mathematical terms this would be 0 . 00000 … extending to infinity with I added on to the end

A number as close to 0 as it is possible to be without it actually being 0 and because this number is infinite then it cannot ever be expressed in finite terms
This is the same problem with the physical for no matter how small you go you can always go smaller without ever reaching the non existent / non physical

There is no such thing. First, if x is a positive real number, then x/2 is a smaller one so x wasn’t the smallest; and since x was an arbitrary positive real, there is no smallest positive real.

Secondly, the meaning of a decimal string is that each digit position represents a negative integer power of 10. So .1234 means 1/10 + 2/100 + 3/1000 + 4/10000, and likewise for infinite decimals. There is no corresponding interpretation of .000…1 where there are infinitely many 0’s preceding the 1. It’s a meaningless expression.

Thank you for the kind words. Mad as in angry? Or mad as in crazy? Haha.

I don’t mean to intimidate anyone or show off in any way. I just try to put things into their proper mathematical context. For example a lot of people have an intuition that there’s a smallest positive real number, but I just gave two demonstrations that this cannot possibly be the case.

Moreover, there is a philosophical reason as well. The real numbers are a continuum. If we visualize them as a string of bowling balls, they’re not a continuum.

Yes definitely.

Just for sake of discussion: this is the kind of thing that makes me mad angry AND mad crazy. I never said that physics is based on mathematical infinity. I said, and I will ask you to go back and verify this for yourself, that all of our admittedly contingent THEORIES of physics are based on mathematical infinity. That is a fact of the world that anyone can verify, and in support of which I gave copious references.

And if you don’t know the difference between physics and THEORIES of physics, it’s this. If you are sitting down right now and gravity disappears, you will float up to the ceiling of the room you’re in. But if our THEORIES of gravity all disappear, you will still be stuck to the earth. The only difference is that you won’t be able to go to school to learn what people think about WHY that is. Do you see the difference?

We are not fooling ourselves that our theories are based on mathematical infinity. That’s a matter of fact, verifiable from anything you can read online or in books.

Are you willing to acknowledge that you profoundly mischaracterized what I said? I’d like to have a congenial conversation too, but that’s not possible when you deliberately claim I said things I did not say. I went to great lengths to point out that I was talking about our THEORIES of physics, and not at all how the world itself may happen to work. Did I not make that clear?

ps – I’m not upset. You don’t have to walk on eggshells. My point is very clear in my own mind but evidently I’m not making my case! I’ll work harder.

Interestingly, not so much. Euclid demonstrated the infinitude of primes two thousand years before we had any concept of the real numbers. And in fact Euclid’s original proof is valid even without any theory of the real numbers today. We do not need to assume the axiom of infinity to prove that there is no largest prime. In other words if there are no infinite sets at all, it’s still true that there’s no largest prime, and Euclid’s proof goes through in its original form.

QM is based on an infinite-dimensional vector space called Hilbert space. It’s infinite-dimensional. Literally and truly, Whether the world itself is, or only our theory is, I can’t say. But QM takes place in infinite-dimensional space and that’s a fact. I gave a link to Hilbert space earlier.

Perhaps that’s more of Leibniz’s view or more of a philosophical view. In calculus, the derivative is the limit of deltaY/deltaX as deltaX goes to zero. What that “means” is a bit elusive but one gets used to it after a while.

Yes, Berkeley made the same point, calling derivatives the “ghosts of departed quantities.” Today we have the theory of limits that makes it all mathematically rigorous; though I agree that it’s perhaps not entirely satisfying philosophically.

The modern theory of limits.

People have been arguing about this since Newton and it’s still going on. There are modern incarnations of infinitesimals.

Hilbert space is infinite-dimensional and that’s what they use to do QM.

I appreciate your kinds words. it is a FACT that all of our theories of physics require actual infinity. Not unboundedness. Actual infinities. Infinities of points in a continuous space in which to do relativity; infinities of wave functions in an infinite-dimensional Hilbert space to do QM. Nobody has a finitary theory of physics. Nobody.

I wonder if you are still conflating physics (how the world is) with theories of physics (our mathematical models). Could that be the case?

Perfectly ok. I’m easily provoked but I get over it quickly. Your philosophical points are valid but the theories themselves require mathematical infinities. Are you saying you don’t believe our theories themselves require mathematical infinity? But they do. I’ve given as many explanations as I can think of.

About the 0/0. Say you’re driving in your car and your speedometer says 50mph. That’s the derivative of your position function. Conceptually it says that if we divide your distance over time for smaller and smaller intervals, the “instantaneous” change in position is 50. But if we could stop time in that one instant, where is the motion? Back to Zeno again. Ancient mystery. And yet your speedometer has no problem measuring your speed.

The map must never be confused for the territory because it is merely an approximation of it and they are therefore not the same

But of course. And I went to great lengths to explain that I am talking about the maps and NOT the territory. Every THEORY of physics is based on mathematical infinity.

You know I wonder if I’m making a point of far greater subtlety that I realize, because now two people are pushing back for (to my mind) no reason.

I agree that I have no idea how the world “really” is, or if the question is even meaningful. But our mathematical theories of physics, the historically contingent theories of Galileo and Kepler and Newton and Lagrange and Maxwell and Einstein and Witten and all these people … these THEORIES are all expressed using infinitary mathematics.

I know nothing of the territory. But the MAPS are based on infinitary mathematics.

I like this analogy even though the continuum or infinite set is made up of infinite individual members like all infinite sets

Well, it’s like the north pole. The north pole is the place where we cannot go any farther north. So the smallest point is the point where it doesn’t make sense to go smaller because there is nowhere to go. Like my pixel analogy.

It’s not a problem of precision, but that there is no there, there.

Physical limits are not like a government insisting we stop, but we cannot go faster than light because the universe would have negative size, which doesn’t make sense. I mean, you could arrive before you left and see yourself off. Size is similar. There can’t be anything smaller than the thing that determines size. Just like there is no standard of temporal measurement applicable to light because light defines time.

Math transcends reality.

[i]When you hear this, you may stop and think, “Surely, if I have a length, then I half it, and I repeat this over and over, I will be able to get to something smaller.” However, this is an occasion where physics doesn’t allow something that mathematics does. For example, think about moving faster than the speed of light. On paper you could apply a force to a mass and accelerate it up and past the speed of light, but we know that in nature that just is not physically possible because the mass of the object (and thus, the energy needed to speed it up) goes towards infinity—both keep growing without any limit. So what we can do on paper, we can’t do in reality.

So, how does a tiny number such as this tie into physics? If two particles were separated by the Planck length, or anything less, then it is impossible to actually tell their positions apart. Moreover, any effects of quantum gravity at this scale (if there are any) are entirely unknown as space itself is not properly defined. In a sense, you could say that, even if we were to develop methods of measurements that took us down to these scales, we would never be able to measure anything smaller despite any sort of improvements to our equipment or methods.[/i] futurism.com/apotd-ngc-1316-2

It dawned on me while talking to James over a year ago that it didn’t makes sense to have an infinity of smaller particles, but putting the reason why into words is hard. Essentially there would be no reference point, no anchor to any kind of reality. If the size of an atom could have been any of the infinity of sizes, then why this one? It’s just too arbitrary on the hierarchy, but if there is a smallest size, then it makes total sense why an atom is the size it is because it’s relative to that smallest size.

Also, if there is an infinite hierarchy of possible sizes, then we need an external standard of measurement to measure what size an atom is because there would be no way to tell how big it would be relative to the universe itself. But if there is a smallest size in the universe, then no such external measurement is required to define the size of an atom (although the size of the smallest point becomes arbitrary relative to an external standard of measurement). Is this making sense? IOW, we need a zero point for an origin in order to have anything.

Infinite computer memory is another analogy. Once something is stored in such memory, how could it be found again?

I see no problem at all. As in an actual computer, each mem location is labeled with a positive integer address: 1, 2, 3, 4, …, and we store and fetch data to and from a given location via its address. How would this change if we simply used all the integers? We store value x at location n; and later, we fetch value x from location n.

In fact a Turing machine is conceptually just like this, and it does not require infinite memory; only unbounded memory. We don’t require that there are an actual infinity of memory cells (or locations on a paper tape, as Turing puts it). Given some positive integer n, the tape has cell n. There are never an actual infinity of cells; only an unbounded array.

You have literally described a TM.

Because infinite memory would be a box with no walls and therefore have no objective reference point for where to begin addressing.

I think we have to be careful not to conflate the unbounded finite with the completed infinite.

If we had infinite cells, which one would we label the zero? Zero has no objective anchor and must be chosen at random. Whichever cell we randomly choose for zero will have infinite cells under and infinite cells above it, so any zero isn’t a true zero, but an arbitrary location to start sequentially ordering. It’s not finite on one end and infinite on the other, but it can only be infinite in both directions because if the idea is to assume that we can always add one, then the zero must itself be a product of that addition, as is every integer.

Half of infinite capacity is absurd to me like a squared circle. Integers extend in both directions, but we cut it in half at zero and pretend we’ve corralled the infinite as if we could somehow hold the end of an infinite rope. We can conceptually cut it in half all we want, but we can’t remove the fact that the negative integers still exist and that zero was merely an arbitrary starting point where we decided to cut it. We could cut it at 5 and say that’s the beginning. Or 437859348 and say that’s the beginning. Or -53478. Or any number. Zero is an arbitrary delineation on an infinite continuum that has neither beginning nor end and is not an objective anchor to actuality, and in actuality we’d be presented with a completed infinite thing and tasked with ordering it rather than sequentially building an infinite thing by adding one forever. Like, if space were infinite, where is the finite edge where all that infinite space begins?

A finite stick of memory can be ordered because it has an edge, but infinite sticks have no edges and starting points must be chosen at random, then when the computer returns for the information, it must choose another random point and it has 1 out of infinity chance of finding the same point as before, so there is no chance it could find the info it deposited. Infinite memory is zero memory: the absence by ubiquitousness.

Let’s say there is infinite time and here we are in the middle of infinite past and infinite future, what is this location in time referenced to? If time is finite with a beginning, then we would just reference this point to the objective beginning, but in infinite time, we could only reference to another arbitrary point that itself is deficient of objective reference. No point in infinite time has distinction from any other. It’s the same for infinite particle sizes.

If particle sizes are arbitrary and there are an infinity of smaller ones, then obviously there are infinite smaller universes just like this one, where you and I are having the same conversation. And if time were infinite, then we’ve already had this conversation an infinite number of times and are bound to keep repeating it forever. Anything that has any probability of happening, must happen infinite times in infinite time. Since we know for sure that this universe happened, then it will happen infinite times on infinite particle-size levels. That’s a little bit ridiculous for me lol. It’s easier for me to believe the answer is deeper and we haven’t found it yet than to be satisfied with infinity as a solution with all its absurd ramifications.

Unbounded memory will always be finite memory that hasn’t yet found a bound (like the money supply is theoretically unbounded but always finite). Infinite memory is the impossibility of a bound and yet in some sort of completed form (a wall-less box).