There are an infinite number of steps but we cannot complete them. The fact that we cannot complete the steps does not mean that they do not exist.

Consider a set of points. We say that the set is continuous if there is a point between any arbitrary pair of points. We say that the set is discrete if there is a minimal distance between a pair of points. In other words, there is no point between such a pair. The example of a continuum is the real numbers and the example of a discrete is the natural numbers (the minimal distance between a pair of points is 1).

OK, so we risk introducing error if we treat spacetime as discrete, — Relativist

Yes, if the spacetime is continuous and we treat it as discrete then we are introducing error.

but if it IS discrete, we introduce no errors by treating it (mathematically) as continuous. — Relativist

If spacetime is discrete we introduce error by treating it as continuous. We however might not be able to observe the error if our measurement devices are not precise enough.

So treat it as continuous and use the math. Problem solved, right? — Relativist

Yes, we can use a continuous model as far as our measurement devices are not precise enough. Otherwise, we have to use a discrete model.

Why do we have to deal with Zeno's paradox? Is there some problem in physics where it makes a difference, or are you like the rest of us navel-gazers around here - and just curious the logical implications? — Relativist

Zeno paradox is a metaphysical problem rather than a physical one. It tells us something about reality without a need for any measurements.

The infinite series entails an unending series of steps. So a final step is logically impossible. — Relativist

That is not a solution but the point of Zeno. If the final step is logically impossible then you cannot complete an infinite series of finite steps therefore you cannot finish the task.

Suppose spacetime is continuous. We still can't distinguish spatial measurements that differ less than a planck length, nor time measurements less than a planck time. — Relativist

I think if spacetime is discrete and our capacity to measure spacetime interval is much higher than Planck length and time then we can treat spacetime continuously, hence we can use the continuous physical models that describe reality well. We however still have to deal with Zeno's and infinite staircases paradoxes.

This suggests to me that we will make no errors by treating space and time as discrete, even if it is continuous. What's your thoughts? — Relativist

If spacetime is continuous then we are dealing with an error in treating space and time as discrete. There are numerical methods that allow us to minimize the error but at the end of the day, we cannot avoid the error at all. In most cases, we are safe if we discretize space and use good numerical methods. In the case of time, however, the error accumulates over time so we can find significant errors in our calculation in the long term. This error in the predicted variables can be catastrophic over time if the system is chaotic, such as the weather processes.

I agree with Sime, and I also gave a solution in that thread that is similar to his. — Relativist

What is your solution to the paradox? Could you explain @sime's solution to me?

More or less. Both demonstrate the fact that limits don't correspond to the completion of an infinite series of finite steps. — Relativist

Exactly right! So what are the options in this situation: (1) Spacetime is discrete or (2) Spacetime is continuous. In the first case, we don't have the problem of infinite division so there are no conceptual problems or paradoxes such as the one of Zeno. The rule of mathematics, Leibniz's calculus is to help us easily calculate things, such as differential and integral, so it is just a useful tool. In the second case, we however need a mathematical formulation that allows us to directly deal with infinity, for example, we should be able to set n equal to infinity, if not we cannot complete an infinite series, so we are dealing with the paradoxes such as the Zeno's or infinite staircases. In simple words, we cannot move and time cannot pass. So let's wait for mathematicians to see if they have a solution for the second case. If there is no solution for (2) then we are left to (1)!

Thank you very much for your post. Unfortunately, my country is under sanction and I cannot purchase any book from Amazon. I am retired and living in the countryside so I don't have access to any library as well. So, I am out of luck when it comes to reading books unless I find a PDF file online.

Yes, there is. But if you want to discuss it, use that thread and tag me. — Relativist

That paradox is nothing more than Zeno's paradox. It simply replaces the distance in Zeno's paradox by time. There is however a problem when you want to discuss Zeno's paradox by standard analysis. To discuss this further, let's consider the following sequence:
f(1)=1/2
f(2)=f(1)/2
...
f(n+1)=f(n)/2
...
where n is a natural number. This sequence is nothing but the sequence that was first mentioned by Zeno. The sum of the sequence then can be found and it is s(n)=1-1/2^n. Although we can use standard analysis to calculate the limit of s(n) when n tends to infinity we are not allowed to consider n to be infinity. That is because n is a member of the natural numbers and all members of natural numbers are finite. @sime in this post discusses the paradox and it seems that he resolved it. It seems to me that he uses nonstandard analysis to resolve the paradox. I am however pretty ignorant of nonstandard analysis so I cannot tell how he resolves the paradox. To my understanding, motion is real and Zeno's paradox is invalid. I however don't have the mathematical tools to discuss Zeno's paradox since my knowledge of analysis is limited to standard analysis. Perhaps, other mathematicians @sime, @fishfry, and @TonesInDeepFreeze can offer us a solution to the problem.

There is no paradox there. The total time which is needed to get to the top is 60 seconds. You might find this link useful.

(1) In open forums like this, there is usually more disinformation and confusion about mathematics than there is information and clarity. Instead, prolific cranks dominate, or discussions center on a few reasonable people trying to get a prolific crank to come to the table of reason. — TonesInDeepFreeze

You can find all sorts of people in any forum. I agree that the number of knowledgeable individuals may vary from one forum to another.

(2) There are no set theory experts in this thread (or, to my knowledge, posting in this forum). — TonesInDeepFreeze

That is all right. You are enough good to teach me a few things in set theory.

(3) Picking up bits and pieces of mathematics, hodge podge, is not an effective, not even a coherent, way to understand concepts that are built from starting assumptions and definitions. This thread itself is evidence of that. — TonesInDeepFreeze

Quite oppositely I learned a few things in this thread. Thank you very much for your time and patience.

That you have limited time for mathematics is all the more reason for not wasting that limited time in routes that lead to dead ends, misinformation and confusion. — TonesInDeepFreeze

I have to disagree. I have a wide range of interests. One of the main reasons that I signed up in this forum was the very good quality of knowledge of posters in this forum such as you. I am an expert in a few fields as well, such as physics, epidemiology, philosophy of mind, and the like. The idea is to share the knowledge that one accumulated over decades with others through discussion in the forums so that all individuals can benefit from it. In this way, one can save lots of time in understanding a topic through discussion with experts and decide where to focus on a topic and how to manage the valuable time.

Ok, I will try to get the book you suggested in this thread.

If your question is whether spacetime is continuous then I have to say that there is an ongoing debate on the topic. I am not an expert on the topics of loop quantum gravity and string theory. I searched on the net a while ago and I found this manuscript which you might like to read. The manuscript is however old, 2003, so it does not reflect the current state of debate.

Thanks for the writing. I will see if I can find any time in the future to study the filter. I have other interests as well rather than mathematics.

Ok, thanks for the clarification.

Thanks for writing. On my reading schedule.

Again, thank you very much for providing the argument and extensive writing. I will read through the argument and try to understand it. I will look for proof of the theorems on the net by consulting ChatBot or reading through books or math forums.

I didn't ask for a proof that shows that aleph_0 is the least infinity but to show that X is the least infinity namely aleph_0.

A mistake on my part and I am sorry for that. I should have written: "To avoid confusion, assume that the cardinality of the set of natural (I wrote real instead of natural) numbers is X. How could one show that X is the least infinity namely aleph_0?"

Ok, that I agree. How about whether the real number represents reality or not? Do you know any references that claim that reality is not continuous? I am currently reading a manuscript about loop quantum gravity and discrete time. I am a condensed matter physicist but I studied particle physics and cosmology to a good depth. I am somehow ignorant of loop quantum gravity and string theory though.

I am aware of that. To avoid confusion, assume that the cardinality of the set of real numbers is X. How could one show that X is the least infinity namely aleph_0?

Are you suggesting this proves real numbers are logically impossible, or are you arguing that there is no valid 1:1 mapping from the set of real numbers to the actual world? — Relativist

I directly attacked the continuum in the mathematical sense. The discussion is ongoing but it seems that the classical continuum exists but suffers from problems which are discussed here. The post is very technical and I have problems understanding it though.

I ask, because it fails to do the former. — Relativist

Do you mean later (instead of former)? If yes, it would be nice of you to elaborate.

Please accept my sincere apology for not replying to your post earlier. I read your post a few times in the past and had a difficult time understanding it. Unfortunately, I forgot to respond to your post. I recalled forgetting to reply to this post when I saw your second post in my thread. I read this post more than ten times trying to understand what you are trying to say but I have to say I am still ignorant in many parts. Please find my reply in the following.

Formally, the classical continuum "exists" in the sense that that it is possible to axiomatically define connected and compact sets of dimensionless points that possesses a model that is unique up to isomorphism thanks to the categoricity of second order logic. — sime

So classical continuum exists. I had to read about second-order logic, first-order logic, and zeroth-order logic trying to make sense of what you are trying to say here. Unfortunately, I don't understand what you are trying to say in the bolded part, the last part of the paragraph. Do you mind elaborating?

But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. — sime

Do you mind elaborating on what problem you specifically have in your mind?

Notably, Dedekind didn't believe in the reality of cuts of the continuum at irrational numbers and only in the completeness of the uninterpreted formal definition of a cut. — sime

Do you by cut mean the exact position of an irrational number for example? What do you mean by the bolded part?

Furthermore, Weyl, Brouwer, Poincare and Peirce all objected to discrete conceptions of the continuum that attempted to derive continuity from discreteness. For those mathematicians and philosophers, the meaning of "continuum" cannot be represented by the modern definition that is in terms of connected and compact sets of dimensionless points. — sime

Ok, thanks for the reference.

E.g, Peirce thought that there shouldn't be an upper bound on the number of points that a continuum can be said to divide into, — sime

My understanding is that there is no upper bound on the number of points that a continuum can be divided into. I however don't understand whether he agrees with the classical notion of continuum or not. If not, what is his point?

whereas for Brouwer the continuum referred not to a set of ideal points, but to a linearly ordered set of potentially infinite but empirically meaningful choice sequences that can never be finished. — sime

Do you mind discussing his results further?

The classical continuum is unredeemable, in that weakening the definition of the reals to allow infinitesimals by removing the second-order least-upper bound principle, does not help if the underlying first-order logic remains classical, since it leads to the same paradoxes of continuity appearing at the level of infinitesimals, resulting in the need for infinitesimal infinitesimals and so on, ad infinitum.... whatever model of the axioms is chosen. — sime

So, one cannot define infinitesimal in the classical continuum. Is that what you are trying to say?

Alternatively, allowing points to have positions that are undecidable, resolves, or rather dissolves, the problem of 'gaps' existing between dimensionless points, in that it is no longer generally the case that points are either separated or not separated, meaning that most of the constructively valid cuts of the continuum occur at imprecise locations for which meta-mathematical extensional antimonies cannot be derived. — sime

What do you mean by antimony here?

Nevertheless this constructively valid subset of the classical continuum remains extensionally uninterpretable, for when cut at any location with a decidable value, we still end up with a standard Dedekind Cut such as (-Inf,0) | [0,Inf) , in which all and only the real numbers less than 0 belong to the left fragment, and with all and only the real numbers equal or greater than 0 belonging to the right fragment, which illustrates that a decidable cut isn't located at any real valued position on the continuum. Ultimately it is this inability of the classical continuum to represent the location of a decidable cut, that is referred to when saying that the volume of a point has "Lebesgue measure zero". And so it is tempting to introduce infinitesimals so that points can have infinitesimal non-zero volume, with their associated cuts located infinitesimally close to the location of a real number. — sime

So you are trying to say that introducing infinitesimal can resolve the problem of cut which is problematic for classical continuum.

The cheapest way to allow new locations for cuts is to axiomatize a new infinitesimal directly, that is defined to be non-zero but smaller in magnitude than every real number and whose square equals 0, as is done in smooth infinitesimal analysis, whose resulting continuum behaves much nicer than the classical continuum for purposes of analysis, even if the infinitesimal isn't extensionally meaningful. The resulting smooth continuum at least enforces that every function and its derivatives at every order is continuous, meaning that the continuum is geometrically much better behaved than the classical continuum that allows pathological functions on its domain that are discontinuous, as well as being geometrically better behaved than Brouwer's intuitionistic continuum that in any case is only supposed to be a model of temporal intuition rather than of spatial intuition, which only enforces functions to have uniform continuity. — sime

What do you mean by temporal and spatial intuition here?

The most straightforward way of getting an extensionally meaningful continuum such as a one dimensional line, is to define it directly in terms of a point-free topology, in an analogous manner to Dedekind's approach, but without demanding that it has enough cuts to be a model of the classical continuum. — sime

I am reading about point-free topology right now so I will comment on this part when I figure out what you mean with point-free topology.

E.g, one can simply define a "line" as referring to a filter, so as to ensure that a line can never be divided an absolutely infinite number of times into lines of zero length, and conversely, one can define a collection of "points" as referring to an ideal, so as to ensure that a union of points can never be grown for an absolutely infinite amount of time into having a volume equaling that of the smallest line. This way, lines and points can be kept apart without either being definable in terms of the other, so that one never arrives at the antimonies you raised above. — sime

I did read your links partially but I couldn't figure out what filter and ideal are. I don't understand how they resolve the problem of the classical continuum too but I buy your words on it.

The set of natural numbers is aleph_0. — TonesInDeepFreeze

What do you mean by this? Do you mean that the set of natural numbers is the set of aleph_0? aleph_0 is a number. How could you treat it as a set?

Every set is one-to-one with itself. So the set of natural numbers is one-to-one with alelph_0, so the cardinality of the set of natural numbers is aleph_0. — TonesInDeepFreeze

I don't understand this argument. How could aleph_0 be a number and a set at the same time?

I am looking for proof that the set of natural numbers that each its member is finite has aleph_0 members.

So you are saying that aleph_0 is not a member of the set of natural numbers yet the number of its members is aleph_0. I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members.

It's not counterintuitive that there exist functions whose domain is the set of positive natural numbers. Any Calculus 1 textbook has such functions all through the book. Moreover, there are functions whose domain is the set of real numbers. Moreover, there are functions whose domain is a proper subset of the set of real numbers. Moreover, for any set whatsoever, there are functions whose domain is that set (except there is only one function whose domain is the empty set). — TonesInDeepFreeze

That I understand and that is not my problem.

To be clear, 'N' there does not stand for the set of natural numbers, so I'll use 'n'':

Yes, {1 ... n} is a set with n number of members and n is a member of {1 ... n}.

That doesn't vitiate that there are other sets with n number of members but such that n is not a member:

{0 1} has 2 members, but 2 is not a member.

{1 4 7} has 3 members, but 3 is not a member. — TonesInDeepFreeze

I understand there are sets with n members, but n is not a member of the sets. That was why I defined the domain D that has this specific property, the number of its members, n, is a member as well.

the set of natural numbers has aleph_0 members, but alelph_0 is not a member. — TonesInDeepFreeze

Let me ask you this question: Are all members of the natural number set finite? If yes, how the number of its members could be aleph_0?

Do you really not understand what a domain of a function is? Or are you trolling me? — TonesInDeepFreeze

I am so sorry that you feel that I am trolling. I respect your time and my time. I appreciate your effort in explaining things to me and I learned lots of things from you that I am grateful for.

This is counter-intuitive to me. Consider a function f with the domain D={1,2,...N} where N is a finite positive integer. The domain has N members and N is a member of the domain of f. Could you please explain what happens when N is aleph_0?

There are relative statements, like "I like cats". There are statements like, "man is mortal" which are objective. Therefore I think the conclusion is that the first category of statements, relative statements, are not part of truth whereas the second category of statements, objective statements, are part of truth.

I think I see where I'm going wrong. A relative truth would be that relative to a society of evangelical Christians, gay marriage is indeed wrong on the basis of their subjective belief that it is wrong. That goes further than just reporting a fact, which is what I did in my post. — ToothyMaw

:up: :100:

I think this post is getting at exactly that - is the property of being true based on facts that are verifiable independent of our feelings, or is the property of being true based on subjective experiences? Or at least, that's what I would like to think. — ToothyMaw

Given the definition of objective and subjective from here, the truth is objective if it is a set of statements that are true and independent of opinion, biase, conscious experience, and the like.

Why is f(aleph_0) meaningless when we know that natural numbers have aleph_0 members?

Have you studied the material of Calculus 1? At least the first week in which the definition of 'limit' is given? — TonesInDeepFreeze

I studied Calculus 1 40 years ago. I am familiar with this notation of the limit.

For the third time: The domain of f is the set of positive natural numbers, therefore, aleph_0 is not in the domain of f, therefore "f(aleph_0)" is meaningless. — TonesInDeepFreeze

What do you mean by infinity when you talk about the limit in this post?

How do you define truth?

There are limits. As an example, consider the sequence 1/2, 1/4, 1/8, 1/16, ...

We all know that the limit of this sequence is 0. You can certainly call this infinite division if you like, as long as you understand what limits are. There are indeed infinitely many elements of the sequence, and you CAN think of this as "infinite division."

You can even think of doing it "all at once" if you like

What it means formally is that the elements of the sequence get (and stay) arbitrarily close to 0.

What it does NOT mean is that there is some kind of magic number that the sequence attains that is a "distance of 0" from 0, but is not 0. That's a faulty intuition.

In fact the formal theory of limits, once one learns it, is the antidote to all our non-rigorous, faulty intuitions about infinite processes.

Does that help, or perhaps refresh your memory? I'm pretty sure that physicists must be exposed to the formal theory of limits at some point. — fishfry

I agree with what you stated.

Yes I understand and agree. But I am a little surprised that you seem to think that a sequence attains some kind of mysterious conclusion that lies at a distance of 0 from its limit, but is distinct from the limit. That's just not right.

Did my mention of limits ring a bell at all? Or raise any issues that we could clarify or focus on?

Because your idea of endless division is perfectly correct. But all that shows is that endlessly halving leads you to the limit of a sequence. But there's no "extra point" in there that's distinct from but at a distance of 0 from the limit.

Let me know if we're on the same page about this. — fishfry

Yes, we are on the same page and thank you very much for your contribution. I learned a lot of things and refreshed my memory. :)

Thank you very much for your contribution to my thread. I learned lots of things. You saved me a lot of time as well.

What do you mean by "infinity" used as a noun?

There is the adjective "is infinite": S is infinite if and only if S is not finite.

And there are various infinite sets, such as:

the least infinite ordinal = {n | n is a natural number} = w = aleph_0

the least infinite cardinal = aleph_0 = w = {n | n is a natural number}

the least infinite cardinal greater than aleph_0 = aleph_1

card(set of functions from w into 2) = 2^aleph_0 = card(the power set of aleph_0)

card(set of functions from aleph_1 into 2) = 2^aleph_1 = card(the power set of aleph_1)

There are various uses of "infinity" as a noun, including such things as infinity as a point in the extended reals or infinity as a hyperreal in nonstandard analysis. But you've not specified.

In any case, how can you ask your question when I explicitly defined the domain of f to be the set of positive natural numbers? (Whatever you mean by "infinity" used as a noun, it is not a member of the set of positive natural numbers.) — TonesInDeepFreeze

By infinity, I mean aleph_0. I thought that the value of f(alep_0)=0 which is why I asked for its value. But after some thinking, I realize that it is not. In fact, one could define a sequence g(n+1)=g(n)/10 where g(0)=1. It is easy to see that for any value of n g(n)<f(n) except n=0 if (f(0)=1. But g(aleph_0)=0.0...1 and we find g(aleph_0) >0 so f(aleph_0)>0 as well. I am sure you can define things better and provide a better argument.

Then just say "finite". But your particulars involve both countable and uncountable cardinals, all of which are greater than any finite cardinal but some of which are greater than any countable cardinal:

aleph_0 is countable

aleph_1 is uncountable

2^(aleph_1) is uncountable and greater than aleph_1 — TonesInDeepFreeze

Just out of curiosity, why aleph_1 is uncountable?

Greater than any countable number or greater than any finite number? — TonesInDeepFreeze

It is better to say greater than any finite number given the definition of a countable set in mathematics.

That's what I understand as a limit (process) - that can be approached but never reached, but if it could be reached, would yield whatever - the process of whatevering being nothing in itself, but useful as a tool. — tim wood

That is a correct interpretation if you divide the interval by two, then another time divide the result by two, ad infinitum. What I have in mind is that I simply divide the interval by 2^infinity in one step. This operation seems to be invalid though to mathematicians.

And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense. — TonesInDeepFreeze

Yes, I got that. Thank you very much for your explanation.