First, again, I don't know what the poster means by "the real world" so I don't know what firm and clear notion there is of an injection from the set of real numbers into "the real world". — TonesInDeepFreeze

The world that is studied by physics. The phenomena around us that are amenable to experiments. Things that have mass, electric charge, velocities, and so forth.

As opposed to conceptual things like numbers, abstract geometric shapes.

I'm not sure what point you are making to ask what is meant by the real world.

The real numbers are a mathematical abstraction. The question is whether it is literally instantiated -- that's the word I prefer -- in the real world.

In other words, is there a true mathematical continuum in the world?

I am not sure what you are trying to get at with this question, since the answer is either obvious, or else you are making some subtle point along the lines of "what is reality?"

Also, the argument "There are no relevant experiments regarding surrounding aspects of the reals, therefore there is no such injection" requires the premise, "If there is such an injection, then there are relevant experiments regarding surrounding aspects of the reals". But how would we rule out that there could be an injection but no relevant experiments regarding surrounding aspects of the reals, or that there could be an injection but no known relevant experiments regarding surrounding aspects of the reals? — TonesInDeepFreeze

Also BTW, "injection" is the word YOU are using. I reject it utterly. An injection is a technical term in set theory. An injection is a type of function between two sets. There is not a shred of evidence that the objects in the real world -- the tables, the chairs, the electrons, the quarks -- obey the axioms of ZF as they pertain to infinite sets. Of course finitary ZF, also known as combinatorics, applies to the objects of the real world. But it's doubtful that anything in the real world is infinite, let alone satisfies the axioms of the real numbers.

So when you say injection, what can you possibly mean? What is your target set? Any injection from the reals must necessarily have an infinite range (or codomain, whatever the hell is the contemporary term. I gather that range and image are different now than when I learned them, and I never heard the word codomain till recently).

Clearly there's no injection from any infinite set to any collection of objects in the real world that we know of. The number of atoms in the observable universe is finite. Start there.

The question was about an injection. — TonesInDeepFreeze

No, that's your word. I use instantiation, in the sense that the von Neumann ordinal we call 12 is instantiated by a carton of eggs. For there to be an injection you'd need a set, and perhaps you can nail that part down for me.

What is the definition of "instantiated in the real world"? Does it just mean that there is the range of an injection from the set of real numbers? — TonesInDeepFreeze

No not at all, since I strongly doubt there is an injection from the real numbers to anything in the physical universe. I doubt there's even an injection from the integers.

But the idea of a physical continuum is that below the Planck length, at the smallest level of reality, we find a copy of the real numbers. A mathematical continuum satisfying the second-order axioms of the reals. The least upper bound property and all that jazz. Metric completeness. All the Cauchy sequences converge. You ain't got nothin' like that in the world as far as anyone knows.

I don't know that that is the case. Moreover, cutting back to the question of an injection, I don't know that that the lack of someone thinking up an experiment would entail that there is no injection. — TonesInDeepFreeze

But it must be so. If there is a set of physical objects (quarks, universes, whatever) with the cardinality of the reals, we can ask what Aleph number it has, and there must be a definite answer. It would be a question amenable to physical experiment, even if we can't do it this week.

Instead of the Large Hadron collider we'd have the Colossal Continuum Counter. What a cool experiment that would be. The Superconducting Colossal Continuum Counter.

I hope you are seeing my point. If there is a set in the world cardinally equivalent to the reals, then we can in principle aspire to count them and see which Aleph they are.

Moreover, would entertaining that there is an injection from the set of natural numbers N into the real world entail that there must be some experiment to conduct? — TonesInDeepFreeze

Oh hell yes! We could count its subsets and see which Aleph they are. Same problem as for the reals, but expressed a little differently.

Or we could verify the axiom of choice. I think a countably infinite set would do, since it has uncountably many subsets.

I'm sure you can either see exactly what I'm talking about, or else you're not seeing what I'm talking about at all. In which case I should await your response.

I don't know that that is true. — TonesInDeepFreeze

Over the years I have Googled around. If someone has proposed an experiment to relate set theory to the world I might well have heard of it. In fact there are a smattering of papers relating set theory to physics, but they'er all behind academic paywalls.

That might be the case. Indeed, even the question alone of the existence of an injection from the set of real numbers into "the real world" doesn't seem to me to have, at least so far, been given a firm and clear meaning. — TonesInDeepFreeze

It has a perfectly obvious meaning. There is a familiar injection from the abstract number 12 to a carton of eggs (a standard dozen). There is an injection from the number 5 to the members of a basketball team.

Nobody has any idea whether there's an injection from a countable set to anything in the real world. But if there is, AND if ZF applies to the world (a darn good question IMO) then there is automatically an uncountable set, namely the powerset of the countably infinite set. And then AC and CH and all the large cardinal axioms become questions of physics.

That might be the case; I don't know. But I don't see that to entertain that there might be an injection entails that there must be an experiment to conduct. But again, the question of the existence of an injection from the set of real numbers into "the real world" doesn't seem to me to have, at least so far, been given a firm and clear meaning. — TonesInDeepFreeze

Please tell me why you say this.

It's perfectly clear what injections from finite sets to the world mean. It's perfectly clear that all finitary combinatorial math applies to the real world.

It's perfectly clear what an injection (or instantiation) of an infinite set would mean.

My point, which I'll bold, since it's really the only thing I have to say, is:

If there is an infinite set in the world, then all the questions of higher set theory become questions of physics, in principle amenable to physical experiment.

Heck, we did the LIGO experiment. If we can detect gravitational waves, why can't we count the points in a continuum?

Answer: Because there is no contnuum in the real world. If there were, the physics postdocs would be all over it.

I wouldn't think that to entertain that there is an injection from the set of reals into the real world entails that there is a physical version of Banach-Tarski. — TonesInDeepFreeze

Of course it would. Well ok I need a bit more. I need a three-dimensional Euclidean space. That's the minimum requirement. The isometry group of Euclidean 3-space contains a copy of the free group on two letters, which is what powers the B-T theorem.

So I'll concede that a mere linear or 2-D continuum is insufficient for B-T.

But again, the notion of such an injection is not definite enough for me to have much of a view anyway (as well as I'm not prepared to discuss details of Banach-Tarski). — TonesInDeepFreeze

All you need is Euclidean 3-space.

I surely don't have a strong opinion on the question of the existence of an injection from the set of real numbers into "the real world", but at least I would want to ponder whether the question is even even meaningful to either affirm or deny. — TonesInDeepFreeze

Well it's a meaningful question, to which I'm prepared to argue that the answer is NO.

Start with one thing:

Also BTW, "injection" is the word YOU are using. — fishfry

The poster asked about a "map into" as a "1:1 relation between".

A word for that is 'injection'.

An injection is a type of function between two sets. — fishfry

That's fine with me. And if you object to saying "injection" rather than "1:1 map" that's fine with me too. I'm not the one asking whether there is a 1:1 mapping (whether you wish to rule out calling that an 'injection') from the set of real numbers into whatever is designated by 'the real world'. Perforce, obviously, I'm not claiming that if there were such a 1:1 relation then its range would be a mathematical set.

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.

Please please please do not refer to AI bots for math proofs. They are so often incorrect. I've tried it a few times, and the bot gives clearly incorrect arguments - petitio principii - using as an assumption what it claims to be proving.

And looking around at forums is also a disorganized and extremely poor way to learn mathematics.

Get a book if you actually want to understand the material.

Thanks for writing. On my reading schedule.

Zeno's paradox: maybe. Do you think it demonstrates a meaningful mapping, or does not?

Ok, thanks for the clarification.

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.

What 1:1 map are you referring to? A 1:1 map from a real interval into points in space? A 1:1 map from a real interval into points of time? A 1:1 map from a real interval into a set of particles?

I don't have much to say about those vis-a-vis Zeno's paradox. I'm only asking what your argument is that Zeno's paradox entails that there is not a map.

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.

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.

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

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.

(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.

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

(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.

I was just proposing the possibility of someone (not me) coming up with a thought experiment. Nothing in particular in mind, but here's an example of the sort of thought experiment I had in mind:

This thread.. This entailed a thought experiment of traversing a countably infinite set stairs in a finite period of time, where the stairs-stepping entails dividing up an interval of time.

It was just an off-hand comment.

(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.

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.

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

(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. — MoK

Yes, and are quite knowledgable in this regard. I never studied anything beyond naive set theory and I have appreciated reading their posts, both instructional (and at times, argumentative). As for reference books, I am compelled to mention one that I have found wonderfully informative and written by a colleague from Colorado College: Introduction to Topology and Modern Analysis (George Simmons)

Continuity is perhaps best approached through elementary topology. Here is what Simmons says:

In the portion of topology which deals with continuous curves and their properties, connectiveness is of great significance, for whatever else a continuous curve may be it is certainly a connected topological space.

Of course, in the case of the reals the previous discussion concerns connectiveness.

That's fine with me. And if you object to saying "injection" rather than "1:1 map" that's fine with me too. I'm not the one asking whether there is a 1:1 mapping (whether you wish to rule out calling that an 'injection') from the set of real numbers into whatever is designated by 'the real world'. Perforce, obviously, I'm not claiming that if there were such a 1:1 relation then its range would be a mathematical set. — TonesInDeepFreeze

Ok. Injection it is. The rest of my point holds. If there is an infinite collection of anything in the physical world; and if ZF applies to the infinite collections of the world; then questions of higher set theory become questions of physics, in principle subject to experiment. That presents many problems for those claiming there are actual infinities in the world.

No one said anything about ZF.

But if it is taken that there only finitely many things in what is designated as 'the real world', and it is regarded that there is no injection of an infinite set into a finite set, then the question is thereby settled, regardless of ZF; also, as far as I can tell, the other poster's call to Zeno's paradox or other supertask paradoxes would be unneeded.

I have a good grasp of the some of the basics of set theory, but I am not very knowledgeable beyond those basics.

Anyway, the idea of someone, who doesn't understand that the set of natural numbers is not a member of itself, trying to grapple with how ultrafilters play into proving the existence of hyperreals is ridiculous.

.

Anyway, the idea of someone, who doesn't understand that the set of natural numbers is not a member of itself, trying to grapple with how ultrafilters play into proving the existence of hyperreals is ridiculous — TonesInDeepFreeze

:up:

No one said anything about ZF. — TonesInDeepFreeze

I did. I said that if there are infinitely many things in the world AND that ZF applies to them, then questions of higher set theory become subject to physical experiment. It's an interesting point.

But if it is taken that there only finitely many things in what is designated as 'the real world', and it is regarded that there is no injection of an infinite set into a finite set, then the question is thereby settled, regardless of ZF; — TonesInDeepFreeze

Of course if there are no infinite sets in the world, then there is surely no continuum in the world.

also, as far as I can tell, the other poster's call to Zeno's paradox or other supertask paradoxes would be unneeded. — TonesInDeepFreeze

I noted that supertask discussions have no bearing on any aspect of reality.

It's fine that you're talking about a question that's different from the one I replied to.

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.

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.

That paradox is nothing more than Zeno's paradox. — MoK

More or less. Both demonstrate the fact that limits don't correspond to the completion of an infinite series of finite steps. I agree with Sime, and I also gave a solution in that thread that is similar to his.

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)!

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.

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?