Mathematics only implies that the list of positive integers goes on forever, usually using an ellipsis(...) It, in fact, doesn't, as such a list only exists in the imagination, which makes it exactly as valid as an imaginary infinite corridor.
It goes on forever in that for any positive integer you can come up with, I can add 1 to it, to get a different positive integer that is bigger than the one you give me. This is what mathematics really mean by the list "go on forever".
The ellipsis is just a shorthand notation. It is actually not how one would rigorously formulate all the positive integers; after all, the next number after "1, 2, 3, ..." might not be 4, it might be 5 for example if you are actually talking about the Fibonacci numbers. The notation using ellipsis may be convenient, but clearly imprecise. Instead of example members plus ellipses, to be rigorous you would just provide a formula that says you start with 1, and then you can keep creating the next integer by adding 1 to the previous. You don't need any ellipses and there is no ambiguity.
Yes, you can call it imaginary in the same way as the imaginary infinite corridor. My question is, it seems like you object to the imaginary status of the list of all positive integers more than you object to the imaginary infinite corridor, is that so and if yes, why is that?
Suppose we have an infinite set that represents real numbers between 0 and 1
What causes my confusion about real numbers belonging to a set at all is this:
Start at 0. Now count to 1, but don't miss out any numbers in between.
Where do you start?
Ok, I think you have a confusion between the concept of a
sequence vs a
set. A sequence is ordered and countable. A set is not ordered and doesn't have to be countable. You can define a set purely based on a formula that can describe, given something, whether that something belongs to the set or not.
As already explained by others, it is actually impossible to list out all the real numbers, or even just the ones between 0 and 1, in any kind of lists. Any attempt to do so will provably result in some real number that is not part of the list you try to make. The set of real numbers can be defined simply by the number being expressible as, for example, a regular integer before the decimal point followed by an infinite sequence of digits 0-9 after the decimal point. Any number expressible in that form is a member of the set.
This is a completely understandable confusion given that the informal ellipsis notation for numeric sequences, is often used in place of a formal formulation, as a way to describe the membership criteria for the set. That is, the set of positive integers is often written down as {1, 2, 3, ...}, but as explained above, it is actually rather informal and imprecise to use this ellipsis notation. You don't have to use ellipses to define an infinite set, and for something uncountable like the real numbers, such a notation is completely useless anyway.
What I don't understand is... how can something that is infinite have "a top-left corner"?
The sequence of natural numbers (1, 2, 3, ...) has a start but no end. It has a minimum member but no maximum; any number you can think of, I can add 1 to it to make a bigger number. So it is unbounded (ie. infinite) but in one direction only.
Other examples abound, such as Dullstar talking about levels being infinite in one direction but finite in another.
Cantor's cardinality measure is a specific way to extend the intuitive concept of "size of a set" from finite sets to infinite sets, where human intuition is far less reliable and certainly lacks rigor.
I can't find a smiley for this. Cantor was a human, so any ideas he may have established are, by definition, based on human intuition!
Right. At the same time, he also took care to ensure his formulations are precise and logically consistent, so that it is usable for further mathematical inquiry. As I mentioned before, a logical inconsistency/paradox (ie. a statement both true and false) is bane to mathematical proofs, because the operations of logic can take even just one inconsistency and allow it to "prove" any other statement in the system to be both true and false as well. A system with unresolved paradoxes cannot be used as the basis for further mathematical inquiry, especially for something like set theory that was considered for possible use as foundational basis of all of mathematics.
It dismays me that somehow the only thing you seem to read into the sentence is that I'm somehow denying the role of human intuition in any discipline. Of course human intuition plays a part in the exploration and understanding of any topic, no one is trying to deny that. But at the same time, surely you can agree that human intuition isn't infallible either? That's really all I'm saying, and somehow you seem to be reading a completely different meaning into it apparently.
Is it wrong for me to question their findings, though, limited though my own understanding may be?
No, but your initial attempts, at least in their wordings, seem to show you haven't even bothered questioning your own understanding of those findings. Shouldn't one first "question and evaluate" one's own understanding of the things one is trying to challenge, lest the attempt only wound up challenging a completely distorted version not congruent to what the findings were actually saying?
It also doesn't help much that between your questions, you talked about things like this
By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective
Somehow seeming to insinuate that anyone who don't share the same confusion as you had on the details of set theory's treatment of infinity, must only be because they are "rigid" and "stubborn", as opposed to maybe other possibilities like, I don't know, maybe they understood it a little better? Interesting that Dullstar also asked some questions here that basically seek for the same clarifications on some of the same things you were a little confused about, and yet somehow he managed to not have to bring up anything about anyone being rigid and stubborn, which is neither here nor there and certainly not helping to actually addressed the questions being raised.
The longer something has been established as "truth", the more necessary it is to question and re-evaluate it.
Sure, but at the same time, it's not like this topic is something that's decreed as unquestioned truth one day by some king or emperor, and then blindly accepted for here on out. That's never how things work in any academic disciplines. Right from the beginning as the initial version of the theory or framework is still being formulated and explored, things are already questioned, discussed and critiqued, and things get revised multiple times in the journey towards the formulation you see today. People often lose sight of this, because from a textbook, of course they skip over right to what ends up being the current "final" formulation of the topic; they are not going to waste ink elaborating on all the various initial earlier formulations, the things that were questioned or critiqued and thus revised, possibly even some things that proved to be false starts and thus abandoned. Even if you read up on additional reference materials that go more into the history and evolution of the topic, they too are liable to summarizing and not necessarily provide the full scope of everything that had happened throughout the evolution of the topic or area of study. So far from it being unquestioned, what you see today generally had already endured years of questions and evaluations; you wound up with what had stood the tests of time.
For a topic that had been around for so long and exposed to so many people, it would be patently silly to assume that the questions one can raise haven't likely already been asked in a similar way before by someone else and then addressed, especially questions from someone who is still learning about the topic. At the same time, don't assume that there aren't questions raised today on the topic either. It's just that given the things that had already stood the test of time, one would likely need to question at a deeper level, at a much more particular part of the theory, in order to truly be asking something that hadn't already been asked and addressed before. It might be hard for the average person to find examples of such questions as they likely resides in relatively obscure academic journals devoted to rather deep specializations into the topic or area of interest, but that doesn't mean no such questions are raised.
I'm not even sure I understand what it means to say to "interact" with "infinity". Isn't it just an abstract concept?
I guess I mean how can we make sense of it, explore it, when it's seemingly unexplorable. As you've touched upon: nothing in the physical world, particularly humans, can experience the infinite except in our imaginations. We seem to agree about this.
I guess I don't really see how it is "unexplorable". Really, if we can talk about it, isn't that enough to make it explorable? Academic studies are full of things that don't have direct physical manifestations in the real world anyway. Geometry may be based on our experiences in the real world, but there is no actual physical manifestation of an ideal geometric line of 0 thickness and infinite length.
At the same time, just because something is grounded in the physical world, don't assume that our intuition and initial understanding of it will necessarily be more reliable and accurate. After all, Aristotle the ancient Greek philosopher believed that it is the natural order for heavier things to fall faster than lighter things. And honestly, that matches our everyday intuition pretty well, so pretty much everyone quickly believed it to be true. It took a surprisingly long time before people finally discovered that, no, in fact that's not true. In the absence of air, all objects would actually fall at the same speed regardless of their weight. It is actually the air providing air resistance that results in the differences one observes in the speed of falling, and in turn the air resistance depends on other properties of the object besides its weight/mass.