Infinity cannot be divided, if it can then it becomes multiple finite objects. Therefore there cannot be multiple Infinities. If infinity has a size, then it is a finite object. If infinity has a boundary of any kind, then it is a finite object.
I’m not entirely sure I understand your comment, but the fact that there are more real numbers than natural numbers can be readily shown using something called cantor’s diagonal argument. It goes something like this:
Suppose the set of real numbers and natural numbers had the same size. Then we could write down an infinite list, where each line represents some real number written in it’s decimal representation. So something like
1: 3.141592652: 1.414213563: 0.24242424
...
This list goes on forever. We will now construct a new real number r as follows: The first number after the decimal point of r shall be different from the first number after the decimal point of the first number in our list, the second shall be different from the second decimal of the second number on the list, and so on (the name diagonal argument comes from this, we consider the entries on the diagonal from top left towards the bottom right).
The key point now is that this constructed real number is different from every single number on the list: After all, we have made sure it differs from each number on the list in at least one place. Therefore, it is impossible to write down the real numbers in such a way that each real number gets its own natural number: There are simply too few natural numbers for this. In particular, there are at least two different “sizes” of infinities.
The set, that has a measurable starting point, is a finite object. If infinity can be measured, then you have created a finite space, one that can be measured, within an infinite space, infinity itself, which cannot be measured. Infinity remains untouched and undivided. The sets that represent infinity are finite objects, that represent an infinite space, a representation which they can never truly achieve.
Maybe we should talk about what “infinite” means. I’d like to propose the following idea: a sequence of things is infinite, if there is always “one more” object to consider. We could also say that for any number of finite steps, there is always another object of the series we haven’t looked at yet.
As an example, the sequence of natural numbers would satisfy this: if I start considering the sequence 1, 2, 3 and so on, if I ever stop after finite time (say 1729 steps), I can always compute +1 to find another element of the sequence I haven’t seen yet.
Also consider the following: the set of all numbers between 0 and 1 is in some sense bounded. However, I can find an infinite sequence of numbers in this set: consider
1/1, 1/2, 1/3, 1/4, …
These numbers are always between 0 and 1, and are infinitely decreasing.
Perhaps the confusion comes from you talking about infinity as in a number which is larger than any real or natural number, while I’m talking about sizes of sets of infinite size. As I had demonstrated earlier, we can show the existence of uncountable infinite vs countably infinite sets, while such distinctions don’t really come up in limit theory and calculus.
Infinity cannot be divided, if it can then it becomes multiple finite objects.
It really depends on what you mean by infinity and division here. The ordinals admit some weaker forms of the division algorithm within ordinal arithmetic (in particular note the part about left division in the link). In fact, even the cardinals have a form of trivial division.
Additionally, infinite sets can often be divided into set theoretic unions of infinite sets fairly easily. For example, the integers (an infinite set) is the union of the set of all integers less than 0 with the set of all integers greater than or equal to 0 (both of these sets are of course infinite). Even in the reals you can divide an arbitrary interval (which is an infinite set in the cardinality sense) into two infinite sets. For example [0,1]=[0,1/2]U[1/2,1].
If infinity has a size, then it is a finite object.
Again, this is not really true with cardinals as cardinals are in some sense a way to assign sizes to sets.
If you mean in terms of senses of distances between points, in the previous link involving the extended reals, there is a section pointing out that the extended reals are metrizable, informally this means we can define a function (called a metric) that measures distances between points in the extended reals that works roughly as we’d expect (such a function is necessarily well defined if either one or both points are positive or negative infinity).
If infinity can be measured, by either size, shape, distance, timespan or lifecycle, then the object being considered infinite is a finite object. Infinity, nothing, and everything follow these same rules. If there are two multiple infinite objects side by side to each other, which means there is a measurable boundry that seperates them, then those objects aren’t infinite, they are finite objects, within an infinite space that contains them. Only the space that contains these objects is infinite. Any infinite numbers that are generated within this infinite space, regardless of where they originated within this space, belong to this single infinity. There is no infinityA or infinityB there is just infinity itself.
My degree is in math. I feel pretty confident in saying that you are tossing around a whole bunch of words without actually knowing what they mean in a mathematical context.
If you disagree, try the following:
What is a function? What is an injective function? What is a surjective function? What is a bijection?
In mathematics, what does it mean for a set to be finite?
In mathematics, what does it mean for a set to be infinite?
I’m willing to continue this conversation if you can explain to me in reasonably rigorous terms what those words mean. I’ll help you do it too. The link I sent you in my previous post that mentions cardinal numbers links you to a wikipedia page that links to articles explaining what finite and infinite sets are in the first paragraph.
To be clear here, your answer for 2 specifically should rely on your answer from 1 as the mathematical definition of a finite set is in terms of functions and bijections.
Here are some bonus questions for you to try also:
In mathematics, what does it mean for a set to be countable?
In mathematics, what does it mean for a set to be uncountable?
A finite universe, the one in which we live, can only produce finite objects. Those finite objects can only produce other finite objects. A finite object cannot create an infinite object, as the act of creation would be a starting point for the object, and if an object has a starting point or an end point, which are really the same thing, then the object is a finite object.
If a set of numbers originates from a starting point and moves away from that point in a seemingly infinite distance, and then you decide to traverse that set in the opposite direction towards the starting point, the starting point becomes an ending point, and if an object, in this case the set itself, has an ending point, it is a finite object. Finite objects cannot create infinite objects because the act of creation would negate their infinity. Infinity is neither created nor ends, nor does it have size, shape, or form.
None of this includes the correct answers to the questions I asked you. I’m not going to read anything else from you until you correctly answer the questions I asked.
I find it interesting that you have a degree in math, and apparently have never questioned a question. As I’ve demonstrated, in the posted problem, the statement “some Infinities are bigger than other infinities” is an illogical statement. The mere statement that there are multiple infinities, negates either objects identification as being infinite, and reduces both objects to finite objects, as the only way these objects can be determined to be seperate from each other is through a boundary that would impose a starting or ending point on each object, which in turn reduces them into finite objects.
I also find it interesting that you resort to gate keeping to try and control a situation that you are frustrated by. I was able to simply and clearly demonstrate my position. I also demonstrated the technique of: solving the problem by defeating its purpose. I’ve also demonstrated the difference in how a mathematician and an engineer attempt to solve a problem.
It seems you are having a hard time comprehending this. I get it’s hard to learn new things. But I can walk you through it.
TL;DR: If an object can be measured, in any way, it’s a finite object. Infinity cannot be measured.
In the posted problem the train tracks themselves are finite objects, as they each have a starting point, the fork the train is in front of.
The train tracks are bound to physical ground, ground that is itself bound to a finite world, a world has a shape, that can be measured, so it is a finite object.
If the shape of the world the train tracks are on is round, then these seemingly infinite tracks will eventually loop back on themselves. If the tracks loop back on themselves, then they must eventually converge as the train starts out the problem on a single track. So neither of the tracks are infinite.
It’s important to understand that the tracks are finite objects, as finite objects exist by different rules then infinity itself.
I’m not arguing that uncountable numbers are a thing. What I am stating is that if those numbers exist within a finite universe, then they have a lifespan, the lifespan of the finite universe that contains them, thus those numbers aren’t infinite, uncountable yes, but not truly infinite. As I have stated many times, finite objects, like the finite universe, can only create other finite objects. Infinity cannot be created, therefore there is only one infinity, infinity itself, all other objects that can be measured are finite objects. This also means if infinity decides to create anything, it can only produce finite objects. Infinity cannot produce another infinity, as the act of creation would be a measurable starting point.
This is why the statement (some infinities are smaller than other infinities) is an illogical statement. If you can measure multiple infinities, then none of those objects are infinite, as one object can be measured to be smaller or larger than the other. And as I keep stating, infinity cannot be measured. If your measurement is uncountable, then the measurement itself is finite.
Infinity cannot be divided, if it can then it becomes multiple finite objects. Therefore there cannot be multiple Infinities. If infinity has a size, then it is a finite object. If infinity has a boundary of any kind, then it is a finite object.
I’m not entirely sure I understand your comment, but the fact that there are more real numbers than natural numbers can be readily shown using something called cantor’s diagonal argument. It goes something like this:
Suppose the set of real numbers and natural numbers had the same size. Then we could write down an infinite list, where each line represents some real number written in it’s decimal representation. So something like
1: 3.14159265 2: 1.41421356 3: 0.24242424 ...This list goes on forever. We will now construct a new real number r as follows: The first number after the decimal point of r shall be different from the first number after the decimal point of the first number in our list, the second shall be different from the second decimal of the second number on the list, and so on (the name diagonal argument comes from this, we consider the entries on the diagonal from top left towards the bottom right).
The key point now is that this constructed real number is different from every single number on the list: After all, we have made sure it differs from each number on the list in at least one place. Therefore, it is impossible to write down the real numbers in such a way that each real number gets its own natural number: There are simply too few natural numbers for this. In particular, there are at least two different “sizes” of infinities.
The set, that has a measurable starting point, is a finite object. If infinity can be measured, then you have created a finite space, one that can be measured, within an infinite space, infinity itself, which cannot be measured. Infinity remains untouched and undivided. The sets that represent infinity are finite objects, that represent an infinite space, a representation which they can never truly achieve.
Maybe we should talk about what “infinite” means. I’d like to propose the following idea: a sequence of things is infinite, if there is always “one more” object to consider. We could also say that for any number of finite steps, there is always another object of the series we haven’t looked at yet.
As an example, the sequence of natural numbers would satisfy this: if I start considering the sequence 1, 2, 3 and so on, if I ever stop after finite time (say 1729 steps), I can always compute +1 to find another element of the sequence I haven’t seen yet.
Also consider the following: the set of all numbers between 0 and 1 is in some sense bounded. However, I can find an infinite sequence of numbers in this set: consider 1/1, 1/2, 1/3, 1/4, …
These numbers are always between 0 and 1, and are infinitely decreasing.
Perhaps the confusion comes from you talking about infinity as in a number which is larger than any real or natural number, while I’m talking about sizes of sets of infinite size. As I had demonstrated earlier, we can show the existence of uncountable infinite vs countably infinite sets, while such distinctions don’t really come up in limit theory and calculus.
It really depends on what you mean by infinity and division here. The ordinals admit some weaker forms of the division algorithm within ordinal arithmetic (in particular note the part about left division in the link). In fact, even the cardinals have a form of trivial division.
Additionally, infinite sets can often be divided into set theoretic unions of infinite sets fairly easily. For example, the integers (an infinite set) is the union of the set of all integers less than 0 with the set of all integers greater than or equal to 0 (both of these sets are of course infinite). Even in the reals you can divide an arbitrary interval (which is an infinite set in the cardinality sense) into two infinite sets. For example [0,1]=[0,1/2]U[1/2,1].
In the cardinality sense this is objectively untrue by Cantor’s theorem or by considering Cantor’s diagonal argument.
Edit: Realized the other commenter pointed out the diagonal argument to you very nicely also. Sorry for retreading the same stuff here.
Within other areas of math we occasionally deal positive and negative infinities that are distinct in certain extensions of the real numbers also.
Again, this is not really true with cardinals as cardinals are in some sense a way to assign sizes to sets.
If you mean in terms of senses of distances between points, in the previous link involving the extended reals, there is a section pointing out that the extended reals are metrizable, informally this means we can define a function (called a metric) that measures distances between points in the extended reals that works roughly as we’d expect (such a function is necessarily well defined if either one or both points are positive or negative infinity).
If infinity can be measured, by either size, shape, distance, timespan or lifecycle, then the object being considered infinite is a finite object. Infinity, nothing, and everything follow these same rules. If there are two multiple infinite objects side by side to each other, which means there is a measurable boundry that seperates them, then those objects aren’t infinite, they are finite objects, within an infinite space that contains them. Only the space that contains these objects is infinite. Any infinite numbers that are generated within this infinite space, regardless of where they originated within this space, belong to this single infinity. There is no infinityA or infinityB there is just infinity itself.
My degree is in math. I feel pretty confident in saying that you are tossing around a whole bunch of words without actually knowing what they mean in a mathematical context.
If you disagree, try the following:
What is a function? What is an injective function? What is a surjective function? What is a bijection?
In mathematics, what does it mean for a set to be finite?
In mathematics, what does it mean for a set to be infinite?
I’m willing to continue this conversation if you can explain to me in reasonably rigorous terms what those words mean. I’ll help you do it too. The link I sent you in my previous post that mentions cardinal numbers links you to a wikipedia page that links to articles explaining what finite and infinite sets are in the first paragraph.
To be clear here, your answer for 2 specifically should rely on your answer from 1 as the mathematical definition of a finite set is in terms of functions and bijections.
Here are some bonus questions for you to try also:
In mathematics, what does it mean for a set to be countable?
In mathematics, what does it mean for a set to be uncountable?
A finite universe, the one in which we live, can only produce finite objects. Those finite objects can only produce other finite objects. A finite object cannot create an infinite object, as the act of creation would be a starting point for the object, and if an object has a starting point or an end point, which are really the same thing, then the object is a finite object.
If a set of numbers originates from a starting point and moves away from that point in a seemingly infinite distance, and then you decide to traverse that set in the opposite direction towards the starting point, the starting point becomes an ending point, and if an object, in this case the set itself, has an ending point, it is a finite object. Finite objects cannot create infinite objects because the act of creation would negate their infinity. Infinity is neither created nor ends, nor does it have size, shape, or form.
None of this includes the correct answers to the questions I asked you. I’m not going to read anything else from you until you correctly answer the questions I asked.
I find it interesting that you have a degree in math, and apparently have never questioned a question. As I’ve demonstrated, in the posted problem, the statement “some Infinities are bigger than other infinities” is an illogical statement. The mere statement that there are multiple infinities, negates either objects identification as being infinite, and reduces both objects to finite objects, as the only way these objects can be determined to be seperate from each other is through a boundary that would impose a starting or ending point on each object, which in turn reduces them into finite objects.
I also find it interesting that you resort to gate keeping to try and control a situation that you are frustrated by. I was able to simply and clearly demonstrate my position. I also demonstrated the technique of: solving the problem by defeating its purpose. I’ve also demonstrated the difference in how a mathematician and an engineer attempt to solve a problem.
To me you have demonstrated:
You don’t know even the most basic definitions of the things you are trying to talk about.
You are possibly too willfully stupid to bother to learn said definitions.
You are capable of babbling incoherently about things you do not understand ad nauseum.
It seems you are having a hard time comprehending this. I get it’s hard to learn new things. But I can walk you through it.
TL;DR: If an object can be measured, in any way, it’s a finite object. Infinity cannot be measured.