Thread: Infinite or not infinite?

Hi, I am new here, I am just an amateur. I got the following discussion with with a friend:

He states that, We do not need to appeal to limit ordinals, infinite cardinals, or any other infinite number in order to construct a model of past-infinite time. All we need are the negatives of the natural numbers---finite numbers, of which there are infinitely many ( he is counting backwards from a t(0) , t(-1), t(-2), ..t(-n)...)

I think he is confusing the fact, that it is a "countable infinite" with the fact that an actual infinite is a limit, not a bunch of numbers or events.(confusing http://mathworld.wolfram.com/CountablyInfinite.html).

I guess I am confused too.

And he gives this following example as a non limit infinite number (w+1)
Not all infinite numbers are limits. For example, consider omega+1.

I think this is incorrect, but I am not sure.
For instance, as far as I know w+1 is defined as (the limit value no natural number sequence can reach) +1

Can anyone clear this up , for me?

thanks!

I'm not sure what you mean by an "actual infinite". The concepts of "countable" and "uncountable" are set concepts and do not involve limits. If you are talking about x "going to infinity" then you are talking about limits. I'm not sure what you mean by "w+ 1". In the very next line, you talk about "omega+ 1". Does "w+ 1" mean (omega+1)? But even then, what do you or he mean by "omega+ 1"? You can't just add a number, like 1, to an infinity.

I see that my post is a mess.

This is the situation:

We were discussing the possibility or impossibility of real infinite ( in real life ):
This was, one of the arguments against real infinite:
You cannot traverse an infinity - If we work from the bottom up in causality then in order for reality to get to this event "now" it would've first needed to traverse an infinite number of event. This is impossible, because in order for someone to traverse an infinity there would always be one more step they needed to walk, otherwise that infinity becomes a countable finite number of steps. To illustrate this, imagine if a friend knocked on your door and you asked, "How far did you walk to get here?" and he responded. "Infinity miles." You would know this to be impossible because if he indeed did walk an infinite number of miles, he would never stop walking those infinite miles, thus would never arrive at a destination. The same can be said for causality, we would never arrive at this causally present event if the universe needed to first traverse an infinite number of causal events.


My friend answer the following:
The relevant sense of traversing an infinity means to have completed an infinite number of tasks (or steps) or to have experienced an infinite number of events. If you mean anything else by the term, then you need to show why traversing an infinity is required in order for time to be past-infinite. So, is it true that in order to have completed an infinite number of tasks "there would always be one more" task needed for the completion? If so, it's not clear why. You claim that otherwise there would only be finitely many tasks. But---again---what makes you think that?


In fact, it's fairly easy to show that these claims are false. All we need to do is show a counter-example. So here's one: Let t(0) denote the present moment, and let t(-1), t(-2), t(-3), ..., denote successive past moments, on ad infinitum. Suppose Xenophon has been alive all this time, experiencing these successive moments passing. Then for any n, at t(-n) Xenophon has already experienced an infinite number of temporal successions, i.e. he has already traversed an infinity. At no point does he need to complete another task in order to complete the traversal. since the traversal has always been completed.

Perhaps I am misunderstanding his point, but I don´t think this example is correct. But i can not pinpoint exactly what is wrong in his example if there is anything wrong with it.

So for the time example, would that be a set of events or, would it be an x going to infinity.

by w+1 I meant (omega+1). for me w+1 sorry I don´t know how to print that symbol omega+1 = {x| x finite or x=omega}
My friends says that not all infinite numbers are limits and offers omega+1 as an example.

Thanks a lot ,for your answers and help.

Here is one of my favorite quotes. It is by Albert Einstein written in his 1923 essay Geometry and Experience.

"As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality."

Originally Posted by ontologicalme

I see that my post is a mess.

This is the situation:

We were discussing the possibility or impossibility of real infinite ( in real life ):
This was, one of the arguments against real infinite:
You cannot traverse an infinity - If we work from the bottom up in causality then in order for reality to get to this event "now" it would've first needed to traverse an infinite number of event. This is impossible, because in order for someone to traverse an infinity there would always be one more step they needed to walk, otherwise that infinity becomes a countable finite number of steps. To illustrate this, imagine if a friend knocked on your door and you asked, "How far did you walk to get here?" and he responded. "Infinity miles." You would know this to be impossible because if he indeed did walk an infinite number of miles, he would never stop walking those infinite miles, thus would never arrive at a destination. The same can be said for causality, we would never arrive at this causally present event if the universe needed to first traverse an infinite number of causal events.

Well, this, at least, is simply wrong. You cannot treat a "infinite number of things", such as an infinite number of steps, that way. On the other hand, there exist an infinite number of points between, say, 0 and 1, but you can cross from 0 to 1 without any trouble.

My friend answer the following:
The relevant sense of traversing an infinity means to have completed an infinite number of tasks (or steps) or to have experienced an infinite number of events. If you mean anything else by the term, then you need to show why traversing an infinity is required in order for time to be past-infinite. So, is it true that in order to have completed an infinite number of tasks "there would always be one more" task needed for the completion? If so, it's not clear why. You claim that otherwise there would only be finitely many tasks. But---again---what makes you think that?


In fact, it's fairly easy to show that these claims are false. All we need to do is show a counter-example. So here's one: Let t(0) denote the present moment, and let t(-1), t(-2), t(-3), ..., denote successive past moments, on ad infinitum. Suppose Xenophon has been alive all this time, experiencing these successive moments passing. Then for any n, at t(-n) Xenophon has already experienced an infinite number of temporal successions, i.e. he has already traversed an infinity. At no point does he need to complete another task in order to complete the traversal. since the traversal has always been completed.

Perhaps I am misunderstanding his point, but I don´t think this example is correct. But i can not pinpoint exactly what is wrong in his example if there is anything wrong with it.

So for the time example, would that be a set of events or, would it be an x going to infinity.

by w+1 I meant (omega+1). for me w+1 sorry I don´t know how to print that symbol omega+1 = {x| x finite or x=omega}
My friends says that not all infinite numbers are limits and offers omega+1 as an example.

Thanks a lot ,for your answers and help.

You are both talking as if "an infinite number of things" can be treated the same as a finite number of things- and that's just not true.

Wow.. I just wrote a research paper on this for my AP Calc class. Got a 97 on that paper.

Originally Posted by zachd77

Wow.. I just wrote a research paper on this for my AP Calc class. Got a 97 on that paper.

Please give us a summery of the points in your paper.

Hi again

Originally Posted by HallsofIvy

Well, this, at least, is simply wrong. You cannot treat a "infinite number of things", such as an infinite number of steps, that way. On the other hand, there exist an infinite number of points between, say, 0 and 1, but you can cross from 0 to 1 without any trouble.



You are both talking as if "an infinite number of things" can be treated the same as a finite number of things- and that's just not true.

For what is worth none of the answers are mine. I am just trying to make sense of them.
the definition for omega+1 I got from a math book. omega+1 = {x| x finite or x=omega}.

I wonder if your example of an infinite number or points, would be applicable to a real space in "reality" with all the talk of " Planck Length", but I get the point.

Thanks a lot for taking the time to answer.

Originally Posted by ontologicalme

I wonder if your example of an infinite number or points, would be applicable to a real space in "reality" with all the talk of " Planck Length", but I get the point.

Did you read reply #4?

Why do you think that mathematics and reality are related at some ontological level? Mathematics is wonderful at giving models for the empirical. But why would you think that the model is real?

Originally Posted by Plato

Did you read reply #4?

Why do you think that mathematics and reality are related at some ontological level? Mathematics is wonderful at giving models for the empirical. But why would you think that the model is real?

You are right, thanks.

Here is an outline concerning my paper topics. I unfortunately had to cut down on the length of the paper due to time constraints.

Topic: Infinity, Vagueness, & Paradoxes

Summary: This paper will explore and discuss the theory of infinity and its importance in mathematics.

Thesis: Despite its vagueness and relatively recent development, the ideas, theories, and paradoxes of infinity are fundamental to the science of mathematics. As a whole, infinity describes not only the very large (or very small), but it also concerns the development of mathematical thought and proof, as well as how one can make one’s own decisions and convey such decisions into argument, reason, and truth.

Part 1: How High Can We Count?
The paper will begin with an explanation of how one can observe and read large numbers, as well as some notable examples of these large numbers. The progression of “the repeating operator” will be discussed (addition, multiplication, exponentiation, tetration), with an emphasis on tetration (a number exponentiated to itself n times). Notations of tetration will be a focus (up-arrow notation). Also, famous large numbers, such as the googol or googolplex will be discussed.

Part 2: The Discrete Infinity (Fun with Set Theory & Cantor).
This section will discuss infinity in a discrete sense. Discrete functions will be discussed as well as definitions of basic set theory. Cardinality of sets will play a huge role as we explore the different “sizes” of infinity. For example, we will prove that different sets have different cardinalities (Cardinality of N, Z, Q, R, etc). We will also explore notations (omega & the alephs) & ways to discuss the first “infinite” ordinal number, and we will also take a look at Cantor’s work.

Part 3: Applications.
This shorter section will pose many problems that I have encountered that may seem interesting. We will look at some geometric applications (infinity geometry & possibly the golden ratio) as well as some physical applications in order to fully grasp this idea known as infinity.

Part 4: Vagueness & Paradoxes
In this part, we will discuss vagueness. We will define vagueness, and discuss a few examples illustrating why infinity is a “vague” concept. The discussion will lead into the introduction of paradoxes, which are sets or groups of statements that eventually lead to a contradiction or a situation that seems to defy the laws of logic. We will also discuss how the two concepts given are related to infinity and question if infinity is also a paradox.

Part 5: Famous Paradoxes & Mathematical Fallacies
The final part of the paper will discuss numerous examples of paradoxes and mathematical fallacies (1=0, indeterminate forms, etc.). Paradoxes discussed will probably be those of Zeno, Hilbert, Galileo, Cantor, Russell, and other notable examples. A list of common mathematical fallacies will be included, and we will discuss their significance & why these fallacies tend to arise quite often.