# Thread: The Concept of the Infinite

1. Originally Posted by Boxing Pythagoras
Did you get a chance to read my really quick set theory primer from post #12? It wouldn't quite be accurate to say that the ordinal number for the week is 7, so much as we would say that Saturday is the element of the days of the week which corresponds to the ordinal 7. The set of days of the week can be put in one-to-one correspondence with the elements of the ordinal number 7, so the week has the cardinality of 7.
Got it.

Nope. Ordinals and Cardinals are not the same sorts of things, despite the fact that we tend to use the same symbols for both when dealing with finite numbers. Again, ordinality tells us about an ordering relationship-- where should this element show up in a list of elements? Cardinality tells us how the elements of one set can be mapped onto another-- can this list be lined up with some other list? So, while ω has cardinality $\aleph _0$, nonetheless we cannot say that $\omega=\aleph _0$. So, ω<ω+1, despite the fact that ω and ω+1 have the same cardinality.
Got it. Cardinality and Ordinality are different kinds of things.

Not a bother, at all. Feel free to Private Message me if you have any questions on how I utilize those links. I'll be happy to help.
Thanks!

Yeah, the way in which mathematics is often taught in schools rather obfuscates its beauty. Honestly, the best way to come to that sense, in my opinion, is to start looking into the History of Mathematics. Seeing the reasoning and work that went into the inventions and discoveries of mathematics, and the people responsible for these, can really make you appreciate the field.
Agreed.

Do you think we have a good enough grasp on the infinite as of right now (even though I realize we're scratching the surface of the surface . . .)?

2. Originally Posted by mattdamore
Excellent summary of infinite ordinals! Quick question, though.

I might have misunderstood your meaning, then, in my other reply to you.

It was my understanding that the set of all natural numbers had a cardinality which was equal to its ordinality. I had thought that "n" becomes "less than" ω only when it's the case that ω+1, or ω*2, or ω2, or ωω? I do remember you saying that cardinals and ordinals are different. Is it that an ordinal and a cardinal could have the same number, but that it's a different kind of number, due to the fact that ordinals and cardinals are different kinds of numbers? Thank you for your patience.
I think I was hinting at your worry here: in terms of ordinals and cardinals being different kinds of numbers. I should have said 'different kinds of things'.

3. Originally Posted by shunyadragon
When you say "not definitive as to the limits and nature of our physical existence", what does this mean exactly? What does it mean for a "concept of the infinite" to not be "definitive as to the limits and nature of our physical existence"?

Yes, math over the millennia has been developed as the tool box with science to understand our physical existence and not define it. This has evolved from counting sheep and goats, to the modern math that science uses to describe the sciences, Quantum Mechanics and Cosmology.

And if it doesn't tell us exactly, does it tell us something inexact about its limits and nature?

No, not its limits, actual infinities and potential infinities, are descriptive of what could be the nature of infinities. They are used in math to describe various aspects of the nature of physical existence within math proofs, theorems, and described in axioms. Infinities were never meant to prove nor demonstrate the limits or our physical existence.

There is no proof that our universe is finite nor infinite, and some way limited nor eternal. It can be assumed by the evidence that it is potentially eternal.

No, within math the 'concept of infinity' is descriptive as part of the tool box of science. Within the context of apologetics cosmological arguments the concepts it cannot be used in the logical arguments fro the finite nature of our physical existence.

I do not believe the conceptual understand of infinities necessarily exists within the of math construction of infinities in axioms, sets and theorems. I understand the discussion between Pythagoras and you, and I bow to Pythagoras as to his deep understanding of the mechanics of math and I will continue to follow the discussion, but I believe the understanding of math and the 'concepts of infinity' go deeper than a course on the nature of set theories and the mechanics of math.

Fine . . . hold off if you wish. I defer to Pythagoras as being more knowledgeable concerning the meahanics of math.

Understanding the nature of math being a part of the 'tool box' of science, and more specifically 'actual infinities' used in physics and cosmology I refer to the following, which I may cite from in later posts.

http://www.iep.utm.edu/infinite/

Yes, actual infinities are defined as sets withing a greater reality such as a physical existence that is either infinite or finite, or in the case that it is potentially eternal.

Simply . . .

Source: https://en.wikipedia.org/wiki/Actual_infinity

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.

I will bow to Pythagoras on this and his knowledge of math for the best explanation.
Thanks for the clarification, Shunyadragon.

4. Originally Posted by mattdamore
Do you think we have a good enough grasp on the infinite as of right now (even though I realize we're scratching the surface of the surface . . .)?
It depends on what we want to use it to discuss. If we're going to talk about the question of an infinite regress posed by apologists and theologians, then I think we're actually better served by discussing a different concept of infinity than that of Cantor's set theory. The transfinite numbers are wonderfully interesting, to be sure, but transfinite arithmetic is different than common arithmetic, and this can quite easily lead into equivocation and confusion.

To that end, allow me to briefly introduce the Hyperreal numbers.

First, some history. The Natural numbers are so named because that number system was the first to be developed and explored, amongst modern number systems. It involves the whole numbers with which one counts-- 0, 1, 2, 3, 4, et cetera. However, the Natural numbers are incomplete. Think, for example, of the equation $5+x=4$. This fairly simple algebra problem has no solution on the Natural numbers, but we can extend the concept of the Natural numbers to give us numbers which are less than zero-- that is to say, numbers which are negative. This extension gives us the Integers. The Integers, themselves, can be extended in order to solve equations like $5\cdot x=4$. For this, we need the Rational numbers. But the Rational numbers are insufficient to solve something like $x^2=5$, so we can extend once again to create the Real numbers.

Just as the Reals extend the Rationals, and the Rationals extend the Integers, and the Integers extend the Naturals; so too can we create an extension for the Real numbers. This extension will allow us to discuss infinite numbers with consistency, using the exact same arithmetic and algebra with which we are already familiar. We call this extension the Hyperreal numbers.

On the Hyperreal number system, there exist numbers which are described as being infinite. A Hyperreal number, K, is called infinite if and only if, for all n such that n is a Natural number, it is true that $|n|<|K|$. Any number which is not infinite is called finite. There are also numbers which are called infinitesimal. A Hyperreal number, ε, is infinitesimal if and only if, for all r such that r is a Real number, it is true that $|\epsilon|<|r|$. Note that this means zero is the only Real number which is infinitesimal.

The Hyperreal numbers have some very familiar properties. For example, given an infinite Hyperreal number, K, it will always be true that $K-1. Note that K-1, K, and K+1 are all infinite numbers, but they are different infinite numbers. In exactly the same way, if K is positive, it will also be true that $\frac{K}{2}. For a more detailed breakdown of the Hyperreals, check out this link, or if you really want to get in depth with the subject, pick up Abraham Robinson's book Non-Standard Analysis. There is no better text on the subject-- after all, it was Robinson who discovered, invented, and developed this number system.

The Hyperreals can be added, subtracted, multiplied, divided, exponentiated, logarithmed, and manipulated in exactly the same ways that the Real numbers can be. They give us a very familiar platform with which to discuss infinities, and as such, they are far more intuitive for use in discussions about infinite regress.

5. Originally Posted by Boxing Pythagoras
It depends on what we want to use it to discuss. If we're going to talk about the question of an infinite regress posed by apologists and theologians, then I think we're actually better served by discussing a different concept of infinity than that of Cantor's set theory. The transfinite numbers are wonderfully interesting, to be sure, but transfinite arithmetic is different than common arithmetic, and this can quite easily lead into equivocation and confusion.

To that end, allow me to briefly introduce the Hyperreal numbers.

First, some history. The Natural numbers are so named because that number system was the first to be developed and explored, amongst modern number systems. It involves the whole numbers with which one counts-- 0, 1, 2, 3, 4, et cetera. However, the Natural numbers are incomplete. Think, for example, of the equation $5+x=4$. This fairly simple algebra problem has no solution on the Natural numbers, but we can extend the concept of the Natural numbers to give us numbers which are less than zero-- that is to say, numbers which are negative. This extension gives us the Integers. The Integers, themselves, can be extended in order to solve equations like $5\cdot x=4$. For this, we need the Rational numbers. But the Rational numbers are insufficient to solve something like $x^2=5$, so we can extend once again to create the Real numbers.

Just as the Reals extend the Rationals, and the Rationals extend the Integers, and the Integers extend the Naturals; so too can we create an extension for the Real numbers. This extension will allow us to discuss infinite numbers with consistency, using the exact same arithmetic and algebra with which we are already familiar. We call this extension the Hyperreal numbers.

On the Hyperreal number system, there exist numbers which are described as being infinite. A Hyperreal number, K, is called infinite if and only if, for all n such that n is a Natural number, it is true that $|n|<|K|$. Any number which is not infinite is called finite. There are also numbers which are called infinitesimal. A Hyperreal number, ε, is infinitesimal if and only if, for all r such that r is a Real number, it is true that $|\epsilon|<|r|$. Note that this means zero is the only Real number which is infinitesimal.

The Hyperreal numbers have some very familiar properties. For example, given an infinite Hyperreal number, K, it will always be true that $K-1. Note that K-1, K, and K+1 are all infinite numbers, but they are different infinite numbers. In exactly the same way, if K is positive, it will also be true that $\frac{K}{2}. For a more detailed breakdown of the Hyperreals, check out this link, or if you really want to get in depth with the subject, pick up Abraham Robinson's book Non-Standard Analysis. There is no better text on the subject-- after all, it was Robinson who discovered, invented, and developed this number system.

The Hyperreals can be added, subtracted, multiplied, divided, exponentiated, logarithmed, and manipulated in exactly the same ways that the Real numbers can be. They give us a very familiar platform with which to discuss infinities, and as such, they are far more intuitive for use in discussions about infinite regress.
I follow you, and I very much appreciate the explanation. Sure! Is it your experience that apologists and theologians neglect to talk about infinite regresses in terms of Hyperreal numbers? Is this a shortcoming in the apologetic? If so, I would really like to dive in to where the apologists are in error with this concept.

6. Originally Posted by mattdamore
Is it your experience that apologists and theologians neglect to talk about infinite regresses in terms of Hyperreal numbers?
It's not so much that they neglect to talk about it as that they are ignorant of the Hyperreals and how these can resolve many of the typically cited questions regarding infinite regress.

For example, I've written two small articles critiquing William Lane Craig's understanding of infinity. Dr. Craig claims that things like Hilbert's Grand Hotel and Al Ghazali's infinite regress of celestial motion reveal absurdities in the concept of the infinite. I show that the proposed absurdities are actually the result of his misconceptions, and that we can easily resolve the issues which he questions by using the Hyperreals.

https://boxingpythagoras.com/2015/10...tand-infinity/
And
https://boxingpythagoras.com/2015/10...finity-part-2/

7. Originally Posted by Boxing Pythagoras
It's not so much that they neglect to talk about it as that they are ignorant of the Hyperreals and how these can resolve many of the typically cited questions regarding infinite regress.

For example, I've written two small articles critiquing William Lane Craig's understanding of infinity. Dr. Craig claims that things like Hilbert's Grand Hotel and Al Ghazali's infinite regress of celestial motion reveal absurdities in the concept of the infinite. I show that the proposed absurdities are actually the result of his misconceptions, and that we can easily resolve the issues which he questions by using the Hyperreals.

https://boxingpythagoras.com/2015/10...tand-infinity/
And
https://boxingpythagoras.com/2015/10...finity-part-2/
Your articles are excellent! I actually do not consider WLC's bad math simply naive misconceptions. I consider it is a dishonest deceptive misuse of math.

8. Originally Posted by shunyadragon
Your articles are excellent! I actually do not consider WLC's bad math simply naive misconceptions. I consider it is a dishonest deceptive misuse of math.
I try to read opposing arguments with a charitable view whenever there is not an overt reason to believe that they actually understand that their views are not generally accepted but present them as the concrete truth, anyway.

In this case, it seems that Craig is simply ignorant of the actual mathematics and philosophy thereof. He doesn't seem to even understand the mathematics which he does discuss, let alone the vast body of infinite mathematics which he does not discuss.

If, on the other hand, he actually understood the math and demonstrated familiarity with the methods by which that math resolves his proposed absurdities, but proceeded to pretend that the notion of the infinite was still unreconcilable or inconsistent, then I might be inclined to think him dishonest. Since this is not the case, the charitable thing is to simply think he is ignorant of the subject matter.

9. Originally Posted by Boxing Pythagoras
It's not so much that they neglect to talk about it as that they are ignorant of the Hyperreals and how these can resolve many of the typically cited questions regarding infinite regress.

For example, I've written two small articles critiquing William Lane Craig's understanding of infinity. Dr. Craig claims that things like Hilbert's Grand Hotel and Al Ghazali's infinite regress of celestial motion reveal absurdities in the concept of the infinite. I show that the proposed absurdities are actually the result of his misconceptions, and that we can easily resolve the issues which he questions by using the Hyperreals.

https://boxingpythagoras.com/2015/10...tand-infinity/
And
https://boxingpythagoras.com/2015/10...finity-part-2/
I appreciate your links. I'll give those articles a look and have something probably by the weekend. Dr. Craig and I have had a professor/student relationship for the past year, and we've dined together as a class. In one-on-one conversation, he's a really nice, down-to-earth, pleasant guy. Perhaps I could bring these things up and see what he thinks. He's currently working on a conceptual analysis of the Christian Atonement, but he's always willing to talk about these things. I'll have to wait until next Fall until he returns to campus, though!

My problem is that I haven't had much exposure to the way "infinity" is used in that particular project Dr. Craig had. So I have a lot of reading to do. But thank you for perhaps unearthing such misconceptions! I trust your allegation of "misconception" comes from a good, productive spirit, as evidenced by the tone of your posts.

10. Originally Posted by shunyadragon
Your articles are excellent! I actually do not consider WLC's bad math simply naive misconceptions. I consider it is a dishonest deceptive misuse of math.
Hmmm. I've known Dr. Craig for the past two semesters at Houston Baptist University, and have gotten to know the wonderful faculty and friends in that particular academic circle, and I can assure you that there's not the least hint - that I've discerned - of deception in any area of academic research and study. I realize I could be wrong, but I'm typically a good reader of character, and after many conversations with Dr. Craig, I can tell you the guy is a very nice, sincere person, who has been very caring toward me, and everyone in our class. I'm sorry you've gotten that impression, however. I really am. It's just hard to hear that said about someone I've known for a year who clearly evidences indicators of a stalwart character. But I don't want to sidetrack the issue from the concept of the "infinite" too much. I just wanted to tell you my perspective, and perhaps we can just agree to disagree.

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