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The Concept of the Infinite

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  • The Concept of the Infinite

    I think there's a tendency to be so enamored with a philosophical argument that we tend to not treat certain concepts with as much rigor as they deserve. One concept that has always fascinated me (but which I wished I knew more) is that of the Infinite. The Infinite is most popularly invoked (in my experience) in the context of various cosmological arguments for God's existence. But what I notice is that there are objections as to how the Infinite is being applied in such contexts. So I'd like the purpose of this thread to be an exploration of this very interesting concept: the Infinite.

    Before I really get into this, I'd like to construct what is called a set-theoretical hierarchy of numbers, from zero to trans-finite numbers, as inspired by A.W. Moore's The Infinite (1990).

    - 0 - [Empty Set] (No sets above here)
    - 1 - [Set of one member]
    - 2 - [Set with two member]
    - 3 - [Set with three members]
    - 5 - [There are 65,536 sets above this point]
    - 6 - [The number of sets above here has 20,000 digits]
    -------------
    At this point, I need to use characters for the Greek alphabet, which I don't have. I'll use an English transliteration, but I apologize if this is cause for confusion. I'll try my best to be as clear as I can.

    - Omega = Aleph null - [Omega is the first infinite ordinal. Aleph null is first infinite cardinal: the set of all natural numbers. The set of all natural numbers is countably infinite. - The sets above this point are all finite.]

    - Omega + 1 - [The rules for addition are different for transfinite numbers. It is actually at this point where Hilbert Hotel becomes an issue. For brevity, I'll stop here.]

    - Omega X 2 - [The rules for multiplication are also different for transfinite numbers. Here and above is suppose to be the last point where it's not necessary to use sets.]

    - Omega2 - [These are limit ordinals, bigger than ordinals from infinite sets.]

    - OmegaOmega - [This is the least ordinal that's bigger than all the natural powers of omega.]

    - OmegaOmegaOmega - [It cannot be expressed by an infinite amount of natural powers of omega.]

    - Epsilon 0 - [Epsilon null. The first inaccessible ordinal.]

    - Aleph 1 - [a. First uncountable ordinal. b. Second infinite cardinal.]

    - Aleph Omega - [First cardinal that is preceded by an infinite amount of cardinals.]

    - Kappa is the first cardinal such that Kappa = Aleph Kappa

    - At this point, the axioms of Zermelo-Fraenkel set theory cannot be used to prove sets beyond this point.

    - Here are supposed to be inaccessible cardinals.

    As I said, I am a rank amateur when it comes to this concept, and I have probably made mistakes above. Any contribution or clarification is most welcome.

  • #2
    Technically speaking you're not doing any construction at all, you're listing a bunch of results without any technical steps involving in getting from one point to another.

    Comment


    • #3
      Originally posted by Leonhard View Post
      Technically speaking you're not doing any construction at all, you're listing a bunch of results without any technical steps involving in getting from one point to another.
      Fair enough. Construction was probably the wrong word. I admitted that much of this is beyond my ken. That's why I started the thread. If anyone could help connect the dots, I'd really appreciate it.

      Comment


      • #4
        Originally posted by mattdamore View Post
        At this point, I need to use characters for the Greek alphabet, which I don't have. I'll use an English transliteration, but I apologize if this is cause for confusion. I'll try my best to be as clear as I can.
        Most fonts support Greek and Hebrew characters, these days, but you'll have to use some extra manner of accessing them-- for example, the Charmap program in Windows or http://typegreek.com/

        Another option, and the one which I prefer, is to use LaTeX formatted images. This tool is very helpful in that regard: http://www.codecogs.com/latex/eqneditor.php

        Omega = Aleph null - [Omega is the first infinite ordinal. Aleph null is first infinite cardinal: the set of all natural numbers. The set of all natural numbers is countably infinite. - The sets above this point are all finite.]
        It is not quite true that . The cardinality of omega is Aleph null, but ordinals and cardinals are very different sorts of numbers. We can't just equate them in this way. For example, it is true that ; however, we know that . Saying that these two numbers equal one another would make our mathematics inconsistent.

        As I said, I am a rank amateur when it comes to this concept, and I have probably made mistakes above. Any contribution or clarification is most welcome.
        In general, it is useful to note the difference between ordinal and cardinal numbers. Ordinals, as their name implies, are a description of how elements of a set may be ordered. Cardinals, on the other hand, are a description of how the elements of one set can be mapped onto another. So, ω describes the first number which is ordinally greater than any Natural number, in transfinite arithmetic. On the other hand, represents the cardinality of any set which can be mapped with 1-to-1 correspondence onto the Natural numbers.
        "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
        --Thomas Bradwardine, De Continuo (c. 1325)

        Comment


        • #5
          Originally posted by Boxing Pythagoras View Post
          Most fonts support Greek and Hebrew characters, these days, but you'll have to use some extra manner of accessing them-- for example, the Charmap program in Windows or http://typegreek.com/

          Another option, and the one which I prefer, is to use LaTeX formatted images. This tool is very helpful in that regard: http://www.codecogs.com/latex/eqneditor.php
          Hello Boxing Pythagoras. I really appreciate those links. That will help tremendously.

          It is not quite true that . The cardinality of omega is Aleph null, but ordinals and cardinals are very different sorts of numbers. We can't just equate them in this way. For example, it is true that ; however, we know that . Saying that these two numbers equal one another would make our mathematics inconsistent.
          I understand. The way I explained myself was probably misleading. When I used the "=" symbol, I had meant it to mean the "is" in "the cardinality of omega is Aleph null." But I do understand that ordinals and cardinals are different sorts of numbers. Thank you for emphasizing that for me. One question, though, on your example. What is the "2" in ? Is it raising to the second power? In this case, is it the point that both have the same cardinality? If so, I see your point that because is the case, cardinality is distinct from ordinality.

          In general, it is useful to note the difference between ordinal and cardinal numbers. Ordinals, as their name implies, are a description of how elements of a set may be ordered. Cardinals, on the other hand, are a description of how the elements of one set can be mapped onto another. So, ω describes the first number which is ordinally greater than any Natural number, in transfinite arithmetic. On the other hand, represents the cardinality of any set which can be mapped with 1-to-1 correspondence onto the Natural numbers.
          Correct. Clear explanations. Ordinals (the name contains "ordin . . .") relate to order. ω "comes after" (ordinal"ly") the last finite cardinal, and is the first infinite cardinal. And because this cardinal is infinite, it can be put into a one-to-one correspondence with the natural numbers.

          P.S. I'm having trouble seeing where the Aleph Null is on the links you provided. Thank you for your insights! You seem to have a lot of knowledge of mathematics.

          Comment


          • #6
            Originally posted by mattdamore View Post
            Hello Boxing Pythagoras. I really appreciate those links. That will help tremendously.
            My pleasure, of course!

            One question, though, on your example. What is the "2" in ? Is it raising to the second power? In this case, is it the point that both have the same cardinality? If so, I see your point that because is the case, cardinality is distinct from ordinality.
            Yep, the superscripted 2's imply power operations on their respective numbers. So, the square of is equal to , while the square of ω is greater than ω.

            Correct. Clear explanations. Ordinals (the name contains "ordin . . .") relate to order. ω "comes after" (ordinal"ly") the last finite cardinal and is the first infinite cardinal.
            Two small corrections, here. Firstly, we're comparing infinite ordinals to finite ordinals in this case, not cardinals to cardinals-- probably just a minor typographical error, there. However, more importantly, there is no "last" finite ordinal. For any finite ordinal, m, it will always be true that there exists another finite ordinal, n, such that m<n. However, for any finite ordinal, n, it is always true that n<ω.

            And because this cardinal is infinite, it can be put into a one-to-one correspondence with the natural numbers.
            In set theory, the ordinals are defined as sets. The ordinal ω is the set which contains all the Natural numbers as its elements. Therefore, the cardinality of ω is the cardinality of the set of Natural numbers, that is .

            P.S. I'm having trouble seeing where the Aleph Null is on the links you provided.
            In LaTeX code, you can denote an Aleph symbol by typing \aleph, and the subscript is denoted by using an underscore. So, for , you would type \aleph _0

            Thank you for your insights! You seem to have a lot of knowledge of mathematics.
            I'm still woefully amateurish, but I do have a particular love for mathematics dealing with infinities.
            "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
            --Thomas Bradwardine, De Continuo (c. 1325)

            Comment


            • #7
              The problem of the 'concept of the infinite' from the perspective of being 'invoked in the context of various cosmological arguments for God's existence' is that the different 'concepts of infinity' are descriptive, as with all math, of our physical existence, and not definitive as to the limits and nature of our physical existence.

              This leads to the problem of apologetic cosmological arguments using the math of 'actual infinities' to define limits of our physical existence. The odd assertion that 'actual infinities' do not exist in the reality of our physical existence, and limit the 'potential infinity of our existence,' is in contradiction with fact that the math of 'actual infinities' is indeed used as part of the science 'tool box,' like all concepts, proofs and axioms, to describe aspects of our physical existence. The nature and application actual infinities sets have no relationship to the question of whether our physical existence is potentially infinite or not,

              This is true of axioms of Zermelo-Fraenkel set theory, which was developed to demonstrate a set theory that is free form paradox's such as Russell's Paradox. It is an important set theory, but one of many, and there are many versions and variations developed since. My question is; How do you propose to use Zermelo-Fraenkel set theory to develop your 'concept of infinities?'
              Last edited by shunyadragon; 02-18-2017, 08:46 AM.
              Glendower: I can call spirits from the vasty deep.
              Hotspur: Why, so can I, or so can any man;
              But will they come when you do call for them? Shakespeare’s Henry IV, Part 1, Act III:

              go with the flow the river knows . . .

              Frank

              I do not know, therefore everything is in pencil.

              Comment


              • #8
                Originally posted by shunyadragon View Post
                The odd assertion that 'actual infinities' do not exist in the reality of our physical existence, and limit the 'potential infinity of our existence,' is in contradiction with fact that the math of 'actual infinities' is indeed used as part of the science 'tool box,' like all concepts, proofs and axioms, to describe aspects of our physical existence.
                I don't think that Matt was attempting to make an assertion one way or the other about the reality of actual infinities. He was simply noting that he was brought to an interest in getting a better understanding of the concept of infinity due to the appearance of that concept in apologetics.

                This is true of axioms of Zermelo-Fraenkel set theory, which was developed to demonstrate a set theory that is free form paradox's such as Russell's Paradox. It is an important set theory, but one of many, and there are many versions and variations developed since. My question is; How do you propose to use Zermelo-Fraenkel set theory to develop your 'concept of infinities?'
                ZFC takes the existence of infinite sets axiomatically. After that, it's just a matter of discovering the properties of such sets.
                "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                --Thomas Bradwardine, De Continuo (c. 1325)

                Comment


                • #9
                  Infinity from our finite point of view can be never. Parallel lines never meet. Parallel lines meet at infinity. That is just looking at infinity from one aspect.
                  . . . the gospel of Christ: for it is the power of God unto salvation to every one that believeth; . . . -- Romans 1:16 KJV

                  . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3-4 KJV

                  Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1 KJV

                  Comment


                  • #10
                    Originally posted by 37818 View Post
                    Infinity from our finite point of view can be never. Parallel lines never meet. Parallel lines meet at infinity. That is just looking at infinity from one aspect.
                    "Never" is quite a different concept than is "infinity."

                    To say, "parallel lines never intersect," is not the equivalent of saying, "parallel lines intersect at infinity." These are, in fact, completely opposite statements.
                    "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                    --Thomas Bradwardine, De Continuo (c. 1325)

                    Comment


                    • #11
                      Originally posted by Boxing Pythagoras View Post
                      I don't think that Matt was attempting to make an assertion one way or the other about the reality of actual infinities. He was simply noting that he was brought to an interest in getting a better understanding of the concept of infinity due to the appearance of that concept in apologetics.
                      OK. No assumption on Matt's purpose, which was unclear.

                      ZFC takes the existence of infinite sets axiomatically. After that, it's just a matter of discovering the properties of such sets.
                      True, I was just trying get context of Matt's direction of discussion.
                      Glendower: I can call spirits from the vasty deep.
                      Hotspur: Why, so can I, or so can any man;
                      But will they come when you do call for them? Shakespeare’s Henry IV, Part 1, Act III:

                      go with the flow the river knows . . .

                      Frank

                      I do not know, therefore everything is in pencil.

                      Comment


                      • #12
                        Here's a really quick, rather imprecise summary of the idea underlying the infinite ordinals. Let's say that we are attempting to construct a set, but we currently have no elements to place in that set. We can still make meaningful reference to such a set-- and, indeed, we do. We call this the Empty Set, denoted as or { }.

                        Now, we want to create another set. Thankfully, we now have something we can place in this new set-- the Empty Set which we created before. So our new set is {}.

                        If we were to make another set, there are now more elements which we can place in it: the Empty Set, and the set containing the Empty Set, denoted {, {}}.

                        We can continue this process over and over. The sets which we create in this manner represent the Natural numbers. The Empty Set represents zero. The set containing zero is one. The set containing zero and one is two. Et cetera, et cetera. There is a very simple way, now, for us to create an ordering on the sets which we have created-- if one set is an element of another set, then the former is "less than" the latter.

                        Now, let's consider the set which contains all of the Natural numbers, ω. Since any Natural number, n, is an element of ω, it is clear that n<ω. Thus, we have created a number which is greater than any of the Natural numbers.
                        "[Mathematics] is the revealer of every genuine truth, for it knows every hidden secret, and bears the key to every subtlety of letters; whoever, then, has the effrontery to pursue physics while neglecting mathematics should know from the start he will never make his entry through the portals of wisdom."
                        --Thomas Bradwardine, De Continuo (c. 1325)

                        Comment


                        • #13
                          Originally posted by Boxing Pythagoras View Post
                          "Never" is quite a different concept than is "infinity."

                          To say, "parallel lines never intersect," is not the equivalent of saying, "parallel lines intersect at infinity." These are, in fact, completely opposite statements.
                          Yeah. It is an issue of language, and what one understands by the words being used. Both statements "never intersecting" and "intersecting at infinity" being understood to be true. Disallowing that is a matter of one's understanding of truth and a different understanding of infinity.
                          . . . the gospel of Christ: for it is the power of God unto salvation to every one that believeth; . . . -- Romans 1:16 KJV

                          . . . that Christ died for our sins according to the scriptures; And that he was buried, and that he rose again the third day according to the scriptures: . . . -- 1 Corinthians 15:3-4 KJV

                          Whosoever believeth that Jesus is the Christ is born of God: . . . -- 1 John 5:1 KJV

                          Comment


                          • #14
                            Originally posted by Boxing Pythagoras View Post
                            Yep, the superscripted 2's imply power operations on their respective numbers. So, the square of is equal to , while the square of ω is greater than ω.
                            Correct. I believe I follow this. Thank you.

                            Two small corrections, here. Firstly, we're comparing infinite ordinals to finite ordinals in this case, not cardinals to cardinals-- probably just a minor typographical error, there. However, more importantly, there is no "last" finite ordinal. For any finite ordinal, m, it will always be true that there exists another finite ordinal, n, such that m<n. However, for any finite ordinal, n, it is always true that n<ω.
                            Yes. I'm understanding the difference between comparing infinite ordinals with finite ordinals. For example, a finite ordinal might be the calendar week according to which Monday is the first day of the week, and so on. Thus, the ordinal number for the calendar week is 7. It seems to me (and please correct me if I'm wrong) that for any finite collection, the ordinal number and the cardinal number are the same. Thus, the calendar week's cardinal number is also 7.

                            On the other hand, an infinite cardinality and an infinite ordinality is odd to me. If infinite ordinality is defined in terms of sets, then the ordinality of the set of all natural numbers would be the same as its cardinality, correct? This would be true up until we added (+1), multiplied (*2) or raised the power of (X2), the infinite ordinality. The reason this happens is because addition, multiplication, and "squaring", introduce additional ordinality. Let me know if I have that right.


                            In set theory, the ordinals are defined as sets. The ordinal ω is the set which contains all the Natural numbers as its elements. Therefore, the cardinality of ω is the cardinality of the set of Natural numbers, that is .
                            Yes. I think I said that above. Let me know if I have this part of it understood. Thanks!

                            In LaTeX code, you can denote an Aleph symbol by typing \aleph, and the subscript is denoted by using an underscore. So, for , you would type \aleph _0
                            I appreciate it. I am not very tech-savvy so I'm still struggling with how to use the links exactly. I don't mean to bother you.

                            I'm still woefully amateurish, but I do have a particular love for mathematics dealing with infinities.
                            I'm relatively new to mathematics, but I do find it particularly arresting. It has a sort of petrified, symphonic quality about it that I wasn't aware of in high school.
                            Last edited by mattdamore; 02-19-2017, 02:08 PM.

                            Comment


                            • #15
                              Originally posted by shunyadragon View Post
                              The problem of the 'concept of the infinite' from the perspective of being 'invoked in the context of various cosmological arguments for God's existence' is that the different 'concepts of infinity' are descriptive, as with all math, of our physical existence, and not definitive as to the limits and nature of our physical existence.
                              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"? Does it mean that the concept of the infinite doesn't tell us exactly what the limits and nature of physical existence are? And if it doesn't tell us exactly, does it tell us something inexact about its limits and nature?

                              You do say, however, that the concept of the infinite (within the context of cosmological arguments, of course) are "descriptive" of our physical existence. I hadn't heard this before. Why do you think this?

                              This leads to the problem of apologetic cosmological arguments using the math of 'actual infinities' to define limits of our physical existence. The odd assertion that 'actual infinities' do not exist in the reality of our physical existence, and limit the 'potential infinity of our existence,' is in contradiction with fact that the math of 'actual infinities' is indeed used as part of the science 'tool box,' like all concepts, proofs and axioms, to describe aspects of our physical existence. The nature and application actual infinities sets have no relationship to the question of whether our physical existence is potentially infinite or not,
                              Hmmm. I'm having trouble following this. Your first proposition seems to imply that Cosmological Arguments commit the mistake of using an Actual Infinite as a Mathematical Concept to "define limits of our physical existence." I'm not sure what this means. I didn't want to get into the Cosmological Arguments per se. I did want to try and do a conceptual analysis of "the infinite", especially as delimited in mathematics. But your proposition does intrigue me. I'm only not sure exactly what it means. I don't know what it means to say that a mathematical concept defines the limits of physical existence, not to mention a mathematical concept involving "the infinite."

                              I'd rather hold off on your second proposition until we get a good hold on the concept itself. It appears as though Boxing Pythagoras is the knowledgable one in this respect. I will say that I also find it hard to comprehend the notion of an actual infinite "limiting the potential infinity of our existence." I would interact with this, and it seems as though you're communicating something important; I'm not confident, however, that my interaction would be productive because I can't understand your meaning.

                              And if you wouldn't mind, I'm also not sure what it means to say that "actual infinities" are a "part of the science 'tool box'". What aspect of our physical existence do scientists illuminate with actual infinities?

                              As for your last proposition, I'm not sure I follow the point. Is your main point that actual infinities are irrelevant to the question as to whether our physical existence is potentially infinite? If it is, that's fine. But for now, I was hoping that we could first perform a conceptual analysis of "the infinite", first.

                              This is true of axioms of Zermelo-Fraenkel set theory, which was developed to demonstrate a set theory that is free form paradox's such as Russell's Paradox. It is an important set theory, but one of many, and there are many versions and variations developed since. My question is; How do you propose to use Zermelo-Fraenkel set theory to develop your 'concept of infinities?'
                              Correct me if I'm wrong, but I believe that Zermelo-Fraenkel set theory defines an infinite set S as that kind of set with a proper subset P, according to which S and P have the same cardinality. Further, it is my understanding that Zermelo-Fraenkel set theory only provides a sort of abstract semantics for talking meaningfully about a purely abstract, mathematical universe of sets. It is my understanding that Zermelo-Fraenkel set theory was never meant to stand as a kind of abstract, geographical map meant to coordinate links between the map and the universe. But I could be wrong. I was actually hoping that we could pick Boxing Pythagoras' brain on this, since I think I may be getting this wrong. Thank you for your input!

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