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Infinite Regress


Sargon

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I have been slowly reading and digesting Ostler's essay rebutting the arguments made by Craig and Copan in "The New Mormon Challenge". It has been a good read so far, but not exactly one you should do in one sitting, not me at least.

http://www.fairlds.org/New_Mormon_Challenge/TNMC01.html

Ostler tries to explain that there is a difference between "infinite numbers" and "finite" or "inductive numbers". I recognize that there are differences, and some of those differences make sense to me. However, Ostler loses me with some of his explanations. I am hoping that someone here who has command of these concepts can help me better understand some of Ostlers' arguments.

I will start with only one small piece of the article at a time, and work slowly if there is someone willing and able to assist me. I'm sure others will be interested also. Thank you.

The most amazing difference between an inductive number and an infinite number is that the rules that apply to finite "numbers" do not apply to infinite "numbers." The word "number" is thus equivocal when used for infinite sets rather than finite sets. Consider the set of all cardinal numbers: 0, 1, 2, 3, 4, 5 . . . . This set has a first member but no last member. It is infinite. Cantor called the smallest of infinite cardinals '0 (aleph-zero). Now this "transfinite number" has some very different properties from finite inductive numbers. We can add or subtract 100,000 to or from '0 and it is the same transfinite number! That is, it still has the property of being an infinite set.

I'm not sure I understand the first sentence that I bolded. Is "aleph-zero" ('0) meant to be the ZERO (0), in the cardinal numbers listed immediately preceding that sentence (0, 1, 2, 3, 4, 5, ....) ?

If this is the case, I'm also not sure I understand what the second bolded portion means exactly. I hope there is an easier way to explain this concept then Ostler's.

Thank you!

Now, I'm off to take my wife to the ward Valentine's Day dance. I can guarantee you that Infinite Logic is the very last thing on her mind right now! :P;)

Sargon

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I have been slowly reading and digesting Ostler's essay rebutting the arguments made by Craig and Copan in "The New Mormon Challenge". It has been a good read so far, but not exactly one you should do in one sitting, not me at least.

http://www.fairlds.org/New_Mormon_Challenge/TNMC01.html

Ostler tries to explain that there is a difference between "infinite numbers" and "finite" or "inductive numbers". I recognize that there are differences, and some of those differences make sense to me. However, Ostler loses me with some of his explanations. I am hoping that someone here who has command of these concepts can help me better understand some of Ostlers' arguments.

I will start with only one small piece of the article at a time, and work slowly if there is someone willing and able to assist me. I'm sure others will be interested also. Thank you.

I'm not sure I understand the first sentence that I bolded. Is "aleph-zero" ('0) meant to be the ZERO (0), in the cardinal numbers listed immediately preceding that sentence (0, 1, 2, 3, 4, 5, ....) ?

If this is the case, I'm also not sure I understand what the second bolded portion means exactly. I hope there is an easier way to explain this concept then Ostler's.

Thank you!

Now, I'm off to take my wife to the ward Valentine's Day dance. I can guarantee you that Infinite Logic is the very last thing on her mind right now! :P;)

Sargon

Sargon--

It's mostly way over my head as well. Not that it will clarify much, I fear, but in set theory, alephs describe "countable" infinite sets vis-a-vis an unbounded, undefined progression forward along the number line (i.e., infinity).

I'm not sure I understand the first sentence that I bolded. Is "aleph-zero" ('0) meant to be the ZERO (0), in the cardinal numbers listed immediately preceding that sentence (0, 1, 2, 3, 4, 5, ....)

I don't think so. As I understand it, aleph-naught is not itself a "number," but the name of the set of the natural numbers. More specifically, aleph-naught just is the cardinality (or, size) of the set of natural numbers. Cantor was concerned to show that infinite sets can, theoretically, have differing cardinalities (or, sizes).

(Where's Tarski when you need him?)

I believe there was some response to Ostler's critique that may shed some light on the matter. I'll see if I can track it down.

CKS

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This from Ostler (and I think it clarifies his point):

The fallacy is that, as the mathematician Cantor has elegantly shown, not all infinite sets must be equal. Cantor bids us to consider two infinite but unequal sets, the set of all ordinal numbers and the set of all even numbers. The coherence of infinite sets that are unequal can be demonstrated by pairing members of each set in a one-to-one correspondence. Even though both sets are infinite, the set of even numbers is only half as large as the set of ordinal numbers.

Yes, theoretically.

But, Ostler's rebuttal hinges upon belief in an actual infinite, which concept, to me, just seems wrong.

At any rate, it seems a rather long, long way around the bend to counter the infinite-regress complaint.

One hopes some mathematical types (who also happen to be realists) will weigh in here.

CKS

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Transfinite numbers are counterintuitive. When Cantor is speaking of Cardinal numbers, he is speaking of the size of a set. The set, {3, 6, 10, 19} contains 4 numbers. Cantor proved that there are precisely as many odd integers as there are even and odd integers combined! The sets are of equal size even though the odd integers are a proper subset of all integers. If you're not confused about that, then something is wrong with you.

It's also a crazy math insult to say, "Yo momma is so big she's isomorphic to a proper subset of herself." (in other words, she is infinitely large because infinite sets are the only ones with this property).

I don't think that an actual infinite is a logical impossibility. That said, actual infinites do seem to work against parsimony. Also, since they are very counterintuitive it is often difficult to accept. That said I'm not sure why they'd be any more difficult to accept than, say, the Trinity which is another concept I find counter-intuitive (although not a logical imposibility so long as the relations are properly understood which I don't claim to have done).

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I don't think that an actual infinite is a logical impossibility. That said, actual infinites do seem to work against parsimony. Also, since they are very counterintuitive it is often difficult to accept. That said I'm not sure why they'd be any more difficult to accept than, say, the Trinity which is another concept I find counter-intuitive (although not a logical imposibility so long as the relations are properly understood which I don't claim to have done).

I believe Stephen Hawking has used infinity only when it cancels out in the final equation, and even then it's controversial. He, at least, seems convinced that actual infinities are impossible.

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I believe Stephen Hawking has used infinity only when it cancels out in the final equation, and even then it's controversial. He, at least, seems convinced that actual infinities are impossible.

IIRC, Stephen Hawking avoids infinities (or singularities) where he can because they often point to deficiencies in the mathematical models of physics. While Hawking has a model of the universe that does not extend into the infinite past (Hawking-Hartle model), I don't think he has discounted the possibility of multiple big bangs and crunches. I think he may be against it, but I don't think he finds it to be an impossibility. Rather, I seem to recall him speaking of horizons--overservational horizons or somesuch. We simply may not be able to say anything about what the universe was like before the big bang.

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Asbestosman, what do you think of Ostler's rebuttal in general (specifically the article that Sargon has brought up)?

That it's over my head too.

IIRC, he also made some arguments based on infinite sets that either extend into the infinite past (all negative integers), extend into the infinite future (all positive integers) or do both (all integers). I wasn't particularly convinced by those ideas because I don't recall anything in set theory making such distinctions because sets do not have an inherent order. That said, one can speak of ordering, but as I understand it, it is not a question of transfinite math.

In my opinion, the problem of infinite regress is not a mathematical problem. It is a logic problem. The only way to win that argument is to demonstrate that your opponents can in fact accept actual infinites. The only way for them to win is to demonstrate that infinite regresses are logically impossible. I do not believe they have done this rigorously, but I am not a logician. My naive understanding is that it comes down to arguing that an infinite regress just feels intuitively wrong. If you start in the infinite past, you will never reach a finite present (except after infinite time with no way to distinguish today from tomorrow in length of time from "the beginning"). But then if there is no beginning, you don't exactly start at the infinite past either--hency the intuitive problem which I'm sure I got wrong and cksalmon could probably point out where.

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It seems like we had a similar conversation (to the OP) here on MADB within the last year or so. Does that sound familiar?

I certainly do remember. As I recall, you were about to accept actual infinites as a logical possibility for God, but then I'd start arguing the other side to convince you that you didn't have to accept them after all. It was a good discussion that really stretched me.

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The fallacy is that, as the mathematician Cantor has elegantly shown, not all infinite sets must be equal. Cantor bids us to consider two infinite but unequal sets, the set of all ordinal numbers and the set of all even numbers. The coherence of infinite sets that are unequal can be demonstrated by pairing members of each set in a one-to-one correspondence. Even though both sets are infinite, the set of even numbers is only half as large as the set of ordinal numbers.

Per Cantor:

the set of all ordinal numbers

{1, 2, 3, 4, 5, 6,...}

extended into infinity is twice as large as the set of all even numbers

{2, 4, 6, 8, 10, 12, ...}

Well and good.

But, here's my intuitive problem with Ostler's use of Cantor here in terms of an actual infinite (and it's certainly not original with me):

Let's exchange all ordinal numbers with apples and all even numbers with oranges.

Thus, we would have the cardinality of the set of all apples:

apple-1.jpg

to compare against the cardinality of the set of all oranges:

orange-1.jpg

Extending both sets infinitely forward, I can't for the life of me see how the two sets are not cardinally isomorphic.

CKS

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the set of all ordinal numbers

{1, 2, 3, 4, 5, 6,...}

extended into infinity is twice as large as the set of all even numbers

{2, 4, 6, 8, 10, 12, ...}

Well and good.

I believe this is incorrect. The sets have the same "size" or cardinality (namely aleph-0). There are precisely as many ordinal numbers as there are positive even numbers despite the fact that the positive even numbers are a proper subset of the ordinal numbers.

Extending both sets infinitely forward, I can't for the life of me see how the two sets are not cardinally isomorphic.

I'm with you on this. Furthermore I don't see what point Ostler was trying to make by saying that one infinite set was twice as large as the other. I fail to see what it would have to do with establishing an actual infinite.

Maybe I misunderstand. Did Ostler make the argument that the set of ordinals is twice as large as the set of evens? I can't seem to find it in his paper.

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Maybe I misunderstand. Did Ostler make the argument that the set of ordinals is twice as large as the set of evens? I can't seem to find it in his paper.

I'm sure you didn't misunderstand. I'll see if I can find what I've misconstrued here.

CKS

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I believe this is incorrect. The sets have the same "size" or cardinality (namely aleph-0). There are precisely as many ordinal numbers as there are positive even numbers despite the fact that the positive even numbers are a proper subset of the ordinal numbers.

I'm with you on this. Furthermore I don't see what point Ostler was trying to make by saying that one infinite set was twice as large as the other. I fail to see what it would have to do with establishing an actual infinite.

Maybe I misunderstand. Did Ostler make the argument that the set of ordinals is twice as large as the set of evens? I can't seem to find it in his paper.

No, he did not. From his paper:

Now an infinite set is one whose proper subset can be put into a one-to-one correspondence with the whole of the set. Consider the set of count numbers and odd numbers:

Count Numbers: 1 2 3 4 5 6 7 8 ....

Odd Numbers: 1 3 5 7 9 11 13 15 ....

Note that no count numbers are left over, so no odd number is paired with more than one count number. There is a one-to-one correspondence. Thus, the sets of count numbers and of odd numbers are both infinite sets. However, take any finite collection of numbers:

{2, 4, 6, 8}

No proper subset of this set can be put into a one-to-one correspondence with the whole. Thus, the set is finite. The set {2, 6, 8} cannot be put into a one-to-one correspondence with the set; there will always be something remaining. Thus, we can adopt the following rules regarding finite and transfinite numbers:

R1 = A finite set has more members than any of its proper subsets

R2 = An infinite set does not have more members than any of its proper subsets and each member of an infinite set can therefore be placed into a one-to-one correspondence

For finite sets, the whole set is always "greater than" a subset consisting of only some but not all of the set's members. For an infinite set, the whole set is not "more than" a proper infinite subset consisting of only some but not all of the set's members.

It is imperative to see that transfinite sets have different properties than finite sets. Thus, when we use terms like "number" and "greater than" and "equal to," they mean something different for transfinite series than for finite series. In transfinite logic, for two transfinite sets to have the same number of members means that "the members of each infinite collection can be placed in a one-to-one correspondence." However, for two inductive numbers to be the equal means that they are "the same number."

As to what Ostler's point was, he explains that as well:

This fact is important because C&C attempt to exploit our intuitions about finite numbers and argue that it is absurd that infinite numbers do not act like finite numbers. Indeed, they refuse to accept the possibility of an actual infinite for the same reason that mathematicians so long refused to accept transfinite numbers - they do not obey the rules that apply to finite sets. But this difference between properties of finite numbers and transfinite numbers arises because transfinites numbers actually define properties of sets and not of individual members of sets as do inductive numbers. Sets often have different properties than their individual members. For example, a large crowd of people is not the same as a crowd of large people.

Ostler argues that C&C's argument isn't valid because they try to apply properties that only belong to finite numbers to infinite sets.

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I believe this is incorrect. The sets have the same "size" or cardinality (namely aleph-0). There are precisely as many ordinal numbers as there are positive even numbers despite the fact that the positive even numbers are a proper subset of the ordinal numbers.

I see. I appreciate the clarification here, Ab.

Ostler's argument seems to me a paper tiger with regard to the infinite regress question. I truly don't see how mathematical possibilities translate into quantifiable real-world entities such as, say, years.

But, I'm no set theoretician and so I would do just as well to say I don't understand Ostler's argument. That's not his problem, I don't think.

CKS

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Ostler's argument seems to me a paper tiger with regard to the infinite regress question. I truly don't see how mathematical possibilities translate into quantifiable real-world entities such as, say, years.

I agree, but the flip side is to ask how we would then disprove the possibility of infinite real-world entities such as years. Qunatum mechanics is counterintuitive, but it has been rigorously supported with experiments. How would we do the same for an infinite regress to either prove or disprove it? I don't know, but I do know that intuition seems unlikely to be a very productive route. It also seems that a mathematical possibility alone would not be sufficient either.

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No, he did not. From his paper:

As to what Ostler's point was, he explains that as well:

I believe I must be thinking of something I read last night when I came across this thread. Chances are that I abused that source as well. I'm trying to track it down again. I defer to and agree with your and Ab's reading of Ostler here.

Ostler argues that C&C's argument isn't valid because they try to apply properties that only belong to finite numbers to infinite sets.

I keep running up against an intuitional block at this point. Why wouldn't we expect that a series of quantifiable, real-world entities, like years, to behave like quantifiable, real-world entities?

It seems as if Ostler is suggesting that we should expect that infinite set theory (a mathematical construct) can be more or less directly transposed into the register of actual entities, like years. Or, that if it can be proven theoretically on paper, we can assume that the truths of the mathematical construct actually obtain in the real world.

If the number of actual years, for instance, extends infinitely into the past, then I can't see how we ever arrived at AD 1975, the year I was born. If there is no Year-1, but merely an infinite regress of years stretching infinitely into the past, how did my birth year arrive?

I realize that's the sort of intuitive argument Ostler is seeking to unhinge, but, heck, I don't get it.

:P

CKS

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I agree, but the flip side is to ask how we would then disprove the possibility of infinite real-world entities such as years. Qunatum mechanics is counterintuitive, but it has been rigorously supported with experiments. How would we do the same for an infinite regress to either prove or disprove it? I don't know, but I do know that intuition seems unlikely to be a very productive route. It also seems that a mathematical possibility alone would not be sufficient either.

When we are discussing what is impossible it would seem that the burden of proof lies with the party that posits the impossibility. Ostler has simply shown that C&C's argument isn't bulletproof, thereby leaving open the possibility that actual infinites may exist. He doesn't show positively that actual infinites must exist, but he doesn't really have to.

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I agree, but the flip side is to ask how we would then disprove the possibility of infinite real-world entities such as years.

If one were to assume creatio ex nihilo, and then conveniently bracket questions about the "before" and "after" of creation, then one would not have to wrestle with the issue of the possibility of an actually-infinite number of real-world years.

If, if, if.

CKS

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One point to the infinite regress of gods I find interesting is the idea of one god functionally preceding another and thus the decendant god is functionally dependant on the antecedent god. This would seem to imply some sort of superiority of the antecedent god.

I've toyed with the idea of an infinitely long circle of gods where each is functionally dependent on all the others in the infinitely long circle. In such a scenario, each god is in some sense his own cause as well as the cause for all other gods.

I think it's a fun idea, but I'm not so sure it's how things really happened.

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I keep running up against an intuitional block at this point. Why wouldn't we expect that a series of quantifiable, real-world entities, like years, to behave like quantifiable, real-world entities?

It is difficult to wrap one's head around, but it seems to me that an infinite real-world entity (actual infinite, that is) is no longer quantifiable.

It seems as if Ostler is suggesting that we should expect that infinite set theory (a mathematical construct) can be more or less directly transposed into the register of actual entities, like years. Or, that if it can be proven theoretically on paper, we can assume that the truths of the mathematical construct actually obtain in the real world.

I think he's shooting for only the possibility that those truths may obtain.

If the number of actual years, for instance, extends infinitely into the past, then I can't see how we ever arrived at AD 1975, the year I was born. If there is no Year-1, but merely an infinite regress of years stretching infinitely into the past, how did my birth year arrive?

Ostler addresses this as well, and I agree with him. The trouble lies in the fact that our brains want to slap a beginning on the infinity, when it is actually beginningless. Your birth year arrived because there was no beginning - the years have always been ticking off.

Ostler begins to explain it this way:

C&C offer a second infinity argument which is even weaker than the first:

2.1 The temporal series of events is a collection formed by successive addition.

2.2 A collection formed by successive addition cannot be an actual infinite.

2.3 Therefore, the temporal series of events cannot be an actual infinite.

Premise 2.1 is not true of all temporal series and certainly is not true of a beginningless past series that terminates in the present. An actual past infinite collection is not "formed by successive addition," as if we could add a finite number of terms together and somehow they sum to an infinite number. Rather, for any term added to the past at any given point in the past, the past is already an infinite collection at that past time and therefore is not formed as an infinite collection by such addition. Indeed, C&C have assumed in premise 2.1 that the "temporal series" of past events has the same properties as a potential infinity rather than an actual infinity, for they assume that an infinity is open ended and completed by adding to it rather than being a completed infinity without a beginning term. One cannot form a collection by adding to it if the collection already exists before the addition. The infinite past already exists as an infinite temporal series before the addition of any term and therefore cannot be formed as an infinite temporal series by addition. I accept that one cannot by beginning with any one member of an infinite set complete an infinite set by successively adding new members to the set. But what follows from this is only that a set that has a first member, that is one that could be a "well-formed set ," cannot be completed as an actual infinite by successive addition. However, premise 2.1 is not true for "not-well-formed" sets that have no first member. The view that the past consists of an infinite series without beginning does not imply that it can become or "be formed as" an infinite collection by successive addition.

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When we are discussing what is impossible it would seem that the burden of proof lies with the party that posits the impossibility. Ostler has simply shown that C&C's argument isn't bulletproof, thereby leaving open the possibility that actual infinites may exist. He doesn't show positively that actual infinites must exist, but he doesn't really have to.

I agree that the burden of proof is theirs. I'm seaking more to settling the question of possibility once and for all.

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If one were to assume creatio ex nihilo, and then conveniently bracket questions about the "before" and "after" of creation, then one would not have to wrestle with the issue of the possibility of an actually-infinite number of real-world years.

If, if, if.

CKS

Again it is important to keep in mind that Ostler is not attacking the idea of creatio ex-nihilo, but rather the assertion of C&C that the creatio ex-materia posited by LDS doctrine is logically impossible. I think Ostler would be happy if we were simply to come to the conclusion that both ideas are logically possible... maybe.

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Ostler addresses this as well, and I agree with him. The trouble lies in the fact that our brains want to slap a beginning on the infinity, when it is actually beginningless. Your birth year arrived because there was no beginning - the years have always been ticking off.

Too rich for my blood tonight, guys.

I'm more than happy to acquiesce to two of my favorite MADB posters, though.

*studying*

Until we meet again,

Chris

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