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Infinite Regress


Sargon

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I don't know if this is really on-topic to what you guys were really trying to get at. But in regard to the "Alpha-zero" and different 'classes' of infinity:

'0 + 1 = '0

'0 + 10000000000 = '0

'0 + '0 = '0

BUT...

'0 to the power of '0 - I believe - gives you '1. The 'next class' of infinity.

To think of the different classes of infinity in practical terms, consider:

A. The set of cardinal numbers { 0, 1, 2, 3, 4, 5 ...}

B. The set of decimal fractions between 0 and 1.

The set A may be infinite, but it is enumerable. i.e. you know where you start, and you know how to get to the next number from where you currently are.

But now consider set B. What is the first 'decimal fraction' in that set?

Is it 0.000001?

0.000000000000001?

0.00000000000000000000000000001?

What you end up with is an infinite regress in just determining what the first member of the set is! Let alone trying to work through the set.

As far as I understand it, this is why set B is a different class of 'infinity' than set A.

That sound right to you asb? Or am I pulling that out of my - ermm - ya know...

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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.

I am not sure if this is what you're getting at, but here's one possibility: "now" isn't really "now".

In other words, Craig and Copan are bothered by the question of how, in an additive series with no beginning, we arrived at "now". The answer could be that "now" is not, in fact, "now". There is no "now". The "you" of yesterday is just as certain as the "you" of today that it exists in the present, but, intuitively, both cannot be true. We might resolve this dilemma by suggesting that all points in time are equally present or, alternatively, that "the present" is a totally artificial construct. According to either of these views, time is not additive at all.

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I am not sure if this is what you're getting at, but here's one possibility: "now" isn't really "now".

In other words, Craig and Copan are bothered by the question of how, in an additive series with no beginning, we arrived at "now". The answer could be that "now" is not, in fact, "now". There is no "now". The "you" of yesterday is just as certain as the "you" of today that it exists in the present, but, intuitively, both cannot be true. We might resolve this dilemma by suggesting that all points in time are equally present or, alternatively, that "the present" is a totally artificial construct. According to either of these views, time is not additive at all.

Ostler distinguishes between an "actual infinite" and a "potential infinite" :

Those who adopt an A-theory of time maintain that there is a genuine distinction between past, present and future and that the past and future are not actual, but only the present moment is actual. On the other hand, those who adopt a B-theory of time maintain that the past, present and future are equally real or actual.

....

Perhaps C&C would maintain that since the past has been real, it has actually existed in reality, it constitutes an actual infinity of events even as regress. However, this assertion is ambiguous. An "actual infinity" can mean: (a) an infinite set of events that are all actual at once, or (cool.gif an infinite set of events some of which have been but are no longer actual. The infinite past does not constitute an infinity of actual events. In particular, the set of past events does not constitute a set of actual events because the past events are no longer actual (assuming an A-theory of time). This distinction becomes important when setting up stories that supposedly show that a beginningless reality is absurd. After all, if the universe (in the sense that it constitutes all that is) has always existed, then it constitutes a beginningless series but in actuality is not a regress of events.

I don't pretend to be up to par with the other participants in this discussion, but Ostler's interpretation of the nature of past events at least seems potentially possible to me. C&C demand that an "actual infinte" is impossible, and Ostler temporarily lends them the benefit of the doubt but then interprets the past as not being "actual", thereby making their argument ineffective.

There is something else that has been overlooked so far in the discussion. Ostler makes an argument that is much easier to comprehend, one that an ameatur like me can easily use. Craig and Copan attempt to illustrate the impossibility of an "actual infinite" by pointing out the absurdities that result from manipulating "Hilbert's Hotel", a fun and mind-blowing excerise. However, Ostler quickly points out that "Hilbert's Hotel" and the absurdities accompanying it cannot apply to the past. The following two paragraphs are in my opinion the best two of the first section of his article.

First, we may ask if the example of Hilbert's Hotel is really an analogy for the type of infinity involved in a beginningless past? The answer is that it is not. It does not mirror the infinite past. First, Hilbert's Hotel has a first room, a beginning term, that is followed by consecutive terms and therefore is a well-formed series. The infinite past is a not-well-formed infinite series. Second, the rooms in the Hotel all exist at once and are actual in the same moment. That is not true of the infinite past. Only the present moment is actual or ontologically real assuming an A-theory of time (which both C&C accept). Thus, the past events do not actually exist to be transposed and reordered as the story of Hilbert's Hotel requires. If Hilbert's Hotel were like the past, it would have only one room that has been occupied by an infinite number of guests in consecutive order. Further, the past cannot be jumbled around like the persons in Hilbert's Hotel for reasons quite unrelated to the problems of infinities - the past is fixed and unchangeable once it occurs. Year 351,067 B.C. cannot be exchanged for year 465,789 B.C. Thus, we cannot take away all of the odd years. We have the infinite series of past events just as they have occurred and in the very order they occurred and we cannot alter them in the way C&C suggest for Hilbert's Hotel to create a supposed absurdity.

For these reasons, the supposed "absurd" stories used by C&C to demonstrate that an actual infinite is absurd simply have no application to the type of infinite order involved in the past without a beginning. All of the supposedly absurd stories, like Hilbert's Hotel, or the Tristram Shandy autobiography,10 all depend critically upon properties of the order that the infinite past does not possess. The past events are not like an infinite number of guests in an infinite number of existing rooms all of which actually exist in the same moment that can be shuffled around and transposed and still maintain the same order of infinite numbers.

C&C's argument is based on the impossible absurdities that result from applying a false analogy to past events.

Sargon

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Hi Sargon,

The Hilbert's Hotel example may not apply to past events except in theory (though I'm not sure that's relevant), but it does apply to a regress of gods. One could treat all the "worlds" on which the various gods live as "hotel rooms" and arrive at the same result.

-Chris

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If you want a good book on how Cantor proved mathematically that there are infinitely many sizes of infinity check out journey through genius. I had to read that for my Math Analysis class. It goes through first that there are different sizes of infinity. Namely the rationals are the same size of infinity as the natural numbers and are denumerable. Denumerable means they are the same size as the natural numbers. Natural numbers being 1,2,3,4,5,6...

However it shifts when you get to the irrational numbers. The irrational numbers are not denumerable because they are a bigger size of infinity. There are actually infinitely many more irrational numbers between 0 and 1 than there are rational numbers. I've heard the comparison before in the book that if you look at the sky at night. If you let the stars be the amount of natural numbers the blackness would be the amount of irrational numbers.

He later goes on to show that there are infinitely many sizes of infinity. Anyway it is a good read so I'd recommend it to anyone who is mathematical minded.

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Hi Sargon,

The Hilbert's Hotel example may not apply to past events except in theory (though I'm not sure that's relevant), but it does apply to a regress of gods. One could treat all the "worlds" on which the various gods live as "hotel rooms" and arrive at the same result.

-Chris

There is also the question of whether it is possible that space could be infinite. I don't know that LDS doctrine necessarily requires that space be infinite, but if one considers an infinite chain of divine beings who have created "worlds without number", then I would guess at least some LDS think that space is infinite.

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A general question for those who are nonLDS and accept classical assumptions about the divine nature (absolute omnipotence among them):

Do you believe that God has the power to create an actually infinite number of anything (including infinite space)?

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Hi Sargon,

The Hilbert's Hotel example may not apply to past events except in theory (though I'm not sure that's relevant), but it does apply to a regress of gods. One could treat all the "worlds" on which the various gods live as "hotel rooms" and arrive at the same result.

-Chris

I disagree. Past events of any kind, be it rock-n-roll concerts or mortal lives on other planets, cannot be exchanged and moved around on the linear timeline. These are not presently "actual" events like the guests and rooms in Hilbert's Hotel. The past, according to C&C's A-model of past events (according to Ostler), is not "actual".

I step away for the weekend and another thread pops up taking over mine!! This is the kind of discussion MADB needs more often.

Sargon

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There is also the question of whether it is possible that space could be infinite. I don't know that LDS doctrine necessarily requires that space be infinite, but if one considers an infinite chain of divine beings who have created "worlds without number", then I would guess at least some LDS think that space is infinite.

I'm not astrophysicist, but I would suggest that yes, space is infinite. Space is quite difficult to define however, so don't ask me.

Asbestosman said:

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've toyed with it as well. IIRC, the prophet Joseph once compared eternity to a ring. The term "one eternal round" comes to mind.

Sargon

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There is also the question of whether it is possible that space could be infinite. I don't know that LDS doctrine necessarily requires that space be infinite, but if one considers an infinite chain of divine beings who have created "worlds without number", then I would guess at least some LDS think that space is infinite.

I consider "worlds without number" to be a phrase and not a mathematical fact. I don't assume there are uncountably many worlds, as it litereally would seem to imply, rather that the set of worlds is unbounded.

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I disagree. Past events of any kind, be it rock-n-roll concerts or mortal lives on other planets, cannot be exchanged and moved around on the linear timeline. These are not presently "actual" events like the guests and rooms in Hilbert's Hotel. The past, according to C&C's A-model of past events (according to Ostler), is not "actual".

Presumably the various deities and the worlds they live on still exist. So, as in Hilbert's Hotel, you could introduce an infinite number of new deities and introduce them into the no-vacancy "LDS Divine Worlds Hotel" and you'd still be able to find vacant worlds for them all.

While I don't think I subscribe to the A-model of time, I do think Ostler is missing the distinction within A-theory between eternalists, growing universe theorists, and presentists. Only presentists believe that the past is not real. The other two categories affirm the reality of the past, and eternalists also affirm the reality of the future.

-Chris

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Presumably the various deities and the worlds they live on still exist. So, as in Hilbert's Hotel, you could introduce an infinite number of new deities and introduce them into the no-vacancy "LDS Divine Worlds Hotel" and you'd still be able to find vacant worlds for them all.

While I don't think I subscribe to the A-model of time, I do think Ostler is missing the distinction within A-theory between eternalists, growing universe theorists, and presentists. Only presentists believe that the past is not real. The other two categories affirm the reality of the past, and eternalists also affirm the reality of the future.

-Chris

Chris,

Assuming the infinite model of deities subscribed to by many LDS, these persons would be existing in various places (worlds?) all simultaneously, comparable to Hilbert's Hotel. However, this exampe of an "actual" infinite is not an appropriate analogy to the past, as Ostler argues. The discussion revolves around whether or not an infinte regress of past events if possible, not whether or not an infinte 'amount' of objects is possible in the "now".

As I understand it the problem is summarized as follows:

1) Hilbert's Hotel describes an "actual infinite" of objects in the now, not in the past

2) Infinte Regress as used to argue against popular Mormonism involves past events, not current objects

I'm probably biting off more than I can chew with this!!

Sargon

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I don't know if this is really on-topic to what you guys were really trying to get at. But in regard to the "Alpha-zero" and different 'classes' of infinity:

'0 + 1 = '0

'0 + 10000000000 = '0

'0 + '0 = '0

BUT...

'0 to the power of '0 - I believe - gives you '1. The 'next class' of infinity.

I'm with you so far, but . . .

To think of the different classes of infinity in practical terms, consider:

A. The set of cardinal numbers { 0, 1, 2, 3, 4, 5 ...}

B. The set of decimal fractions between 0 and 1.

The set A may be infinite, but it is enumerable. i.e. you know where you start, and you know how to get to the next number from where you currently are.

But now consider set B. What is the first 'decimal fraction' in that set?

Is it 0.000001?

0.000000000000001?

0.00000000000000000000000000001?

What you end up with is an infinite regress in just determining what the first member of the set is! Let alone trying to work through the set.

As far as I understand it, this is why set B is a different class of 'infinity' than set A.

That sound right to you asb? Or am I pulling that out of my - ermm - ya know...

The set B is also enumerable. What set B doesn't have, however, is a smallest member if one orders the set in the usual way. So the set A has different properties than the set A even though both are the same size.

Yet, here is one way to enumerate set B:

Convert each element of set B into its fractional representation of N / D

To represent each as an integer K, let K = 2^N * 3^D

In fact, one can do this for all rational numbers. The set of rationals is countable. Oddly, so is the set of square roots of rationals.

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