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Paper II

God & Time I: God as Beginningless and Endless – God’s Knowledge of Infinite Numbers

Interacting with William Lane Craig’s model

In memory of Harry Leonard Fisher (1923-2008) to whom, although he did not believe in the actual infinite, I am indebted for my interest in it.

Anthony P Stone MA, DPhil (Oxon)

Version 3.3

ABSTRACT. An abstract actual infinite is demonstrated in a finite context, and a beginningless sequence of intervals of order type ω* is derived from it. With plausible assumptions about God’s knowledge and abilities, this sequence could represent God’s beginningless time up to some stage, contrary to W L Craig’s arguments. Taking God’s time to have no metric, there is no infinite past. I avoid speculation about the detailed structure of God’s time by describing it as a growing sequence of arbitrary intervals of little happening, to be named ‘kairons’. Kairons from some stage before creation are indexed by ordinal numbers, making it is possible to argue that Craig has no basis for asking, Why didn’t God create the world sooner?
It is proposed that God’s endless time is represented by the mathematical Absolute Infinite, which is not a number but a marker greater than all ordinal numbers. Following other writers, the basic postulate is that God’s time derives from God’s life. An easy explanation of ordinals is given in a linked file.

1 Introduction. 2 Beginningless. 3 Endlessly crossing horizons. 4 Concluding remarks. 5 Notes and references. § Easy explanation of ordinals

1. Introduction
In recent years, many writers have thought of God as having some sort of temporality. William Lane Craig has proposed a model in which God is timeless without creation and temporal with creation[1]. (‘Without creation’ is expressible in some models as ‘before creation’.) It is basic that:

A. God is beginningless and endless.
B. If God is temporal, then God’s beginninglessness can be passed through, but God’s endlessness cannot be passed through.

In my model, God’s time is ordered but has no metric (i.e. it is not measured)[a]. God is not ‘in time’, but God’s time is in God, as an expression of his life. There is no pressure of time if it is not measured, and neither is there any sense of delay[2]. (If God is temporal in this way, it does not amount to Open Theism, because there might be only one possible world which God is actualizing[3].) In paper II I show a model of the relation between God’s time and physical time. That is where the question of divine sovereignty and human freedom comes in. [omit:In a later paper I hope to deal with the concept of temporal metrics as discussed by various authors.]
 _ Craig is unwilling to accept the infinite as anything more than abstract mathematics. I shall give reasons to believe that God knows infinite mathematical objects (specifically, the endless sequence of infinite ordinal numbers). The ‘Absolute Infinite’ Ω is identified as a ‘marker’ rather than an ordinal.

2. Beginningless
Craig generally thinks of ‘time’ as measured. He believes that:

C. An infinite number of intervals of time, each with distinct endpoints, cannot have passed.

Under C, time and the physical world had a beginning. This implies that God has not existed for an infinite time although by A, God is beginningless[4]. The temporality of God then becomes a problem for Craig.
 _ Craig presents three points in favour of C, the first two of which were updated by Copan and Craig[5]. I shall argue that C is false for God’s time. Put briefly, Craig’s points are (1) An actual infinite is impossible, (2) An actually infinite past sequence cannot be formed by successive steps, (3) Why didn’t God create the world sooner?

Point (1). An actual infinite is impossible. Craig mostly argues against the existence of an actual infinite in the physical world. Those arguments therefore do not apply to God’s knowledge and activities. However, Copan and Craig do not accept the argument that the existence of abstract objects such as numbers implies that an actual infinite can exist. They argue that accepting the existence of such abstract objects requires that the truth of Platonism or realism be established first. Immediately after that they note that a way out is to attribute knowledge of abstract objects to God,[6] which is what I do (see below).  _ Copan and Craig earlier agree that infinite set theory and transfinite arithmetic is perfectly acceptable as abstract mathematics, but not in the physical world[7]. However, they still use one argument, connected with the idea of a ‘potential’ infinite, which would deny even the abstract existence of the actual infinite. A finite mind can think of the natural numbers 0, 1, 2, ... up to any value n, and know that these are followed by n + 1, ... . A finite mind cannot know the endless whole. Hence from the point of view of a finite mind, the natural numbers form a potential infinite. Craig has denied the existence of a corresponding actual infinite, even if the elements are abstract mathematical objects. This argument applies to pure mathematics. Copan and Craig explain the argument in this way, referring to selecting points in which to divide a line[8, edited]:

D. the property of being susceptible of division without end does not entail the property of being composed of an infinite number of points where division may be made.

Craig expresses this as a modal proposition[9]:

E. From the truth that “Possibly, there is some point at which a line is divided,” it does not follow that “There is some point at which the line is possibly divided.” The same is the case for numbers and counting.

Craig has also used this argument to deny God’s knowledge of the relevant actual infinites[10]. Proposition E denies two particular examples of the Barcan formula – which is, in fact, a theorem in the modal system S5 with quantification[11]. The Barcan formula causes difficulties when entities do not exist at all times. However, for God’s knowledge of actual, permanent, infinite classes which to a finite mind are only potentially infinite, I claim that the Barcan formula does apply.

I shall now demonstrate how transfinite ordinals can be brought into play. The infinite ordinal ω sometimes appears in a finite setting.

Fig.1. omega

Fig. 1 shows an endless sequence of rational numbers 0, 1/2, 2/3, ..., n/(n + 1), ... between the rational numbers 0, 1. These might be rational points in a geometrical line, or just a sequence of numbers.  _ Let the rational 0 have the ordinal 0; ½ have the ordinal 1; ...; n/(n + 1) have the ordinal n, and so on endlessly. This is not a temporal process of thought, but a permanently existing situation. What ordinal should the rational 1 have? If it is some finite ordinal m, then this is already taken by the rational m/(m + 1). Hence 1 must have the first transfinite ordinal, usually denoted by ω.
 _ Here the property of being endlessly divisible at values n/(n + 1) does lead to the property of having an actual infinite number of points with those values. This refutes D.
 _ One might also remark that D only states the negative result of not proving something. It does not disprove it. It may also be noted that further transfinite ordinals can be demonstrated in the same way.

Fig.2. omega2

Fig. 2 shows an extension of the previous sequence to similar numbers between the rationals 1, 2. Since 1 has the ordinal ω, the number 1½ next in the sequence has the ordinal ω + 1. Then the number 2 has to have the ordinal ω + ω. Hence these transfinite ordinals exist in the abstract. Ordinals are more suitable than cardinals for describing infinite sequences, because each element in a well-ordered set is indexed by a unique ordinal. Diagrams similar to Figs. 1, 2 were used by Rudy Rucker[12].

Point (2). An actually infinite past sequence cannot be formed by successive addition. This is equivalent to C for God’s past time. On the other hand, I postulate

F. If there is a mathematical construction for an infinite object, then God knows that object, as infinite.

The finite mind cannot know what it is like for God to know such things. Of course, F simply helps us to find out some of the things which God knows. I also postulate:

G. If God knows an infinite sequence, then God can perform successive actions indexed by all the members of that sequence (if it is appropriate for God to do so)[13].

For example, God knows the ordinal ω by F, so G tells us that God can perform endless actions indexed by 0, 1, 2, ... . Similarly, God can perform beginningless actions in the opposite direction, indexed by ..., 2, 1, 0. God also knows the ordinal ω + 1. Hence God can perform actions indexed by 0, 1, 2, ..., ω. It also follows from F that if God knows an infinite object, then God’s mind is (in simple language) at least as big as that object.
 _ After arguing that the world is not infinitely old, Craig considers the remaining two logical possibilities for the nature of God’s time without creation (before concluding that God is timeless in that situation): (I) ordered but not measured, (II) not ordered and not measured[14].
 _ With (I), if the instant a is temporally before b, and b before c, then Craig notes that even without a metric for time, two successive intervals ab, bc form an interval ac containing ab (using geometrical notation). Implicitly using the principle that ‘the smaller fits into the bigger’, Craig believes that ac is bigger than ab. In that case God’s beginningless time would have infinite length. The trouble with this argument is that if there is no metric, there is no ‘bigger’ in the sense of a longer length of time, and therefore no such thing as an infinite time is involved.
 _ Craig, however, moves on to (II), under which any interval of time is indistinguishable from a single point, and concludes that God is timeless without creation
 _ With the help of the sequence in Fig. 1, I can show that an infinite sequence of intervals can exist without any problem in God’s time, assumed to be ordered but not measured. Necessarily, God’s time is a sequence of some irreducible elements. A sequence of intervals of God’s beginningless time may then be indexed by some infinite ordinal τ going back into the past from some particular interval Ξ (such as that including the beginning of creation). The beginningless sequence of intervals up to Ξ then has order type τ*. By B, the infinite order type τ* of intervals must be such that God can pass through it. God can certainly pass through any one interval. Then by F, G, it enough if there is a mathematical construction for such an order type (or, from G, if God knows such an order type as an abstract object).
 _ I shall now show that we can construct an infinite sequence of intervals, of order type ω*, with distinct endpoints, but having no metric. This is shown as follows. Taken in the reverse direction, the infinite sequence in Fig. 1 is

... n/(n+1),... , 2/3, 1/2, 0.

This sequence is obviously order-isomorphic to the following sequence of half-open[b] intervals, where the elements are rational numbers:

... (bn, n/(n+1)], ... , (b2, 2/3], (b1, 1/2], (b0, 0].

We may choose the b’s so that the sequence is:

S:  _ ... ( (n+1)/(n+2), n/(n+1)], ..., (3/4, 2/3], (2/3, 1/2], (1/2, 0].

S has order type ω*, and the endpoints of every individual interval in S are different.
 _ Now consider a sequence of arbitrary elements a0, a1, a2, ... of order type ω, where a0 < a1 < a2 < ... , and there is no metric on the set of a’s. Then the mapping n/(n+1) ↔ an, (n = 0, 1, 2, ...), makes S order-isomorphic with the sequence T given by:

T:  _ ... , (an+1, an], ... , (a3, a2], (a2, a1], (a1, a0],

(This isomorphism applies to the endpoints of the intervals, irrespective of the elements within the intervals.) Every interval of T has distinct endpoints, with no gaps between the intervals, and no measure is involved.
 _ By F, God knows T, and then G shows that God can do what Craig denies – pass through an actually infinite beginningless sequence of intervals of God’s time up to creation, with order type ω* but without an infinite length of past time.

Description of God’s time
. Before dealing with Craig’s third point, I need to explain what description of God’s time I wish to use. God’s time may be seen as an ordered class of basic elements, but how should we describe them? We have no reason to assume that they are like the real numbers. Are these elements continuous; dense but not continuous (like the rational numbers); nowhere dense but not discrete (like Cantor’s tertiary set); discrete; or a mixture of these - or something different again? And what is their cardinality? - it could be very large; or different in different subintervals. Accepting Alister McGrath’s argument against unwarranted speculation in theology[15], we need a description of God’s time which is independent of all these possibilities.

Fig.3. kairons

Since God’s time arises from his ongoing life, I shall describe God’s time only up to God’s present, which has changing contents. There exist many abstract partitions of the elements of God’s time into sequences of half-open intervals, where each interval corresponds to relatively little happening. Before creation, any one of these partitions may be used to describe God’s time. Call its intervals kairons, after the Greek kairos, ‘time, opportunity’. Call the particular kairon still being completed, the active kairon. At some stage before creation, let the kairons already completed formed a beginningless sequence like T, of order type ω*, which we may index by:

... , -2, -1.

Index the active kairon at that stage by 0, and let the partition form a growing sequence, well-ordered at each stage. The development is shown in Fig. 3, where the kairons (completed and active) are indexed by the ordinals

0, 1, 2, ... , τ.

Then the instant of creation (or its beginning) was during some kairon of index τc, say. The beginningless sequence of kairons up to creation is:

..., -2, -1, 0, 1, 2, ... , τ.

This description of God’s time is not totally precise, but total precision would require knowledge we do not have. Kairons are merely intervals of God’s time, and their internal structure is not known.

Point (3). Why didn’t God create the world sooner? Craig calls God’s time ‘metaphysical time’ and takes it to be Newtonian absolute time, which has measure. His third point is that if God is temporal before creation, he delayed creating the world after an infinite time had elapsed, for which he should have had a good reason; but Craig denies that there is any such reason. “Outside God,” he says, is only metaphysical time with all its instants alike, so God had no reason to choose one rather than another. Then, after an infinite time, why did God wait? Again, in God “there seems to be no change” leading to creation at a particular time, and it does not seem possible “that God should acquire some reason to create” as metaphysical time advanced[16].
 _ I disagree with Craig on all these arguments. I take God’s time to be in God and arising from God’s life, so God’s time need not be empty of incident. God’s time is not measured, but each kairon is individually significant because it is distinguished by its own ordinal, whatever cardinality is involved. (If a different partition is employed, we still have kairons with their individual ordinals.) It was therefore not problematical for God to delay at any stage. God’s choice of the particular circumstances for creation could have been eternal. While we do not know God’s mind and the intercommunion of the Trinity, God could have acquired a reason for the choice through this[17].

Hence I deny that there is any problem with a beginningless past time for God, with God’s time being ordered but not measured. There is then no need for Craig’s postulate that God is timeless without creation. The finite age of the universe can be argued from astrophysics[18].
 _ Alan Padgett has presented a different model, in which God’s time before creation is not measured and has no change[19].

3. Endlessly crossing horizons
From 0 to ω. At first sight, the finite ordinals 0, 1, 2, ... appear to stretch to an unreachable and uncrossable horizon. We found, however, that beyond the finite ordinals is something different: the first transfinite ordinal ω. Then ω contains all the endless finite ordinals, is not itself finite, and any finite ordinal is less than ω.

From ω to Ω. The transfinite ordinals ω, ..., ω + ω, ... each constitute an horizon, and they go on endlessly to ordinals of larger and larger infinite cardinality. There cannot be any stopping point, because crossing each new horizon to an ordinal Θ leads on to Θ + 1, etc., as before. If there is a collection, it can be given a name and have a successor. This situation may be called absolute endlessness. There is no largest ordinal.
 _ Following Cantor[20], the symbol Ω is used to represent all the finite and transfinite ordinals, called the “Absolute Infinite”. ω represents the set of all finite ordinals by being identified with that set, thus being the first ordinal greater than any of them. Things work differently for the Absolute Infinite. Usually, the collection of all ordinals less than a certain ordinal α is α. If Ω is the collection of all ordinals, it is true that Ω is greater than all ordinals. We cannot, however, then call Ω itself an ordinal, because this leads to the Burali-Forti paradox: (the collection) Ω > Ω (an ordinal)[21]. Hence Ω is something different – which will now be identified.
 _ The situation is like using the Wallis symbol[22] ∞ for the “infinity” greater than all real numbers. The non-negative reals form the half-closed[c] set

[0, ) = {x : 0 ≤ x < ∞}.

Here ∞ is not a number, but the ordering of the numbers is extended to include ∞, as greater than all the reals. The symbol ∞ may therefore be called a marker. In the same way, Ω is a marker, not an ordinal but greater than all the ordinals. The ordering of ordinal numbers by class inclusion is extended to Ω. With this understanding the collection of all ordinals forms a half-closed interval of ordinals:

Ω = [0, Ω).

By F, God’s knowledge of infinite mathematical objects will then include all the ordinals within absolute endlessness, but known as endless (i.e., as if God knew all the finite ordinals within ω without knowing ω itself). G then entails that God is able to pass through ordinals endlessly without any of the previous limits, since absolute endlessness is not itself an ordinal of any kind. By B, God’s endlessness cannot be completely passed through. Karl Barth has warned us that God is not undestood by starting with human ideas.[22a] Hence I suggest that God’s endlessness may be represented by the absolute endlessness of the Absolute Infinite, but only for the purposes of the model.

4. Concluding remarks
This paper has shown that in God there can be beginningless and endless time, ordered and not measured. This paper also suggests that God knows the Absolute Infinite as absolute endlessness of infinite ordinal numbers.
 _ Timothy Pennings[23] has discussed the mathematical infinite in relation to theology. Robert Russell[24] has also explored the significance of the Absolute Infinite Ω for our understanding of God.

Thanks are due to Rudy Rucker and an anonymous referee of a related unpublished paper for alerting me to inconsistencies. Any remaining errors are my own.

5. Notes and References
[a] An example of an ordered sequence with no measure is a matrilineal line of descent without any dates.
[b] The half-open interval (a, b] is {x : a < x ≤ b}.
[c] The half-closed interval [a, b) is {x : a ≤ x < b}.

[1] William Lane Craig, Time and Eternity: Exploring God’s Relationship to Time (Wheaton, IL: Crossway, 2001); Craig, “Timelessness and Omnitemporality,” in Gregory E. Ganssle, ed., God & Time: Four Views (Downers Grove, IL: InterVarsity, 2001), pp. 129-160, and references there.
[2] Thomas F. Torrance, The Christian Doctrine of God, One Being Three Persons (Edinburgh: T & T Clark, 1996; repr. 2001), pp. 241-242, follows Barth’s concept of God’s dynamic uncreated time (not identical to my view). J. R. Lucas, The Future: An Essay on God, Temporality and Truth (Oxford: Blackwell, 1989), p. 213, says God is the source of time, and in A Treatise on Time and Space (London: Methuen, 1973), p.307, that for God, time does not press. Alan G. Padgett, God, Eternity, and the Nature of Time (New York: St Martin’s Press, 1992; repr., Eugene, OR: Wipf and Stock, 2000), p. 123, says that time depends on Gods life and does not press for God.
[3] E.g. Calvin says that “what God has determined must necessarily take place”. Calvin: Institutes of the Christian Religion, vol. 1. The Library of Christian Classics, vol. XX (Louisville, KY: Westminster John Knox Press, 1960), 1.16.9.
[4] Craig, Time and Eternity, p. 233.
[5] Ibid., pp. 220-233; Paul Copan and William Lane Craig, Creation out of Nothing: A Biblical, Philosophical, and Scientific Exploration (Grand Rapids, MI: Baker Academic, 2004), pp. 200-217.
[6] Copan and Craig, Creation, pp. 205-6.
[7] Ibid., p. 201.
[8] Ibid., p. 205.
[9] Craig, Time and Eternity, p. 225. The assertion that this modal operator shift “seems to be” invalid is also found in Copan and Craig, Creation, p. 205.
[10] Craig, “Time and Infinity,” International Philosophical Quarterly 31 (1991), pp. 387-401, at pp. 391-392.
[11] Menzel, Christopher, “Actualism”, The Stanford Encyclopedia of Philosophy (Spring 2008 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/spr2008/entries/actualism/. The Barcan formula is also called the Barcan schema.
[12] Rudy Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite (Boston: Birkäuser, 1982; repr. Princeton, NJ: Princeton University Press, 1995), p. 67. The edition published by Bantam has slightly different pagination.
[13] On God’s omnipotence, Richard Swinburne, The Christian God (Oxford: Clarendon Press, 1994) p. 130, says “whatever God can conceive, he can bring about”. Postulate F also agrees with Swinburne’s definition of God’s omniscience as “knowledge at each period of time ... of all propositions that it is logically possible that he entertain then and that, if entertained by God then, are true, and that it is logically possible for God to know then without the possibility of error”, Ibid., p.133.
[14] Craig, Time and Eternity, pp. 233-236. See also criticism by Padgett in Ganssle, God & Time, p. 127.
[15] Alister E. McGrath, A Scientific Theology, vol. 2 (Edinburgh: T & T Clark, 2002), pp. 268-279.
[16] Craig, Time and Eternity, pp. 46, 229-232.
[17] Others who differ from Craig here include Nicholas Wolterstorff, “Response to William Lane Craig,” in Ganssle, God & Time, p. 172; Garrett J. DeWeese, God and the Nature of Time (Aldershot, England: Ashgate, 2004), pp. 265-273; Dean W. Zimmerman, “God inside Time and before Creation,” in Gregory E. Ganssle and David M. Woodruff, eds., God and Time: Essays on the Divine Nature (New York: Oxford University Press, 2002), pp. 75-94. The Trinity is mentioned by DeWeese, Nature of Time, p. 271; DeWeese, “Atemporal, Sempiternal, or Ommitemporal: God’s Temporal Mode of Being,” in Ganssle and Woodruff, God and Time, p. 57; Zimmerman, “God inside Time,” pp. 78-79.
[18] E.g., Copan & Craig, Creation, pp. 219-248.
[19] Padgett, “Eternity as Relative Timelessness,” in Ganssle, God & Time, p. 109.
[20] Joseph Warren Dauben, Georg Cantor / His Mathematics and Philosophy of the Infinite. (Princeton, NJ: Princeton University Press, 1979), p. 243.
[21] Rucker, Infinity, pp. 223-224.
[22] John Wallis, Arithmetica infinitorum (Oxford, 1655, 1656).
[22a] Karl Barth, Church Dogmatics, II.1. (Study edn., London & New York: T&T Clark, 2009), p.465.
[23] Pennings, Timothy J. “Infinity and the Absolute: Insights into Our World, Our Faith, and Ourselves,” Christian Scholars Review 23 (1993), pp. 159-180.
[24] Robert J. Russell, “The God Who Infinitely Transcends Infinity,” in John Marks Templeton, ed., How Large is God? (Philadelphia, PA: Templeton Foundation Press, 1997), pp. 137-165.

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Version 3.2  (with new text in red Copyright (C) Anthony P. Stone 2008. This material may be freely used, provided the author is acknowledged.
Last updated: 9 March 2013