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God & Time: Chronons and the AB-theory of time

Anthony P Stone MA, DPhil (Oxon)

Version 1.3, using the term AB-chronon
1. Introduction 2. God's time 3. Discrete orderings of the rationals 4. The rational AB-chronon _____
5. Physical time 6. The real AB-chronon. Other AB-chronons 7. Remarks 8. Notes and references

1. Introduction
This paper describes a model for physical time derived from God's time. It concentrates strictly on a particular structure for 'AB-chronons' as elements of physical time. In physical time, AB-chronons are internally tenseless and externally tensed. God's time is tensed. As one way of dealing with dense instants in physical time, it is assumed that the instants within an AB-chronon are actualized by God in a discrete order. Such orders are guaranteed for any set by Zermelo's theorem.[1] This is done chronon by chronon, and there is no assumption that AB-chronons are of equal duration. Other aspects of the model are presented briefly (hopefully they will obtain a fuller exposition in other papers).

In the title, 'the AB-theory' refers to any theory using a combination of tensed (or A-theory) and tenseless (or B-theory) time. The model described is therefore one particular AB-theory.

2. God's time
For many years I wanted to know what drives the advance of physical time. Eventually I came to hold that the cause is God's dynamic life. Boethius took a different view, defining God's timeless eternity by saying that

eternity is the perfect simultaneous possession of the whole of unbounded life.[2]

However, many writers have argued that God is not wholly atemporal,[3] and I take this in the following way. God's life is dynamic, beginningless and endless. It includes God's thinking, and the interaction of the three Persons of the Trinity. Contra Boethius, God does not have his life all at once, but as a growing sequence. This sequence constitutes God's time. For simplicity, I take the sequence to be totally ordered. It then gives tense to God's time with past, present, and future in the usual way.

Under this definition, God's time is relational: the changes in God's life produce change in God's time. I take such change to be the essence of time. If some sort of time has measure, that is secondary. I hold that God's time is best described without any measure. This is partly because if the instants of God's time (whatever form they take) have larger cardinality than the real numbers, then there is a problem in finding a large enough set of 'distances' on which addition is defined. In the model, physical time is related to God's time in a complicated way, as will be seen.

3. Discrete orderings of the rationals.
Suppose the time variable t is a rational number, and consider the interval 0 < t ≤ 1, denoted as usual by (0, 1]. The arithmetical ordering of t is dense, meaning that between any pair of rationals, for instance 1/2 = 4/8 and 3/2 = 6/8, there is another rational, such as 5/8.[4] There are, however, other orderings which are discrete, such as:

O1: 1/2, 2/2 = 1, 1/3, 2/3, (omit 3/3 = 1), 1/4, ..., 1/n, ..., (n-1)/n , ...,

where n is a natural number.[5]. The denominators of numbers in O1 are monotonically increasing. Suppose the rational t´ is arithmetically very near to a rational t which has a relatively small denominator. Then t - t´ has a very large denominator. t´ will also have a very large denominator, and will therefore be very far from t in the ordering O1.

4. The rational AB-chronon
Suppose physical time is like a dense interval of rational numbers, with each number corresponding to a single instant of physical time. Then after one instant there is no next instant. It therefore looks as if God cannot actualize states of the world one by one in succession. This is true for temporal succession, but not if God actualizes the states in a discrete order such as O1.

In this model, an AB-chronon is a very short time interval, such as Ψ = (a, a + 10-40], say, where a is rational. Then the rational variable q = (t - a)1040 is in (0, 1], and may be ordered by O1. The times are then

t = a + 10-40 q.

The proposal for this particular model of a rational AB-chronon is that God actualizes instantaneous states of the world (i.e., total states of affairs) for times t by taking q in the order O1 in God's time. A few stages are shown in Fig. 1. To create one AB-chronon, God actualizes an infinite number of states. Since infinite numbers are logically possible, God's omniscience and omnipotence as defined by Richard Swinburne[6] allow this.

Fig. 1. A rational AB-chronon tenseless in physical time: arithmetical order in black, order of creation in red.

A state of the world will include memories of the past. Hence when a state is actualized within a discrete order, there is no way for a sentient being to know whether those memories have actually been experienced or not.

It is also clear that in this scheme, all the states of the world in the AB-chronon Ψ need to be known to God. Hence there are no alternative actualizable states for an agent to choose between. Within Ψ, actualization of states has tense in God's time. Physical time within a AB-chronon is tenseless, in the sense that the happening of states is not in the temporal order of physical time (although it is in the temporal order of God's time). A completed AB-chronon may be divided into succesive subintervals under the earlier-later ordering; but this is not a past-future ordering in God's time, and hence not in physical time. Consequently, an AB-chronon is indivisible in advancing physical time.

After an AB-chronon is completed, all its states have been actualized. The extremely short duration of chronons makes this almost the same as the continuous actualization of states, although in reality it is different.

Rational AB-chronons may be based on different subintervals of rationals and employ different discrete orderings. It is important to note that Ψ has only used the values of rationals, and not required any metric.

5. Physical time
To have physical time constructed of AB-chronons allows it to be indexed by ordinal numbers (conveniently beginning with 0). Fig. 2 shows physical time from creation (with or without a first instant in the 0th chronon), and extending to the ordinal xi +1.[7]. AB-chronons are created in order, giving physical time its dynamic quality, which clearly derives from the dynamic of God's time. The sequence of AB-chronons is advancing, thus allowing for an open future. Hence physical time is tensed in the large.

Fig. 2. Physical time with AB-chronons, and their tensed ordinals.

There is no need for God to actualize states of the world continuously in God's time - if it is continuous. It is possible that there is an interval of God's time between successive elements of the discrete ordering O1 (or whatever ordering is in use).

The physical time derived in this model gives a universe awaiting scientific investigation. The result is thus a kind of low-level theory of a temporal world with an advancing universal present.[8] This is the world waiting for time measure and scientific theories to develop.

6. The real AB-chronon. Other AB-chronons.
If physical time is like an interval of the real numbers, this is dense and we do not know how to show a discrete ordering. But by Zermelo's theorem, a discrete ordering does exist. Hence AB-chronons may be considered to be created by God as before, and used in similar models of physical time.

If physical time is best described by some larger set than the real numbers, Zermelo's theorem again applies and a model may be constructed in the same way as before.

7. Remarks
Why should God actualize states successively? This may not be necessary, but it is a characteristic of this particular model.

If the whole of time is tenseless, then it is possible to give an interpretation of the world similar to the construction of a tenseless AB-chronon: sentient beings experience times, but not in the order of physical time.

The idea behind the tenseless AB-chronon occurred to me some time after a personal discussion on the B-theory of time with (as I recall) Professor Harvey Brown at the conference on Einstein, God and Time, The Clarendon Laboratory, Oxford, England, 12-15 September 2005, for which I am grateful. Errors of any sort in the present paper are my own.

8. Notes and references
[1] E.g., Waclaw Sierpinski, Cardinal and Ordinal Numbers (Warsaw: Panstwowe Wydawnictwo Naukowe, 1958), p. 411.

[2] Boethius, The Consolation of Philosophy, 5.6.4.

[3] See, e.g., the contributors to Gregory E. Ganssle, ed., God & Time: Four Views (Downers Grove, IL: InterVarsity, 2001); Gregory E. Ganssle and David M. Woodruff, eds., God and Time: Essays on the Divine Nature (Oxford: O.U.P., 2002).

[4] Hence there is an infinite set of rationals between them.

[5] The rational fractions in O1 are in their lowest terms. There are many such discrete orderings, because we can reposition any number of rationals and still have a discrete ordering.

[6] Richard Swinburne, The Coherence of Theism. (Oxford: Clarendon Press, revised edn 1993, pp. 165, 180f.

[7] For a universe of finite age, xi will be finite.

[8] In relativity theories, a universal present may be accommodated on the supposition that there is a suitable family of spacelike hypersurfaces. See Anthony P. Stone, "A Program Model of Becoming". Physics Essays 10 ((1997), pp. 150-63.

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Copyright (C) Anthony P. Stone 2008. This material may be freely used, provided the author is acknowledged.

Last updated: 25 July 2009