In the previous blog entry I posed the question, is God's omniscience about events (temporal events) infinite? It turns out it God's knowledge of events is infinite, given the following

(1) Before the creation of the universe there was no time, no events.

(2) The universe is finite.

(3) Time is a series of intervals, possibly the length of Plank time -- (or any small interval of time). This would mean between any two intervals of time, whether it be seconds, years, or millennia, there are a finite number of the smallest possible time intervals between those periods.

(4) In the new heavens and new earth, time continues to be Plank intervals, which continues forever.

We should make some definition of event:

(5) An event is a configuration in the universe that is associated with an individual time interval.

Now, there are some issues with defining events like this, but let's run with this for now.

To get a little formal, we have:

A set of time intervals, T, which contain the smallest consecutive intervals of time, t

A set of configurations, C, which contains configurations c

A set of events, E, where each of the elements, e

Premise: God's omniscient knowledge of events, which is contained in set E, is complete.

An observation. The set of events, E, does not have to have unique configurations for each association with a T (time) element. For instance, configuration c

From statement (4), the set T, time intervals, goes on forever.

Definition: Finite Set. One way to define a set A is finite if A is either empty or there is a way to map the elements of set A with a subset of the natural numbers = {1, 2, 3, ..., k}, where k is a specific number in the sequence of counting the numbers in the natural number set and each element of a of A uniquely maps to the numbers 1, 2, 3, ..., k.

Definition: Infinite Set. A set B is said to be infinite if the set is not finite.

So, let's say we have a set of 16 minute intervals -- this is just a goofy example -- of {16, 32, 48, 64}. We can map all the elements to the natural numbers this way:

16 maps to 1

32 maps to 2

48 maps to 3

64 maps to 4

In fact, we can have the rule for each element in the 16 minute interval set, divide it by 16, and that is the element in the natural number set. Because we found we can map the 16 minute interval set to the first four elements of the natural number set, the set is finite.

Our original set of time intervals, T, is infinite. This is because if T was finite, there would be a way to uniquely map all the elements of T to a subset of the natural numbers of size n. But that would violate that the set T goes on forever because it would stop at time corresponding to the natural number n.

What about the set of events, E? All of the events of E are of the form (c

Therefore, the set E is not finite, but infinite. Since God's knowledge of set E is complete, there is a unique one to one mapping of God's knowledge to the set of E, thus demonstrating that God's knowledge of events is infinite.

Notes:

The picture is from the Hubble website. It is the Hourglass Nebula.

The definition of infinity is based on one of the definitions from Elements of Set Theory, Second Edition, by Peter W. Zehna and Robert L. Johnson (1972, Allyn and Bacon, Inc, Boston, MA), pp. 101-107.

(1) Before the creation of the universe there was no time, no events.

(2) The universe is finite.

(3) Time is a series of intervals, possibly the length of Plank time -- (or any small interval of time). This would mean between any two intervals of time, whether it be seconds, years, or millennia, there are a finite number of the smallest possible time intervals between those periods.

(4) In the new heavens and new earth, time continues to be Plank intervals, which continues forever.

We should make some definition of event:

(5) An event is a configuration in the universe that is associated with an individual time interval.

Now, there are some issues with defining events like this, but let's run with this for now.

To get a little formal, we have:

A set of time intervals, T, which contain the smallest consecutive intervals of time, t

_{1}, t_{2}, t_{3}, ... . This corresponds to all the time intervals of statements (3) and (4), the time intervals since the creation of the universe on into the new heavens and earth.A set of configurations, C, which contains configurations c

_{1}, c_{2}, c_{3}, ... . A configuration is an arrangement of matter and energy in the universe. The set C contains all the configurations of matter and energy in the universe from its beginning, on into all the future, including the new heavens and earth.A set of events, E, where each of the elements, e

_{1}, e_{2}, e_{3}, ... is a set of pairs of configurations associated with times, such that e_{1}= (c_{1}, t_{1}), e_{2}= (c_{2}, t_{2}), e_{3}= (c_{3}, t_{3}), ... . This set E corresponds to all the events since the creation of the universe on into the future. Each time interval in T, t_{j}, corresponds with some configuration c_{k}in C to make the element (c_{k}, t_{j}) which is in the set of events E.Premise: God's omniscient knowledge of events, which is contained in set E, is complete.

An observation. The set of events, E, does not have to have unique configurations for each association with a T (time) element. For instance, configuration c

_{3134}can be associated with time t_{16034}and associated with time t_{920431}. So (c_{3134}, t_{16034}) and (c_{3134}, t_{920431}) can be valid members of the event set E.From statement (4), the set T, time intervals, goes on forever.

Definition: Finite Set. One way to define a set A is finite if A is either empty or there is a way to map the elements of set A with a subset of the natural numbers = {1, 2, 3, ..., k}, where k is a specific number in the sequence of counting the numbers in the natural number set and each element of a of A uniquely maps to the numbers 1, 2, 3, ..., k.

Definition: Infinite Set. A set B is said to be infinite if the set is not finite.

So, let's say we have a set of 16 minute intervals -- this is just a goofy example -- of {16, 32, 48, 64}. We can map all the elements to the natural numbers this way:

16 maps to 1

32 maps to 2

48 maps to 3

64 maps to 4

In fact, we can have the rule for each element in the 16 minute interval set, divide it by 16, and that is the element in the natural number set. Because we found we can map the 16 minute interval set to the first four elements of the natural number set, the set is finite.

Our original set of time intervals, T, is infinite. This is because if T was finite, there would be a way to uniquely map all the elements of T to a subset of the natural numbers of size n. But that would violate that the set T goes on forever because it would stop at time corresponding to the natural number n.

What about the set of events, E? All of the events of E are of the form (c

_{k}, t_{j}), where t_{j}is an element of T. We have constructed E in such a way that all the elements of T uniquely map into E. We saw that each element of T maps uniquely to the natural numbers in such a way that there is no number in the natural numbers that would be the last number of the mapping. We have a way to maps all the elements of E uniquely onto T, i.e., (c_{k}, t_{j}) in E maps to t_{j}in T, which in turn maps to k in the natural numbers. The set E cannot be finite, because if it were, there would be some number m, such that all the elements of E would map to 1, 2, 3, ..., m. But that would mean that all the elements of T would map uniquely in the same way, making it finite, which contradicts that we saw it was infinite.Therefore, the set E is not finite, but infinite. Since God's knowledge of set E is complete, there is a unique one to one mapping of God's knowledge to the set of E, thus demonstrating that God's knowledge of events is infinite.

Notes:

The picture is from the Hubble website. It is the Hourglass Nebula.

The definition of infinity is based on one of the definitions from Elements of Set Theory, Second Edition, by Peter W. Zehna and Robert L. Johnson (1972, Allyn and Bacon, Inc, Boston, MA), pp. 101-107.