Worlds Apart: On the Possibility of An Actual Infinity

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Worlds Apart: On the Possibility of An Actual Infinity
OP - February 21, 2013, 03:44 PM

I found this interesting:

Quote
§1 The Kalam Cosmological Argument and the Finitude of the Past
Cosmological arguments attempt to prove the existence of God by appeal to the necessity of a first cause. Schematically, a cosmological argument will thus appear as:
(1) All contingent beings have a cause of existence.
(2) There can be no infinite causal chains.
(3) Therefore, there must be some non-contingent First Cause.
Cosmological arguments come in two species, depending on their justification of the second premiss. Non-temporal cosmological arguments, such as those of Aristotle and Aquinas, view causation as requiring explanatory or conceptual priority, and thus insist that there can be no infinite regresses in such priority. Temporal cosmological arguments, also called kalam cosmological arguments due to their historical roots in Islamic kalam philosophers such as Abu Yusuf Ya'qub b. Ishaq al-Kindi and Abu Ali al-Hussain ibn Sina, view causation as requiring temporal priority, and thus insist that there can be no infinite temporal regresses.1
The kalam cosmological argument thus requires some supporting argument showing the incoherence of an infinite temporal regress of causally related events. William Lane Craig, in "The Finitude of the Past and the Existence of God"2, attempts to provide such an argument:
(4) An actual infinite cannot exist.
(5) An infinite temporal regress of events is an actual infinite.
(6) Therefore an infinite temporal regress of events cannot exist. (9)
I will not be concerned here with the general status of cosmological arguments, kalam or otherwise, or with contesting Craig's assumption that an infinite past would (unlike an infinite future) constitute a problematic actual infinity. I am rather concerned with Craig's general working principle, embodied in (4) above, that actual infinities are impossible.

Full Paper:

https://webspace.utexas.edu/deverj/personal/papers/worlds.pdf
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Worlds Apart: On the Possibility of An Actual Infinity
Reply #1 - March 04, 2013, 07:42 PM

If

(4) An actual infinite cannot exist.
(5) An infinite temporal regress of events is an actual infinite.

are true, then:

- Time is discrete

Because if time were not discrete, it is always possible to find an infinite chain of causal events between two instants A and B.

Which also implies that:

- Instant velocity does not exist

Because:
Instant Velocity requires differentiability of time and space
Differentiability requires continuity
Continuity requires non-discreteness

If instant velocity does not exist, I am not sure how/why relativity works.

Do not look directly at the operational end of the device.
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Worlds Apart: On the Possibility of An Actual Infinity
Reply #2 - March 04, 2013, 07:50 PM

Example of an infinite chain of causal events in non-discrete time and space:

Somebody shoots an arrow and it travels from point X to point Y in 1 second.

In order for the arrow to reach its destination at time 1s, it must first reach a point somewhere in between at time 0.5s
So the event at 0.5s is a cause for the event at 1s.
But In order for the arrow to reach its position at time 0.5s, it must first reach a point somewhere in between at time 0.25s.
So the event at 0.25s is a cause for the event at 0.5s.
If time "can always be split" (i.e. is not discrete), then you can go on indefinitely and have an infinite chain of events.

Edit: oops, I forgot an important "not".

Do not look directly at the operational end of the device.
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