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Hypothetical proposition(of the form “if p, then q” – without asserting the truth ofporq):

If space-time S is not infinitely divisible, then space-time S cannot be infinite in extent.

Proof:

If space-time S is not infinitely divisible, then there exists a smallest possible increment of S, s. [The precise size of s does not matter; it only matters that it is finite.]

Since S is built from finite space-time components s, S requires an infinite amount of time (t∞) to become infinite in extent. [note that s is a.]space-timecomponent, not simply a component of a geometric space

At any time t, we can assert that t∞ has not yet been reached (since t∞ is infinite).

Therefore, at any time t we can assert that space-time S is not infinite in extent.

Footnote: the number of dimensions of S (or equivalently, s) does not matter; it only matters that at least one is a time dimension.

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# Non-Infinite Space-Time

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