On Thursday, February 27, 2025 at 3:24:18 PM UTC-7 Brent Meeker wrote:
On 2/26/2025 8:41 PM, Alan Grayson wrote:
Yes and yes. If the universe is infinite then the ratio of its size to
that of any finite subset is infinite, no matter how large or small the
subset is. Imagine the infinite set of the integers. Consider the finite
subset {0,1,2,3,4,5,6,7,8,9,10,...,1e12}. It's size is obviously 1e12.
Now shrink the universe by striking every tenth number. Your subset is now
{0,1,2,3,4,5,6,7,8,9,11,...,1e12-1} and it's size is 1e12-1. But the
universe is still infinite.
Brentc
I know enough about set theory to have easily generated what you write
above. But math isn't physics. If the finite observable universe converges
to a singularity, we have a hypothetical universe which is not physically
possible, whether finite or infinite. So I am not sure how we can
distingush between an infinite and finite universe. Set theory does not
help. AG
If you can grasp that, why can't you grasp Cantor's theory of infinite
sets. I and others have said over and over that the singularity is a
prediction of GR which assumes spacetime is a continuum. Quantum mechanics
almost certainly modifies the physics short of infinite density.
Brent
Cantor's theory of infinite sets is now referred to as Naive Set Theory,
ever since 1901 when Russell published his paradox. It is replaced by ZFC
set theory. I just meant that one doesn't need GR to predict a singularity
for a contracting universe. AG
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