I can't speak to non-pointed set-theoretic probability theory. I know about ETCS (elementary theory of the category of sets) so I understand that sets don't require points. But as to probability, I can't say. However if you take finite line segments as sets, you seem to lose intersections. Are these closed or open segments? You have a reference for this interpretation of probability theory?
— fishfry
Let's look at Kolmogorov axioms here: (http://mathworld.wolfram.com/KolmogorovsAxioms.html)
Everything that is needed is a set W
W
, some Qi
Q
i
, that can be "anything", a function Qi
Q
i
from the Qi
Q
i
to real numbers, and a function "complement" on the Qi
Q
i
.
Let's consider as our probability space the segment [0, 1].
I can take for Qi
Q
i
the closed sets included in [0, 1] made of countable number of non overlapping segments with non zero length, and for W
W
the set of all these sets. The complement of a Qi
Q
i
will be the closure of the remaining part of [a, b] when I remove the Qi
Q
i
. There are no Qi
Q
i
of zero measure (and this is very reasonable for a probability theory: every event that can happen must have a non zero probability to happen).
The complement of a Qi
Q
i
overlaps with Qi
Q
i
only on the end points, and that is compatible with the axioms: the sum of measures adds up to 1.
The elements of W
W
are simply ordered pairs of real numbers instead of single real numbers, but everything works at the same way: from the point of view of set theory two segments are equal if and only if the extremes are equal: no mention of overlapping segments at all.
The definition of overlapping segments is the usual one: the higher number of the first pair is bigger than the lower number than the second pair.
There is no need to consider infinite sets of points, and for probability theory there is no need to speak about points at all: probability theory does not need zero-measure events, and no physical possible event has zero probability.
P.S. This is only a very simple example to show that it's not contradictory to define probability without points. Pointless topology is much more general than this and makes use of the concept of "locales" (https://en.wikipedia.org/wiki/Pointless_topology) — Mephist
↪fishfry Can’t a real number go on and on for infinity? Or is that an irrational number...? — Noah Te Stroete
I guess I got confused because you guys were talking about numbers as points, where I was thinking about zero-dimension points on a two-dimensional line. This stuff to me right now is very esoteric, as I don’t remember the terminology for the different kinds of numbers. I was always better at calculations like an engineer than I was so much interested in or ever had any exposure to theory. — Noah Te Stroete
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