"A line is infinitely divisible" which is a finitely describable definition of a rule
with
"A line has an infinite number of segments" which cannot be represented in our syntax. — sime
And I cannot think of a compelling reason to see the axiom of infinity is anything other than a meaningless syntactical rule for manipulating finite syntax that represents nothing and lacks real world application , with the possible exception of representing things that are not infinite. — sime
If "possible" means logically possible (or non-contradictory) alone, then no, not everything logically possible is bound to be the case. An analogy:
1. in an infinitude of numbers, there are every kinds of numbers
2. there are infinite whole positive numbers {1, 2, 3, ...}
3. therefore there are negative numbers among them (from 1)
4. contradiction, 1 is wrong (however intuitive it may seem)
Same argument for the negative whole numbers {..., -3, -2, -1} and 1, the even numbers {0, 2, 4, 6, ...} and π, etc. — jorndoe
As long as you insist on confusing math with physics, people are compelled to push back. Contemporary physics does not allow for infinite divisibility of matter or time. The question isn't even meaningful since there's a certain point past which we can't measure space or time. Math does allow infinite divisibility, but math isn't physics. I suspect you know this, and I'm not sure why you are pushing this line of argument.
In an infinite universe, can you tell me why any particular specific sequence of heads and tails won't eventually show up? — T Clark
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