I'm simply saying that how I cannot see how the finite syntax of mathematical statements can represent a super-task so i cannot see why mathematics should recognise zeno's paradox. Surely the mathematical answer isn't to 'solve' the paradox but to reject it, by showing that it cannot be derived or represented unless there is an equivocation of

"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.

Of course, set theory invented the "axiom of infinity" to express the idea of "countable infinite sets". But there is a big difference between expressing the syntax of an idea vs the representing the semantics of the idea. 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.

So to my thinking, we can colloquially talk about the probability of infinite sequences as 'names' for our syntax and we can point to associated syntactical expressions of limits and mutter words like "measure zero". But philosophers shouldn't take those words literally.

"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

I have read what you have written in this thread up to this point, and I still don't see what difference you are getting at here.

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

What makes you think so? The mathematics that is usually thought of as relying on such notions - mathematical analysis, linear algebra, etc. - is extremely useful for describing the real world. One could make the argument that the same could be accomplished without recourse to infinities - that's what the finitist project is about. But whatever one thinks of the successes and the prospects of that project, it can't take away the fact that standard mathematics has many real-world applications.

I think the inquiry requires further demarcation to be answerable.

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.

It goes further than that. As it turns out, ∞ is ambiguous if you will. In fact, there are infinite different kinds of ∞, of all things.

If, on the other hand, we're talking our (physical) universe alone, then things become much more complicated. We don't know exactly what our universe is, let alone what's (physically) possible for our universe.

Infinite divisibility and having infinite segments can be interpreted in the abstract set theoretic universe. Its your subjective opinion that you dont accept the set theoretic interpretation as "representation".

From your finitist formalist physicalist standpoint you cant even speak about the original problem because concepts like everything or possibility or probability are similar to divisibility in a sense they are built on concepts of infinity and unobservable events.
In fact tossing a coin infinite times is a supertask therefore you shouldnt accept it.

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

I don't think that counterexample works. You defined a set that is composed solely of positive numbers. By definition, that excludes the possibility of any negative numbers. And the same goes for your other examples as well. 1 being in the set of negative numbers isn't a logical possibility due to 1 not being negative and by the same logic neither is π in the set of even numbers.

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.

Um, that's incorrect. There's nothing impermissible about time being infinitely divisible. Whether anything can be infinitely divided, well, I don't know. Space probably is infinitely divisible.

So many assumptions in these types of discussions.
Is the universe infinite?
Is the universe eternal?
Our current best guess is our universe had a beginning (big bang). Given that 90% of the mass of the universe consists of “dark matter” do we really know if the universe is infinite or eternal or does it recycle (expand and contract)?

Is the universe composed of a set of independent events? Not really, I would think given notions of causality.

Is space infinitely divisible? Not if you’re a fan of quantum qravity or just plain old quantum mechanics

Is time infinitely divisible? Not if you think time is just a derivative of change and change consists of a series of quantum events (or occasions of experience if you prefer process philosophy views).

Do mathematical equations represent reality? Not if you think maths model reality and are an idealized and abstracted concept.

So unless you can agree on a few first premises, we will all just be talking past each other and working from profoundly different initial assumptions.

You can always do the, “let us assume the universe is infinite and eternal, then would all possibilities not only occur but repeat in their exact configuration”, a version of “eternal recurrence” Nietzsche and others.

"Whoever thou mayest be, beloved stranger, whom I meet here for the first time, avail thyself of this happy hour and of the stillness around us, and above us, and let me tell thee something of the thought which has suddenly risen before me like a star which would fain shed down its rays upon thee and every one, as befits the nature of light. - Fellow man! Your whole life, like a sandglass, will always be reversed and will ever run out again, - a long minute of time will elapse until all those conditions out of which you were evolved return in the wheel of the cosmic process. And then you will find every pain and every pleasure, every friend and every enemy, every hope and every error, every blade of grass and every ray of sunshine once more, and the whole fabric of things which make up your life. This ring in which you are but a grain will glitter afresh forever. And in every one of these cycles of human life there will be one hour where, for the first time one man, and then many, will perceive the mighty thought of the eternal recurrence of all things:- and for mankind this is always the hour of Noon". Nietzsche

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In an infinite universe, can you tell me why any particular specific sequence of heads and tails won't eventually show up? — T Clark

The sequence in which every coin is heads.

If that shows up, then the sequence in which every coin is tails does not.

This argument is not dissimilar to the Diagonal argument.

It's clear from the diagonal argument that even in an infinite sequence not all possibilities will occur.

Zeno's paradox is resolved by differential calculus. Done and dusted.