'It seems', is the key expression. The proposition is indeed counter-intuitive. But that is not the same as 'logically impossible'.

Math is firmly based in logic. Given the assumption that each successive step takes half the time of the one before, an infinite sequence can be traversed in a finite time. That is a valid mathematical statement, therefore it is also logically valid.

Our intuitions are based on/derived from real world experience, including the experiences of our ancestors as encoded in the basic organization of our brains. So they are unreliable as indicators of logical possibility in this discussion, which goes beyond any obvious mapping into perceivable reality.

You are still laboring under Zeno's fallacy (also known, inaccurately, as a 'paradox' ). By that same argument, Achilles will never catch up with the tortoise.

EDIT: The thing you are missing is that the scenarios, whether as described by Zeno in slicing up the distance into ever smaller pieces, or your description of 'mincing' time slices, are artificial constructs, which don't have to match the constraints of reality or or intuitions.

There is no paradox here in logic or math terms. The sum of the infinite series Zeno's analysis generated is solved in high school math, and it is finite, and no calculus is involved. What is your problem?

The Greeks did not have the math tools that were developed later to solve such problems, and had this hang-up about infinite series, which you appear to share.

There are genuinely tricky problems involving infinities, but this is not one of them. It does not involve any regress issues, so I don't see the relevance of  much of your comment here. No recursive algorithm is required to solve it. Calculus is much more rigorous than you seem to be trying to imply.

You appear to keep conflating the mathematical handling of infinities and infinite series with the very real difficulties of trying to apply ideas of infinity to practical reality.

You 'traverse an infinity' every time you move a finite distance, since mathematically there are mathematically an actually infinite number of points in a finite line, but that doesn't seem to raise the same mental alarm bells.

Zeno's 'paradox' is about traversing a perfectly finite physical distance. Just because you can conceptually define an algorithm which notionally subdivides that finite distance into an infinite number of non-zero segments, does not point to any physical, non-mathematical, problem.

Insofar as there is some sort of 'paradox', it lies in that concept of an infinite number of non-zero segments adding to a finite sum. No 'traversing' involved or implied.

Thompson's Lamp is not directly about the sum of a series, although it partly relies on the same concept of an infinite series having a finite sum to allow an infinite number of operations to appear to be possible within a finite time, it is about the question of whether such an infinite sequence can be unambiguously even or odd. And of course the definition of the timing could well be modified to specify that, without compromising the other properties of the infinite sequence.

Hilbert's Hotel doesn't involve the sum of a series at all, it is about Cantor's idea that all 'transfinite' numbers of the same class being in some sense the same size, so that adding a quantity of the same or lower order to a transfinite number doesn't change its order, or 'size'.

From Wikipedia:

"Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an arbitrary sequence of rational numbers, as given objects."

Which seems to indicate that the natural numbers, a discrete set, is an actual infinity.

Quotes from here forward are from ubuntuAnyone.

"One cannot rationalize 2 without believing in a pragmatic sense 3..."

No, no, no.  One merely has to recognize that the probability of 3 being true is non-zero.  Then Pascal's "reasoning" leads to a belief in 2.

By your reasoning, Pascal's Wager would require believing in a pragmatic sense 1 in order to justify a belief in 1, which is just circular and not at all what he meant.

I can believe that God does not exist without believing that it is impossible for God to exist.  That's the stance taken by the counter-wager.

"How can anything other than a positive infinity if I have a positive probability greater than 0 and an infinite greater than 0?"

If the probability is effectively 1/infinity, or infinitesimal.

"Start counting all numbers iteratively between zero and one. Let me know when you're finished."

I can't halve my time with each iteration while counting.  I can with movement though.  I just passed all those numbers on my number line; in route to typing this sentence no less.  Zeno's "paradox" be damned.

Also, you are really confusing the term "never."  This statement really illustrates it:

"But even if I half the the time it takes to count something as the previous iteration, I'd never converge on the time. If the first iteration took one second, then the convergence would be two seconds, but this is not the case, because one would mince each time slice never actually reaching two seconds."

You start "mincing" the time slice, but then you say "never actually reaching two seconds."  This never is different from the never you used earlier when you claimed that you'd never be able to traverse the infinity.  When traversing the infinity, you appear to be claiming that you will never traverse it no matter how much time you use.  But in this case, you're talking about converging on a time, so that use of the word never doesn't make any sense.  You've jumped to using never to mean "never converge no matter how many iterations you perform," which is different from "never converge no matter how much time you use."  It is by jumping between these uses of the term "never" that you are reaching a false conclusion.  You reach a conclusion about iterations and then attempt to switch the meaning of never in order to apply that conclusion to time.

Traversing is only involved in the description, the specification of each scenario.

Iteration is NOT involved in the solution - otherwise one could not have a solution of such scenarios.

The clearest example of that is the sum of a geometric series:

The sum of a geometric series starting at A₀, where A_n+1 = A_n * r, is 

S = A₀ ( 1 - rⁿ⁺¹ ) / ( 1 - r )

Now this next step is as close as we get to 'traversing' the sequence, but actually uses "in the limit" style argument, as in calculus:

for any arbitrarily small number x, with r < 1,

there exists a value n such that rⁿ < x;

and if na > nb, r^na < r^nb,

which is the basic justification for the following statements:

For n = infinity and r  < 1 

rⁿ = 0

therefore

S_inf = A₀/( 1 - r)

No actual iteration or traversal required.

Iteration is only usable in programming when there is a clear terminating condition, which by definition is not the case in infinite sequences.

EDIT: sorry for this side-track....

Right... one can produce mathematical solutions without traversals...I never denied that. But as I've said, the mathematics are not doing traversals...they are abstracting the results. My concerns are not strictly mathematical as I've mentioned before such as justification for beliefs and cause and effect relationships.

Wow, BobSpence may have put it better than I could.

You seem to be hung up on "rationalizing 2 requires belief in 3," when it would only really require belief that 3 were possible.  I can believe that something is possible without believing that it's true.  Agnostics of both the theist and atheist type do this.

Infinity * infinitesimal.  The point is that the answer varies depending on the nature of the infinity and the infinitesimal.  Sometimes its zero, sometimes it's finite nonzero, sometimes its divergent.  Just because there's an infinity doesn't mean you're guaranteed a positive result because you could get zero.  That's the fallacy in Pascal's Wager.

"There is no consensus on whether or not transfinite sets are discrete or not..."

Do you mean that there's no consensus on whether or not the set of transfinite numbers is discrete or not?  I'm aware of that, and if IRC it goes deeper than that because it's part of the Incompleteness Theorem or something.

Anyway, it sounds like you might be saying that there's no consensus on whether or not any set with a transfinite cardinality (which would be any infinite set) is discrete or not, but I've never heard of any such lack of consensus when it comes to the rationals, integers, irrationals, real, or complex numbers.  In fact, demonstrating that the real numbers were not discrete was what led Cantor to the idea of cardinality in the first place.

I'm not talking about the set of transfinite numbers though.  I'm talking about Wikipedia using a universally-recognized discrete set (the rational numbers) as an example of an actual infinity, which you claimed needed to be continuous.  The set of transfinite numbers was never involved.

"If you read back over the discussion, I wrote about dismissing time altogether."

And that's exactly the problem.  Once you dismiss time, you can no longer conclude anything about time.  You say "never" as if Hercules will never catch the tortoise no matter much time he spends chasing it, but the whole point is that "never no matter how many iterations" doesn't logically imply "never no matter how much time."

I agree that no matter how many iterations you perform you will never reach the limit, but that does not mean that no matter how much time you spend you will never reach the limit.  That's the error in Zeno's supposed paradox, the error that makes it not really a paradox.

Pascal's wager is pragmatic justification for belief in a deity. In a pragmatic sense, one's belief is based on consequences. In effect, if one believes 2 because of the consequences of 3, then one is believing 3. The one thing the wager does not allow is for is agnosticm. Abstaining for a bet in a pragmatic sense is an implicit bet for 2. In other words, one is a garanteed loser if he or she does not bet. Proposing that God rewards those who don't bet and acting accordingly is a bet on such a belief, nevertheless.

Positive infinitesimal values (which infinitesimal probability would be) are greater than 0 and finite. That would mean that any such value multiplied times an infinity results in an infinity. Even an infintesimal probability for the existence of a god is warranted by an infinite payoff.

I think this whole thing has got muddled... Perhaps there needs to be clarification.

Time removed, one would never reach the end logically. That was the point. The paradox is that one could theoretically do an infinite task in a finite time, but the infinite task (that is an infinite number of iterations) itself cannot logically be completed. One cannot confirm one without violating the logic of the other and is why it's called a paradox. The mathematical solution is not the issue.