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Is it coherent to suggest that it is possible to iterate, one-by-one, through every single item in an infinite set? Some have suggested that it is possible to iterate (or count) completely through an infinite set with no start (or lower bound), making infinite regress a genuinely possible reality.

Mathematically, is this possible?

I don't know much about math proofs, so the more basic (with the least symbols) you can keep your answer, the more likely I will understand and appreciate it.

Thank you very much!

ADDENDUM

When I write an infinite loop into my computer code, the code begins to execute and will never complete it's looping (unless it crashes or I stop it). I am wondering if it is mathematically possible, adding one to another, to ever arrive at the completion of an infinite set, like the set of negative integers, or will it continue permanently?

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    If the infinite set countable, i.e. in bijection with $\mathbb{N}$, then you can. For instance, if $X$ is an infinite set and if $f : \mathbb{N} \rightarrow X$ is a bijection, then $f(1), f(2), f(3), ..$ is an enumeration of $X$.2012-09-14
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    Sets a priori have no order on them whatever, so no set has a "lower bound"; that is, until you give them additional structure. It is an axiom that every set can be endowed with an order in which it has such a "lower bound".2012-09-14
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    I guess you need to familiarize yourself with the concepts of [countable sets](http://en.wikipedia.org/wiki/Countable_set) and [uncountable sets](http://en.wikipedia.org/wiki/Uncountable_set).2012-09-14
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    Using the axiom of choice, every set has an ordinal $\alpha$ which is in bijection with it. Let $f : \alpha \rightarrow X$ be such a bijection. $(f(\beta))_{\beta < \alpha}$ is an $\alpha$ enumeration of $X$. Although, I don't know if you would consider an enumeration of length $\omega_1$ to be "iterable".2012-09-14
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    I apologize for not being clear. I wasn't wondering if it could be mapped to N. I was wondering if it could be completely iterated through. It seems like iterating through an infinite set would necessarily be ongoing and a permanent task.2012-09-14
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    @Tim: that depends on what you mean by "iterated through". As suggested by William, assuming axiom of choice we can use well-ordering principle to "iterate through" an arbitrary set in the sense that we can use transfinite induction and/or recursion to do something with it or show something about it. But you certainly can't "iterate through" an infinite set in a finite number of steps.2012-09-14
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    I seems that completely iterating through seems to be a mapping from $\mathbb{N}$ to your set. When you iterate, there is a first thing, a second thing, third thing ... This is exactly what $f(1), f(2), f(3), ...$ means.2012-09-14
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    @Tim Are you thinking of things like mathematical induction, where we show that, where P(n) is a proposition about the number n, if P(1) is true and we show that P(k) implies P(k+1), then we've shown that P(n) holds for every n in \mathbb{N}? There is a sense in which that sort of a proof invokes the idea of iteration over an infinite set.2012-09-14
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    I was not. I realize it can be demonstrated mathematically that certain characteristics apply to every item in a set, without actually "touching" every item. I was wondering more if it were actually possible to "touch" every single item in an infinite set, one after another. Perhaps a physics or philosophy forum would be a better place to pursue this... Thanks everyone!2012-09-14

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