This re-instated what passes for faith in humanity

[wakim]

A set is an unordered collection of elements. To say that a set "begins" with something is contrary to the very notion of a set. To say that a set even has direction is also so much nonsense.
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A brief browsing through a couple mathematical text books at hand (granted incomplete in reference to set theory), a couple of dictionaries, and a brief Google search yield no definition of “set” that matches yours, in that a set necessarily be “an unordered collection of elements.” The R-H Dictionary, to offer a typical example, defines a mathematical set to be “[A] collection of objects or elements classed together.”

The “directions” I was referring to in the example {. . ., -2, -1, 0, 1, 2, 3, . . .} was from smaller to larger – I presume that you’ll grant that this set does indeed possess some elements that are smaller than others, and that it is ordered from left to right to display that. Further, isn’t it an essential property of infinity that it be infinite in regards to either increase or decrease of the property it describes? and thus that infinity must possess direction? For even if something is infinitely the same, how can it be so except in regard to an increase of time?

A set is unordered. A listing of them may have some "order" to it, but that's your doing.
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“If” begins a conditional sentence, as in my writing “ If one supposed an ordered set...”. If one doesn’t grant the premise supposed by the “if” then there no point to arguing against the “then”, as, in a conditional sentence, the “then” is predicated upon the “if”.

If I would accept your assertion that a set may only have “order,” not order, then I wonder how I would know whether a set is actually more than just a “set”? In other words, why is the classifying of elements into a group and calling it a set more than just “your doing”? In yet other words: The organization of elements into any given set is done to reflect some general classifying principle (ex. a group of all integers, a group of all flatware, a group of some interval, etc.). How do I know that the very act of organizing items into collections called sets isn’t equally as arbitrary as you claim the order of the elements in those sets must be? If the choice of items to include or exclude from a set is also arbitrary (thus the set has “organization”, not organization), then what one has isn’t a set, it is a “set”- a thing that is arbitrary and therefore reflective of nothing more than “your doing.” If a “set” is nothing more than “your doing,” then how can any conclusion be expected to be drawn from it that is other than only a reflection of “your doing”, including, therefore, the assertion that it is unordered?

The question of: “If a set contains an infinite number of integers, mustn't in contain every quantity of integers smaller than the infinite? In other words, if a set of 100 items were examined, mustn't it also be found to contain 99 items, and 98 items, and so on? If a set of 100 items did not contain a first item, how could it contain a second?”...

Integers can be negative. So, no.
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The question is about quantity, not value; how do negative integers pertain? In other words, doesn’t the set {100, 101, ...} contain elements that, if one were to enumerate them, would form the set {1,2,3,...}? just as the set {..., -101, -100} likewise would? or any similar set? Is it possible for a set that contains 100 elements, regardless of what the elements are, to not also contain 99 elements, 98, and so on? If I construe correctly your objection it would seem to rest on the claim that a set may contain a negative quantity of elements.

The concept seems sufficiently simple that I think a truly monstrous simplification exists (e.g. "I believe that effect has a cause, in much the same way that any listing of a set has a first element."), so if you could make a post like that, it'd be neat to read it.
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It appears from the summary that you offer of my argument that even given the daunting length that my previous post grew to, to my regret it failed to cogently convey my intent. While your proffered summary, that "I believe that effect has a cause, in much the same way that any listing of a set has a first element,” may be, and I have no cause to doubt you, a “neat” sentence to read, it isn’t my argument.