This re-instated what passes for faith in humanity

[Thecla]

I can recall playground arguments that ultimately devolved into meaninglessness once infinity was allowed to be treated as a number of finite magnitude: "Uh-huh." "Nuh-huh." "Uh-huh, uh-huh." "Nuh-huh, nuh-huh, nuh-huh." "Uh-huh infinity!" "Nuh-huh infinity plus one!"

Well, for cardinal numbers "infinity+1" (or even "infinity+infinity") is the same as "infinity", wheras for ordinal numbers "infinity+1" really is a bigger "infinity", so I hope they were using ordinal numbers. ;) Of course, then there's infinity+2. On the other hand, 'the largest infinity' turns out to be a self-contradictory concept, but I doubt that would stop anyone on the playground ("largest infinity plus one").

I will grant, however, that if one defines "infinity" to mean something different, then naturally different conclusions will follow from that.

I certainly grant that the word "infinity" might be used in lots of different ways. But IMO all of the non-mathematical uses -- whatever their poetic, metaphysical, or religious value -- don't lend themselves well to rational argument.

The question was, if I recall, whether a set than contains some quantity of elements, whether infinite in magnitude or not, could exist that did not contain a first element?  "Thus, how if one claims a set of infinite items, can it not contain a first item?"

I came in after that, but since you ask (WARNING: feel free to read no further unless you wish)...it really depends on what you mean: The set of integers (positive or negative) has no first element -- if by that you mean a smallest element. On the other hand, you could imagine reording them as, for example, 0, -1,1,-2,2,-3,3... and then they would have a first element (0).

The set of all real numbers also has no smallest element, with its usual ordering (or even the set of real numbers 0 < x < 1). You could imagine reording them by picking one number at a time, say as

pi, sqrt{2}, -25.1010010001..., -9/7,..

until they're exhausted (an exhausting procedure requiring an uncountably infinite number of choices), and then they would have pi as a first element; furthermore every subset of the real numbers would also have a first element (e.g. - 9/7 would be the first element of the rational numbers). The assumption that you can so order any set, however big, is called the well-ordering principle, and it's rather controversial (being equivalent to the axiom of choice).

The point of contention that you drew was over "So what is infinite can be larger than what is infinite? [Isn't there] contradiction in stating that a thing may be larger than itself?" This point seems incidental to the question, and I don't see why, if contested, it may not be discarded without loss to the cogency of the argument.

It may well be discarded -- I simply couldn't stand to see such imprecision about mathematically related concepts without responding. ;)

Perhaps you'd be inclined to grant that if one accepts infinity to mean "immeasurably great, unlimited, boundless, endless" that the point of contradiction is valid? and, in general, that a thing may not be larger than itself?

I honestly don't know whether "immeasurably great, unlimited, boundless, endless" really defines a meaningful concept, or whether -- in whatever context this definition is supposed to apply ---the statement that "a thing may not be larger than itself" is tautological, true, false, or meaningless.