Yep. Yet the limit is not something the sequence is chasing, but a property of the sequence as a whole...? — Banno

It depends on whether you are referring to a recursive sequence or to a choice-sequence.

A recursive-sequence is an algorithm for generating a sequence prefix of any finite length, where a limit refers to a convergence property of the algorithm, as opposed to referring to a property of any prefix that is generated by using the algorithm.

On the other hand, a choice-sequence S is an unfinishable sequence of choices that is both

- Dedekind finite - meaning we don't have an injection N --> S.
- Of unbounded length - meaning we don't have an injection S --> {0,1,2,..n} for any finite n.

Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm.

Unfortunately, ZFC grounded classical mathematics cannot formally recognize potentially infinite (live) sequences due to the axiom of Choice that "finishes" them. Hence there is a clash between common-sense mathematical intuition (i.e. intuitionism) on the one hand, that correctly thinks of infinite sequences as referring to either unfinishable processes or algorithms, versus the formal straight-jacket imposed by the timeless world of ZFC, that cannot express the notion of a live process approaching a limit.

By default, classical mathematics is implied when talking about calculus, and even though ZFC isn't explictly assumed in textbook discussions of calculus, the logic they appeal to when discussing logical concepts such as limits, is classical in which calculus proofs are inductive proofs, which aren't applicable when reasoning about choice sequences, whose proof-theory is coinductive.

(I've never read a textbook definition of a limit as a two-player game - but they nevertheless informally appeal to such games when encouraging students to rote learn - a short term pedegogical payoff leading to long-term confusion after the students forget the game-theoretic reasoning behind the proofs)

So I enjoy these chances to exercise my math muscles a bit more directly than usual, and I take deep offense at Metaphysician Undercover's repeated dismissal of mathematics as a tissue of lies, half-truths, and obfuscations. — Srap Tasmaner

I don't understand this feeling of offense. This is philosophy, and what we do is critical thinking, and therefore criticize. What I don't get, is that many people think it's acceptable, even warranted and expected, that we criticize metaphysical principles, yet some of the same people believe it's for some reason unacceptable, and offensive to criticize mathematical principles. Where is the consistency in this type of attitude?

What I apprehend here is that some people take mathematics as a sort of religion. So in the same way that some people get seriously offended when their "God" is criticized, some others get seriously offended when their "mathematics" is criticized.

Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm. — sime

This is what @Banno seems to be in denial of. The intent behind creating the infinite sequence, is to create an infinite sequence. This implies that the so-called "limit", as defined by Banno, is prior to the sequence, as a requirement for the creation of the sequence.

On the other hand, we could look at the infinite extension of pi, as an unintentional infinite sequence. Notice, that now there is no "limit". This exposes the nature of "the limit", it is a concept which serves the purpose of creating an infinite sequence. When an infinite sequence is created unintentionally, there is no "limit".

This leads to a question about the intentionality of the infinity which is the natural numbers. If this is an unintentional infinite sequence, we ought to assume that there is no limit. But if it is intentional, then there ought to be some sort of limit, as the source of its creation.

Grammar Psychology tricking so many here. :lol:

Infinity is a word, that presents the concept of "indefinite continuation" in terms of a beings and unity. Just like we think of the numbers 1 and 2 in terms of being and unity, these things in themselves... so obviously that there is infinitely many things in themselves between these two things in themselves... as grammatical objects these things in themselves are seen as limits. People will see "infinity" as the thing in itself (only 1 infinity), or they will see that infinite meta regress between two things in themselves (Zeno's paradox [infinite infinities]).

Infinity isn't a known truth in terms of indefinite continuation in reality.

It's only possible in meta.

Grammar Psychology tricking so many here. :lol: — DifferentiatingEgg

Care to remedy the confusion, per the unspoken goal and purpose of most philosophies, or merely take pot-shots from a place whose elevation and understanding I'd frankly question. :wink:

Was editing as you were typing.

But there ya go.

Grammar Psychology trick fuckin yall...