I have never used infinity as anything more than unboundedness. — jgill
The inverse relation between points and continua is that the point is nothing and the continuum is something. — keystone
In terms of length, the point is exactly 0 and the line is some positive number. If you're talking about something of infinitesimal length, you are talking about some tiny line segment. If that's the case then you are only talking about continua. — keystone
Nothing cannot be used to measure something (0+0+0+0... always equals 0). Whereas something can be use to measure nothing (e.g. 5-5=0). There is an imbalance here in this relationship suggesting that continua are more fundamental. — keystone
The structure of a continuum is not defined by points, it is defined by an equation(s). Aside from being impossible, you would never try to provide an infinite list of points to completely describe a line (Cantor). You would just provide an equation - a finite string of characters to perfectly describe how points would emerge if cuts are made. — keystone
I don't understand how continua + equations are vague. If I say I'm thinking of a plot containing the curves x=0, y=0, and y=x^2 you know exactly what I'm thinking of. — keystone
The OP mentions Aristotle's distinction of actual vs. potential infinities. The Wikipedia page on the subject doesn't explain the difference between the two all that well. — Agent Smith
I find it hard to imagine that something (an n-dimensional continuum) can be constructed from nothing (0-dimensional points). — keystone
I do feel that there is a little bit of Cantor's nonsense implied in any view that supports actual infinities. — keystone
I've heard people say that the paradoxes entwined with actual infinities are beautifully mysterious...I just think they demonstrate the flaws of the concept of actual infinity. — keystone
Why can't we just say that pi is not a number? Instead, it is an algorithm — keystone
A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum. — keystone
I think I have a grasp of how real numbers play into accepted set theory — keystone
it is challenging for me to envision the existence of a set of all natural numbers. Without assuming its existence, accepted set theory doesn't get far off the ground. — keystone
∞ isn't and object — Agent Smith
Cantor lost his mind (theia mania) and spent his later years in a lunatic asylum for instance. These concepts & paradoxes of which there are many seem to have a deletorious effect on the brain/mind - constantly mulling over them may lead to a nervous breakdown. — Agent Smith
I'll leave you to discuss with the other experts. Good day. — Agent Smith
infinite object, something beyond our comprehension — keystone
logicism, the ideology that there is a single correct logical definition of a mathematical object — sime
I have nothing more to contribute. — Agent Smith
Yeah, you often sign out of a conversation with that snarky "Good day", while not ingesting a single bit of the information and explanation given to you. — TonesInDeepFreeze
you would never try to provide an infinite list of points to completely describe a line (Cantor) — keystone
maybe we need someone else, an arbiter — Agent Smith
you're too technical for my taste. — Agent Smith
I have never used infinity as anything more than unboundedness. — jgill
In Calculus 1 classes, there is not a concern that the subject be axiomatized. But if we are concerned with having the subject axiomatized, then the ordinary mathematical context is one in which there are infinite sets. — TonesInDeepFreeze
The aleph numbers differ from the infinity ( ∞ {\displaystyle \,\infty \,} {\displaystyle \,\infty \,}) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.
I don't want to dive into what happens at the Planck scale, — keystone
I'm not sure how your post relates to my quote. — jgill
It's just that conversations involving cardinalities beyond ℵ1 don't usually occur in classical or even much of modern analysis. — jgill
But maybe you didn't mean that you don't use those sets. But that you do use them, but you don't use the extended real line with its points of infinity? As instead you simply deploy the fact that the reals are unbounded? — TonesInDeepFreeze
(3) What is the difference you have in mind between classical and modern? Ordinary contemporary analysis is classical analysis. — TonesInDeepFreeze
Let's imagine a line where cuts have been made to mark all rational points (I don't believe this is possible, but let's go with it for now). I believe you cannot mark any more points on this line. If you throw a dart in between the rational points then you will hit an indivisible line segment. That is as discrete as it gets, and even then the line is securely continuous. — keystone
The cuts are 0-dimensional so they are illusions of convenience. If you throw a dart at the line you will always hit the line, never the cut. The cuts have measure 0 after all. — keystone
The basic set theoretic structure of the reals underlies almost everything I have done, but I haven't used infinity as a "point" — jgill
Classical means the tools of analysis like limits, differentiation and integration and all those entail. Nitty gritty. Actual specific results vs broad generalities. — jgill
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