The following is not a formal, finalized definition, but I hope sets the stage for the discussion — keystone
Finite object: Finite in the sense that its complete set of attributes can be fully described without invoking infinite processes. — keystone
Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous, and (2) points. A composite 1D object is the union of these fundamental objects and is continuous if, when duplicates are removed, the following conditions are met:
Points are connected to 0–2 curves (but not to other points).
Curves are connected to 0–2 points (but not to other curves).
No objects are disconnected from the composite structure. — keystone
Potential for arbitrarily fine partitioning: The continuum can be subdivided into an increasingly refined composite object made up of arbitrarily many fundamental elements, maintaining its continuity. — keystone
Characterized by the cardinality 2^aleph_0: The partitioning process can be described algorithmically, such that no algorithm can be devised allowing for further division. Although this algorithm would not halt if executed, the structure of the algorithm itself reveals that the potential for infinite subdivision aligns with the cardinality 2^aleph_0. — keystone
I have refrained from providing examples or illustrations for the sake of brevity, though they could help clarify my position. — keystone
A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii — TonesInDeepFreeze
Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous — keystone
the real numbers are defined as the continuum. They can be proven to exist within set theory, but that has no bearing on what's true in the real world. — fishfry
Some things obviously apply to the world. It is often said that there are no perfect shapes in the world. But we can mentally draw a perfect shape WITHIN any object although there it is surrounded by OTHER matter. The shape does exist as a part of another thing — Gregory
So we begin by defining such curves as "inherently continuous". That seems to solve the problem. Why proceed? Why dabble with sets of points that may fill up a curve - or not?
Just chop up such curves and there we are. Bend them a bit to go to two or more dimensions. — jgill
You asked me to define continua, which I assume was prompted by my earlier claim that "the continua" [...] — keystone
I haven’t yet reached the point of formalizing my ideas into a logical system. — keystone
However, my primary concern is the continua used in basic calculus — keystone
I wonder if delving into topology might be more of a distraction. — keystone
I also question whether your definition of a continuum is unnecessarily complicated. — keystone
For reference, even ChatGPT struggled to make sense of it — keystone
This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Here’s a breakdown of the key components:
Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:
T is a subset of the power set of X (denoted PX).
The whole set X and the empty set must be in T.
Any union of sets in T must also be in T.
Any finite intersection of sets in T must also be in T.
Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.
Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.
Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.
Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.
Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!" — ChatGPT
that math is rooted in actual infinities. I just don’t see how applied mathematics has any need for or use of actual infinities. I see an alternative. — keystone
In any case, I’ll provide the requested definitions in my next post. — keystone
I can tell you've taken care to read my words closely. — keystone
you've encountered far more actual continua — keystone
an unpartitioned continuum — keystone
How is that any less of a leap than starting with curves, which are inherently continuous? — keystone
This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Here’s a breakdown of the key components:
Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:
T is a subset of the power set of X (denoted PX).
The whole set X and the empty set must be in T.
Any union of sets in T must also be in T.
Any finite intersection of sets in T must also be in T.
Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.
Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.
Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.
Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.
Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!" — ChatGPT
(1) 'the continuum' is noun that names the set of real numbers. (2) 'is a continuum' is an adjective that we talking about defining. — TonesInDeepFreeze
Then that is mathematics. Or do you mean the study of physical phenomena that calculus is used for? — TonesInDeepFreeze
I can't make heads or tails of whatever it is you're trying to say....your notions are not of that basic kind but rather are intricate enough that they need more than ostensive definitions. — TonesInDeepFreeze
Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction? — TonesInDeepFreeze
A topological space C is a continuum if and only if C is compact, connected and Hausdorff. — TonesInDeepFreeze
When you posted a purported "proof" by ChatGPT, I even exactly showed you that ChatGPT doesn't know what it is doing*. * You didn't acknowledge that. — TonesInDeepFreeze
My definitions are very straightforward use of just the basic logical symbols and a few basic symbols of set theory. And if you don't understand the notation...then you can ask. — TonesInDeepFreeze
The original post in this thread purported to prove that the continuum does not exist. The argument wasn't that it doesn't exist in nature but that it doesn't exist mathematically. — TonesInDeepFreeze
Then I would be right to ask you to defined the undefined terminology in those definitions, and again until (1) You finally fail to reach primitives or (2) You end up in a circle or (3) You do reach primitives. — TonesInDeepFreeze
As I understand, you reject using infinite sets. But you say that we encounter continua. So continua are finite? — TonesInDeepFreeze
So, I don't know why ChatGPT "struggled to make sense of it" for you. — TonesInDeepFreeze
Ostensive indications and understandings not from explicit definition but rather from gleaning in context are fine and useful, mainly at the stage of basic intuitions. — TonesInDeepFreeze
I will be interested in what younger and more agile brains make of this. :chin: — jgill
I've never been able to see where it is you are going. Maybe it's just me, old and weary of days. — jgill
misinterpretation of their meaning — keystone
I think it makes sense to focus on the topics that are absolutely essential for understanding continua — keystone
I suspect I'd have to read at least one textbook to truly understand what you mean by "A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii." — keystone
you stated that 'is a continuum' is an adjective, which it isn’t. — keystone
my only qualm with set theory is the philosophy underlying it (centered around actual infinities). — keystone
Engineering claringly uses math as if it applies to reality. — Gregory
You seem to be saying there is nothing contradictory about continuums or that there would only be such only if they were in the real world. — Gregory
So then there is something about physical matter that in its properties is not entirely mathematical as we understand that. — Gregory
That may be true, although I would like to hear reasons why some day. Where do we draw the line when applying math to matter? How do we know we've gone too far? — Gregory
String theory vs loop quantum gravity. One has little points that are really strings (1 dimension in 0 dimension?) And the other discrete space. The biggest question in physics (quantum gravity) wants to settle the question of the continuum. They don't want to just throw their hands up — Gregory
using ChatGPT as an academic source is grounds for a warning. — fdrake
My ideas are half-baked and I'm not great at communicating them in a way digestible to mathematicians. — keystone
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