Sets - Revision history

Lfox: Redirected page to Set

2024-01-21T01:57:04Z

Redirected page to Set

← Older revision		Revision as of 01:57, 21 January 2024
Line 1:		Line 1:
	~~A '''set''' is a collection of existents, considered as a~~ [[~~unit~~]].		#REDIRECT [[Set]]

	~~== Infinite sets ==~~
	Objective Mathematics says that infinite sets are an invalid [[notion]]. In short, this is because an infinite set does not refer to anything that can be perceived. The concept in Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

	~~=== But what about extended objects? ===~~
	There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

	To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

	We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.

	Note: Objective Mathematics does not use the notion of "the set of all points on O," because of the aforementioned subtlety. However, Objective Mathematics does use the notion of "''any'' point on O." Standard mathematics does not draw a sharp distinction between those two notions. See also the related article on [[the distinction between "any" and "all."]]

	~~=== But what about all the ''practical'' infinite sets? ===~~
	Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept\|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.

Lfox: /* Infinite Sets */

2024-01-20T04:25:39Z

Infinite Sets

← Older revision		Revision as of 04:25, 20 January 2024
Line 1:		Line 1:
	A '''set''' is a collection of existents, considered as a [[unit]].		A '''set''' is a collection of existents, considered as a [[unit]].

	== Infinite ~~Sets~~ ==		== Infinite sets ==
	Objective Mathematics says that infinite sets are an invalid [[notion]]. In short, this is because an infinite set does not refer to anything that can be perceived. The concept in Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].		Objective Mathematics says that infinite sets are an invalid [[notion]]. In short, this is because an infinite set does not refer to anything that can be perceived. The concept in Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

Lfox: Created page with "A '''set''' is a collection of existents, considered as a unit. == Infinite Sets == Objective Mathematics says that infinite sets are an invalid notion. In short, this is because an infinite set does not refer to anything that can be perceived. The concept in Objective Mathematics which is closest to that of an infinite set is that of a concept. === But what about extended objects? === There are an unlimited number of points on an extended object. It..."

2024-01-18T23:03:08Z

Created page with "A '''set''' is a collection of existents, considered as a unit. == Infinite Sets == Objective Mathematics says that infinite sets are an invalid notion. In short, this is because an infinite set does not refer to anything that can be perceived. The concept in Objective Mathematics which is closest to that of an infinite set is that of a concept. === But what about extended objects? === There are an unlimited number of points on an extended object. It..."

New page

A '''set''' is a collection of existents, considered as a [[unit]].

== Infinite Sets ==
Objective Mathematics says that infinite sets are an invalid [[notion]]. In short, this is because an infinite set does not refer to anything that can be perceived. The concept in Objective Mathematics which is closest to that of an infinite set is that of a [[concept]].

=== But what about extended objects? ===
There are an unlimited number of points on an [[extended object]]. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a [[point]]. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific ''point'' on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are ''physical'', i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many ''could'' exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.

Note: Objective Mathematics does not use the notion of "the set of all points on O," because of the aforementioned subtlety. However, Objective Mathematics does use the notion of "''any'' point on O." Standard mathematics does not draw a sharp distinction between those two notions. See also the related article on [[the distinction between "any" and "all."]]

=== But what about all the ''practical'' infinite sets? ===
Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all [[integers]], the set of all [[fractions]], infinite [[sequences]], etc. Objective Mathematics accepts that integers, fractions, and sequences are [[Concept|concepts]], and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.