Are Finite Sets and the Set of All Integers in R^2 Closed?

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Discussion Overview

The discussion revolves around the properties of finite sets and the set of all integers in R², specifically whether these sets are closed. Participants explore definitions and implications related to limit points and closed sets, engaging in a mix of mathematical reasoning and logical interpretation.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant asserts that the set of all integers has no limit points, suggesting that it is not closed based on the definition that a set is closed if all its limit points are contained within it.
  • Another participant counters that if a set has no limit points, it is vacuously true that all limit points are in the set, thus claiming it is closed.
  • A request for clarification on the term 'vacuously true' is made, leading to an explanation that such statements are considered true because they do not apply to any elements.
  • Further discussion touches on the distinction between mathematical conventions and logical interpretations, with one participant emphasizing that logical validity must be satisfied for certain conventions to hold.
  • Participants express frustration over technical issues affecting their ability to contribute lengthy posts, indicating a shared experience of challenges in online discussions.
  • One participant notes that the complement of both the integers and finite sets in R² is open, which may relate to their closed status.

Areas of Agreement / Disagreement

Participants express differing views on whether finite sets and the set of all integers are closed, with no consensus reached. The discussion includes competing interpretations of mathematical logic and the implications of having no limit points.

Contextual Notes

There are unresolved aspects regarding the definitions of closed sets and limit points, as well as the application of vacuous truths in different contexts. The discussion also reflects a mix of mathematical and logical reasoning that may not align uniformly.

jwqwerty
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i have one simple question

if we a consider subsets of R^2 which are: a finite set and set of all integers,
then aren't a finite set and set of all integers not closed? For instance for set of all integers, it do not have any limit points. thus by definition of closed (E is closed if all limit points of E are points of E) the set is not closed. Or did i do something wrong?
 
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Yes, but the sequence which would converge to certain limit in E point need to be composed from points that are in E. So the sequences you need to consider are composed entirely of integers.
 
If a set has no limit points, then it is vacuously true that all of the limit points are in the set, so the set is closed.
 
Citan Uzuki said:
If a set has no limit points, then it is vacuously true that all of the limit points are in the set, so the set is closed.

Sorry but could you tell me further about 'vacuously true'? I have never encountered that word. Thank you.
 
It means the statement is true because it doesn't actually say anything. If P is any property whatsoever, it is vacuously true that [itex]\forall x \in \emptyset, P(x)[/itex]. The reason why this statement is true is usually illustrated by looking at its negation -- if it were not true, then there would be some x in the empty set for which P(x) does not hold. But that is impossible, since there are no x in the empty set.

As applied to this particular problem, if E is a set with no limit points, then E will be closed, because for E not to be closed, there would have to be a limit point of E which is not in E. But there are no limit points of E, in E or otherwise. So E is closed.
 
Citan...nice try.

I cannot tell much about limit points and stuff...but i can tell about elementary mathematical logic...Ha.

I just took about 15 minutes typing slowly to show you why you cannot use 'that reasoning' here...

but when i hit 'submit reply'...guess what...Yeap...i lost all my hard work...i wasn't connected...typing is slow and quite frustrating for me.

I'll retype the situation again tomorrow.
 
jwqwerty said:
i have one simple question

if we a consider subsets of R^2 which are: a finite set and set of all integers,
then aren't a finite set and set of all integers not closed? For instance for set of all integers, it do not have any limit points. thus by definition of closed (E is closed if all limit points of E are points of E) the set is not closed. Or did i do something wrong?

The complement of the integers in R^2 is an open set. So is the complement of a finite set.
 
Again i just lost about 20min worth of hard typing.

i was connected when i started typing and by the time i had finished guess what?

i wasn't connected any more.
 
So i'll do an injustice to my post by just saying.

Citan is correct by a Mathematical convention about the use of the null set.

But, in Logic this convention's use is quite stricter...i.e...many things have to be satisfied before it can be used...unlike as in Mathematics.

Example...P(x) has to be a logically valid formula.

...etc...
In Logic the convention says that:

every interpretation of a formal language is a model of the empty set.Sorry for the Philosophy Detour guys.
Frege...Russell and The Boys...Ha.
 
Last edited:
  • #10
zmodnz said:
Again i just lost about 20min worth of hard typing.

i was connected when i started typing and by the time i had finished guess what?

i wasn't connected any more.

When you type a long post into a web form (on this or any other site), do a Copy before hitting Submit. That way when something goes wrong, you still have your post. After a while this will become second nature.
 
  • #11
zmodnz said:
So i'll do an injustice to my post by just saying.

Citan is correct by a Mathematical convention about the use of the null set.

But, in Logic this convention's use is quite stricter...i.e...many things have to be satisfied before it can be used...unlike as in Mathematics.

Example...P(x) has to be a logically valid formula.

...etc...



In Logic the convention says that:

every interpretation of a formal language is a model of the empty set.


Sorry for the Philosophy Detour guys.
Frege...Russell and The Boys...Ha.


Huh?? Vacuous truths are also "allowed" in "regular logic".. What is so philosophical about this?
 
  • #12
No, i wasn't using philosophical to describe 'vacuous truths.

I was using philosophical to describe what many many (almost all) modern mathematicians call Logic....i wonder if the Intuitionist 'believes in' the convention of vacuous truth...i wouldn't be surprised if they don't...ha.
 
  • #13
Oh yeah!

Thanks for reminding me that we can copy & paste, Steve.

I forgot about that.

Fancy that.
 

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