What is Complementary Logic and its Role in Set Theory?

[Hurkyl]

I use open interval in its standard meaning, which is: {} and (__} are out of the scope of anything that exist between them.

That is not the standard meaning of "open interval".

 

[Organic]

Hi Hurkyl,

http://mathworld.wolfram.com/OpenInterval.html

Please tell me what is the difference between the above and what i wrote.

Thank you.

 

[Hurkyl]

{} and (__} are out of the scope of anything that exist between them.

Problem 1: "scope" is not a standard mathematical term.

Problem 2: you have not supplied the ordering required by the term "between".

Problem 3: this statement bears no resemblance to the definition of an open interval.


The definition is:

$$(a, b) := \{x | a < x \wedge x < b\}$$

(where $<$ is a total ordering)


So, by the standard definition:

$$(\{\}, \{\_\}) := \{x | \{\} < x \wedge x < \{\_\} \}$$

You tell me how what you said bears any resemblance to the definition.

Also note that a piece is still missing; the standard definition requires one to supply a total ordering, $<$. Please clarify what this ordering is.

 

[Organic]

Dear Hurkyl,

Please tell me how can I use this latex notations in my posts?

As you can see it is easy to write my idea by using standard way.

The total ordering is clearly shown in the first post of this thread.

Can you write it in a standard way?


Yours,


Organic

 

[master_coda]

Originally posted by Organic
The total ordering is clearly shown in the first post of this thread.

I certainly don't see anything about a total ordering in the first thread. You should clarify what ordering you want, instead of just instructing us to re-read your posts and pdfs.


As for LaTeX, there's a post that explains how to use it in the General Physics forum.

 

[phoenixthoth]

i think with what organic has in mind, the partial ordering is that induced by the subset relation for, i think he means {} to be the empty set at {__} the universal set so that for all sets x,
{}<x<{__}. however, since not all sets are comparable using subset relation, it's not a total order. however, perhaps this is a kind of weak (not meant in a bad way) open interval. it can be likened to a lattice with {} at the bottom and {__} at the top. i was briefly trying to do set theory this way but I'm not sure how to do the subsets axiom with meet ^ and join v in such a way as to remove russell's paradox.

 

[Organic]

Thank you Dear phoenixthoth, master_coda, Hurkyl an moshek.


I continue this thread here:


https://www.physicsforums.com/showthread.php?s=&threadid=11688


Organic