Is This Series of Implications Logically Valid?

[Settia]

I have a series of things that imply the next thing in the series.

$$\cdots k_{2} \rightarrow k_{1} \rightarrow k_{0}$$

Is it logically valid to represent this series of implications as a contious series of subsets, like the following:

$$\{k_{0} , \{k_{1} , \{k_{2}, \{\cdots\}\}\}\}$$

Am I being too vague?

EDIT: Oops, I meant the title to be "Is this logically valid?"

 

[micromass]

For me, you're being to vague (: Maybe someone else sees what you mean.

Why exactly do you want to do this?

 

[Settia]

Technically I am not doing this for mathematics; instead its for a philosophy essay I am working on for my class. I am using set theory as a tool to help progress through the ideas. There is a point where I do what I have shown above, but I am uncertain if the set I showed follows from the series of implications.

 

[micromass]

Ah yes, I see what you mean. Well it is certainly logically valid to represent a series of implications by a series of inclusions... Although this is not really standard practice.

However, the notation

$$\{k_1,\{k_2,\{k_3,\hdots\}\}\}$$

doesn't really make sense to me. I would use the notation

$$\{x~\vert~k_1\}\supseteq \{x~\vert~k_2\}\supseteq \{x~\vert~k_3\}\supseteq ...$$

 

[Hurkyl]

Generally speaking, if you can write down any mathematical object, rules for extracting information from that object, and can demonstrate the properties of the object and rules reflect the thing you want to study, then you have a good representation for the object of study.

(of course, you have to be careful to use the representation properly. Representing something as a set, for example, doesn't mean that set-theoretic operation have any relevance to your object of study!)


That said, the typical set theories (ZFC and similar) forbid the kind of set you want to write -- the specific problem is that infinite chains of membership aren't guaranteed to exist, or even expressly forbidden.

It's more typical to represent an infinite sequence of things as, well, an infinite sequence. :tongue: A function whose domain is the natural numbers, and whose values are the individual terms of the sequence.

 

[JonF]

The representation you give doesn’t express anything about your series of implication.

Consider the relationship: rational being on Earth → man → mammal → doesn't lay eggs.
This is different from the set: {doesn't lay eggs,{mammal,{man,{rational being on Earth}}}}. This is just a set, with no relation connecting the objects in it.

I think what you are looking for is the relationship between http://en.wikipedia.org/wiki/Equivalence_class" . You could look at each implication as the antecedent partitioning the consequent and forming equivalence classes imbedded in each other.

and get something like: [rational being on Earth]₁ $$\subset$$ [man]₂ $$\subset$$ [mammal]₃ $$\subset$$ [doesn't lay eggs]₄

 

[Settia]

Thanks for everyone's replies.

I was concerned about expressing something that was not intended by representing the implications in the set as I asked. The reason for representing the implications as a set, is to examine the countability of the set as the sequence of implications changes. In the essay I am concerned with the sequence breaking up and the sequence is no longer countable, thus it is no longer a sequence.

I decided to take a different approach and ignore this idea. JonF, you are correct, I did want partitions. Both JonF and Hurkyl gave me some new ideas off topic from this thread.:tongue:

 

[Settia]

Thanks for everyone's help. I received an A on my essay. ^_^