Here is another logical question that I've been trying to solve for the last 3 days, and feeling terribly dumb about this.

In the following, the actual meaning of “metric space,” of “converge,” and of “subsequence” is not important. Consider the following definition: A metric space X is compact if every sequence on X has a subsequence that converges to some point of X. 1. Let X be a given metric space and S(X) denote the set of sequences on X. Introduce appropriate predicates to formalize symbolically the statement that X is compact.

Okay, so this is how I tried to tackle it: it is clear that the general form of the statement is going to be a conditional, of the form: "If every sequence on X has a subsequence..., then there exists a metric space X with property compactness".

My first difficulty is deciding on what the universe of discourse should be. One would think that 'the set of all metric spaces' should do the job (and it goes very well for the consequent: [∃X C(X)], where C(X) would be the predicate 'X has the property of compactness'), but when I look at the antecedent, the universe of discourse I seem to be needing there is something like "the set of all sequences". Then I imagine the antecedent would work like this: let CS(x): x is a (sub)sequence that converges on some point in X). Then:

[∀x∈ S(X) ∃x ∈ Cs(x))]

And connecting the two, [∀x∈ S(X) ∃x ∈ Cs(x))] → [∃X C(X)]

I don't feel too satisfied with this (and don't think you can use two universes of discourse even if you are using two variables). Can you have a hint on what I am doing wrong?

Thanks in advance for your time and words.

  • 1
    $\begingroup$ Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. $\endgroup$ – Shaun Apr 9 at 16:59
  • $\begingroup$ The usual way to formalize such statements is to use predicates instead of "universes" So your statement would look like "$\forall X,d$ if $(X,d)$ is a metric space then $(X,d)$ is compact $\iff \forall {x_n}$ (if ${x_n}$ is a sequence in $X$ then exists ${y_n}$ (${y_n}$ is a subsequence of ${x_n}$ and exists $x$ in $X$ such that ${y_n}$ converges to $x$)). Also, you can visit the logic room. chat.stackexchange.com/rooms/44058/logic $\endgroup$ – famesyasd Apr 9 at 17:09
  • $\begingroup$ Here, "is a metric space", "is compact", "is a sequence in", "is a subsequence of" are all predicates, i.e. properties $\endgroup$ – famesyasd Apr 9 at 17:11
  • $\begingroup$ Thanks for the comments! Yes, I know I have to work with predicates, but being aware of what the universe of discourse is would greatly facilitate, I feel, my deciding on the predicates and their relations. I am afraid I don't understand some of your explanation (what is d?). I think this exercise is meant to be relatively simple (that is, doable with just the basics of propositional logic) $\endgroup$ – Manuel Del Río Rodríguez Apr 9 at 21:24
  • $\begingroup$ You can always take the universe of discourse to be a set theory universe, and your predicates are definable on it. $\endgroup$ – spaceisdarkgreen Apr 9 at 23:20

Hint: Mathematicians never talk about "universes of discourse." In a formal mathematical statement with several quantifiers, every quantifier can be restricted to a different "universe of discourse" (usually a set), e.g. $\forall a, b \in X: \exists d\in R: \cdots$ where $X$ might be a topological space and $R$ the set of real numbers.


Okay. I think the answer should be like this: Given - Let X: "The set of all metric spaces"; let S(X): " The set of all sequences on X" Predicates: C(x): "x is compact"; Sub(y): "y is a subsequence on X"; Co(y): "y converges to some point of X", then:

[∀x∈S(X) ∃y: (y∈Sub(y) ∧ Co(y))] → [∃x: (x∈X ∧ C(x))]

And this would read as: "If, for all/any x that is a member of the set of all sequences on X there exists a y such that y is a member of the subset of subsequences on X and converges to some point of X, then there exists some x such that x belongs to the set of all metric spaces and has the property of being compact".

Would this be correct?


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