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A question about empty sets

Anonymous Fri 27 Mar 13:40:23 2020 No.11506313 View Reply Original Report

Quoted By:

If you have an empty set A, what can be said about this:

"For all x that are elements of A: [statement]".

Is the [statement] true or not?

I'm getting a bit fucked, because the first part is false.

Is it vacuously true?

Anonymous

Anonymous Fri 27 Mar 2020 13:50:57 No.11506330 Report

Quoted By: >>11506341

By the conventions of most logics, the full statement
>"For all x that are elements of A: [statement]"
is true.

>Is the [statement] true or not?
For the empty set (the situation you ask for), you can't instantiate any term an so there's no statement.

Example: Let X denote the empty set.
Informally, consider it the set of natural numbers (in set theoretical foundations, everything is a set, so there's only one type and this statement is rather redundant)
Now the following is true
>for all elements n in X, it's the case that n^2 < -7.

The above is true, but there's no term t in X such that you could even utter t^2 < -7.

You can't instantiate any element of X.
The statement which, by convention of vacuous statements, is true, only uses a bound (natural number) variable n to close the syntactic body (also called matrix) "n^2 < -7" that has a free position (index) "n".

Anonymous

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Anonymous Fri 27 Mar 2020 13:57:53 No.11506341 Report

Quoted By:

>>11506330
To may contrast this with a positive example, let Y be the set of all natural numbers bigger than 3 (formally, Y:={x | x>3})

Then one may prove
>for all n in Y, it's the case that n^5 > 2

Once proven, we're allowed to instantiate any element of Y and thus prove individual cases.
E.g. t=80 is in Y, so having the above theorem, we can instantiate
80^5 > 2
(we have now thus proven a simpler theorem, a special case that also has a proof that doesn't need the induction axiom. We could have just proven it by rewriting mutliplication and using m=m being true for all m by axiom)

But again, we can't instantiate any t in X. So there's no standalone
>[statement]
in that case

https://en.wikipedia.org/wiki/Universal_instantiation

Anonymous

Anonymous Fri 27 Mar 2020 22:08:14 No.11507297 Report

Quoted By:

foundations without empty set are better

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