Thread: reformulate "there are infinitely many primes"

I was given this excersise as homework and to a certain extent i do understand what they are asking, but don't know where to start.

Consider "there are infinitely many primes". Find a suitable universe U of elements so that this theorem can be reformulated in the form : A (is a subset of) B


We were also told that we dont need to prove anything here.

Thank you for any suggestions.

I am not sure if this is the idea of the exercise, but if you formulate "there are infinitely many primes" as for some property P, then it can also be stated as .

Originally Posted by Arturo_026

I was given this excersise as homework and to a certain extent i do understand what they are asking, but don't know where to start.

Consider "there are infinitely many primes". Find a suitable universe U of elements so that this theorem can be reformulated in the form : A (is a subset of) B


We were also told that we dont need to prove anything here.

Thank you for any suggestions.

Would this homework be for a grade?

Originally Posted by Ackbeet

Would this homework be for a grade?

Not directly. We're graded more on effort, but mainly on midterms and final.

A simple set up I came up with (which was similar to his answer) was:

Let U be the Set of all Sets (even though it's understood that there's no set of all sets), and let B be the set of all infinite sets. Moreover, let A be a subset of this set but also having the condition of being the set of primes.

Therefore A (is subset of) B (is subset of) U. ... sorry for not writting the symbol.

Does this sound right?

Originally Posted by Arturo_026

Let U be the Set of all Sets (even though it's understood that there's no set of all sets), and let B be the set of all infinite sets. Moreover, let A be a subset of this set but also having the condition of being the set of primes.

Therefore A (is subset of) B (is subset of) U. ... sorry for not writting the symbol.

Why not learn to post in symbols? You can use LaTeX tags
[tex]A\subseteq B [/tex] gives

Originally Posted by Arturo_026

Let U be the Set of all Sets (even though it's understood that there's no set of all sets), and let B be the set of all infinite sets. Moreover, let A be a subset of this set but also having the condition of being the set of primes.

Therefore A (is subset of) B (is subset of) U.

It is sufficient to take U to be the powerset of natural numbers. However, with A and B defined as above, A is infinite iff , not iff .