I don't think you have a good grasp of expressing your problem in an object-oriented way. For example, this part
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class Person
{
private:
vector<string> this_name;
vector<int> this_number;
vector<string> this_address;
//... rest of code
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would seem to indicate that each person you would create would have a collection of names, a collection of numbers, and a collection of addresses. While I could see where a person could have multiple names, numbers, and addresses, is that really something you want to capture in this program? I would start off with having each person has only one name, one number, and one address.
And this part in main:
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int main()
{
Person Name;
Person Address;
Person PhoneNumer;
//...
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Is Name a person? Is Address a person? Is PhoneNumber a person? The way you've declared them here, then yes they are. But this object model doesn't seem to fit your problem. I would expect something more like:
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int main()
{
Person alice;
Person marty;
Person dave;
//...
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Then you can use your setters (or, better yet, use a constructor) to initialize each person's full name, phone number, and address.