Does this type of classification exist? Im fairly new to data science and trying to see if a type of classification exists for my needs.

I understand that a classification into 2 categories will look something like this:



You have 2 desired outcomes and you try to build a model that classifies as 0 or 1. If these models are not 100% accurate then you will:

a) Miss some true values (outside edges of circles)
b) Get some of the wrong values in each category (overlaps between circles)

However, I am looking for something more like this:



In this case, I want to predict only 1, and I dont mind if some 0s are included but want to make sure that as many as possible 1s are predicted.

In my mind, this is effectively just widening the orange circle (the classification of 1s) in the picture.
How can i achieve this?
Instead of formulating the problem with Venn diagrams you could also look at a simple two by two table. Usually the problem is formulated graphically in a different way (see picture below from Wikipedia page). If you are interested in just predicting the occurrence of the value 1 you are just focusing on the sensitivity (or true positive rate) of your classification algorithm.
This is rater straightforward within a ROC analysis framework, you could just select a minimun value for the classifier threshold. However, this comes at the cost of very low specificity. You should also consider the cost-benefit ratio of your sensitivity/specificity results. 

https://en.m.wikipedia.org/wiki/Sensitivity_and_specificity