What is the myth? The myth quite simply is that the there was an American Dream that could be realised or was more likely to be realized than in other countries (especially the 'old' home countries of Northern Europe). Formally this could be rendered as:
The USA allowed for (significantly) greater social mobility than in other countries (particularly those in Western Europe from which many of the early immigrants came). This mobility might be only in any a specific area, for example referring only to the extreme case of progress from rags to riches, or it might be the more common situation of poor immigrant to self-respecting independent yeoman farmer.