Reasoning about infinity is tricky to do consistently, and being un-comfortable with an idea is not the same thing as that idea being illogical. It is kind of a tangent, but I'm reminded of a puzzle posed by Martin Gardner:

The Hotel Infinity has an infinite number of rooms. Each room is numbered: 1, 2, 3, etc. A guest arrives late one night. The desk clerk greets the guest, but says "I'm sorry sir, but all our rooms are full." It is a dark and stormy night, the guest is tired, and doesn't want to look for another hotel, so he tells the clerk: "Look, move the person in room 1 into room 2, move the person in room 2 into room 3, move the person in room 3 into room 4, ..., and so on. Now you have a vacanct room." The clerk agrees and so is able to accomodate the guest.

After making the arrangments and returning to the desk the clerk is horrified to find an infinite number of new guests! "No problem" the new guests point out, "Just move the person in room 1 to room 2, move the person in room 2 to room 4, the person in room 3 to room 6, and so on". After this shuffle all of the odd rooms are free, and since there are an infinite number of odd numbers all of the new guests can be housed.

Would it help you could to think of the natural numbers (1, 2, 3, ...) as an example of an entity with a finite origin but infinite extent?