Can a totally ordered set with a last element but no first element exist, or is this contradictory? An example of such a set would be a set that is ordered from largest to smallest, with there being no largest element: "...... > 5 > 4 > 3 > 2 > 1" or vice versa such as - "....... < -3 < -2 < -1 < 0".
This question arises from the fact that I got into an exchange with a friend of mine regarding infinite regresses, which I asserted, in terms of causation, are logically coherent. He stated that the mathematical equivalent of such is absurd, stating the following (my paraphrasing of what I understood him to be saying):
"If you were to stipulate A as the last element such that A cannot exist within the set unless it is proceeded by B, then this is to stipulate an infinite number existing, which doesn't exist. The reason for this is that if each element depends on the element prior to it, and the elements go back infinitely, then A relies on an element infinitely prior to it, that element can either be infinite or finite, if it is finite, then it is not infinitely prior to A, meaning that it (A) relies on an actually infinite number or index, which is NaN. What I am trying to point out, is that if A is considered the last element, which each element depending on the last ad infinitum, then A must be dependent on an element infinitely prior to itself (seeing that dependency is transitive), and the index of that element must either be finite or infinite, but it cannot be finite, meaning that it must be infinite, but infinity is not a number, resulting in a problem."
This seems to make sense, but something about it feels very off, I am not a mathematician nor am I intimately familiar with mathematics (I haven't engaged with anything beyond two basic introductory discrete math classes), could someone please clarify this?