Alright.
[s]I’m inclined to think the word “endless” actually means “without an end”. If “mindless” means “without a mind”, it’s only sane to assume that “endless” means “without an end”. (And not, as you say, without a beginning.)
Google appears to be in agreement with me:
But there’s a bigger problem than that.
Even if the word “endless” means “without a beginning”, I don’t see a way to apply it to sets since the word “beginning”, just like the word “end”, is not defined with respect to sets.[/s]
I misread the above quote which is why the above is struck through.
The only part that does not need to be struck through is this last bit:
Take the set of ternary digits ({1, 2, 3}). What’s the beginning of it? And what’s the end of it? Note that sets have no order, so you can’t say the beginning of this set is (1) and the end of it is (3).
So if “infinite” means “endless”, and endless means either “without an end” or “without a beginning”, what does it mean to say that a set has no end/beginning?
It’s actually sequences, not sets, that possess the ability to have a beginning and an and.
The sequence ((1, 2, 3)) starts with (1) and ends with (3). It has a beginning (the first element) and an end (the last element.)
There are sequences that have a beginning but no end e.g. ((1, 2, 3, \dotso)). Such are necessarily infinite.
Then there are sequences that have an end but no beginning e.g. ((\dotso, 3, 2, 1)). Such are necessarily infinite just as well.
And then there are sequences that have no end and no beginning e.g. ((\dotso, -3, -2, -1, 0, +1, +2, +3, \dotso)). These are necessarily infinite just as well.
But there are ALSO infinite sequences that have a beginning and an end e.g. ((1, 3, 5, \dotso, 6, 4, 2)).
Depending on how you define the term “infinite”, it might be a contradiction to say that such sequences are “infinite”. But you can’t deny that they exhibt the same property that what we nowadays call “infinite sets” do – the number of their elements is greater than every integer.
But there’s an even bigger problem than this. You’re actually arguing that the word “infinite” means more than “endless”. You are actually saying that the word “infinite” means “a never-ending process of increase”.
The problem with such a claim is that neither sets nor sequences exist in time. They do not occupy time. Thus, they can’t be one thing at one point in time and another thing (or the same exact thing) at another point in time. They have no temporal existence. Since they have no temporal existence, they cannot change (since change is a difference between two points in time.) Since they cannot change, they cannot increase in size. Their size is fixed. The set of natural numbers (N = {1, 2, 3, \dotso}) isn’t ONE size at one point in time and ANOTHER size at another point in time.
Sequences and sets, being highly abstract concepts, have no temporal existence. But there are special (non-mathematical) kinds of sets and sequences that do have such a property.
For example, the set of apples on one’s table is something that exists through time. Another example is a train – a sequence of wagons. These things have temporal existence but they AREN’T mathematical entities. And this thread is ENTIRELY about mathematics. So there’s no point in discussing them (other than to distract.)
Yet another completely irrelevant concept (:
(Which also does not mean what you think it means.)