Philosophically, you need to make them mathematically identical in a way that doesn’t require a first element.

But I’m still not convinced that they aren’t different in a way that matters. You talked about starting at 1, for example, but since you are generating the set of all integers — including the negative ones — you’d need two functions: one that adds onto 1 to get all the positive ones, and one that subtracts from 1 to get all the negative ones. This means that the set really does run infinitely in both directions from your starting point. In theory, you could pick any integer and generate infinitely in both directions. You CAN’T do that for my dependent set.

This leads to another question: the reason we can start at 1 is that we can determine that 1 is an integer without having to generate anything in the set first. If we choose 1, that’s not an arbitrary choice, but is in fact us choosing something that is an integer. That’s why we could in theory pick ANY integer as the starting point, but can’t do that for my example.

Note that the first argument DOESN’T apply to the set of positive or negative integers, but then those sets DO have a set starting point. The second argument — which just supports my initial argument — DOES apply to them.

The big mathematical question, though, is this: if I pick a number at random — say 53 — how would you determine if that is or isn’t a integer? I doubt you’d generate or even reference the set specifically itself. But in my example, if you wanted to see if it was a, say, Stoicistic number, you’d have to reference the set, generate it, and stipulate the first number in the set.

]]>You are correct that I didn’t disprove your argument in favor of the cosmological argument. I mostly argue that there isn’t any mathematical difference between the set of all integers and the infinite sets you call “dependent”. The cosmological consequences is something I’d have to think more about, before I say anything. I’ve studied mathematics, but I’m pretty new to philosophy.

Thanks for a nice discussion!

]]>An integer (from the Latin integer meaning “whole”)[note 1] is a number that can be written without a fractional component.

So, conceptually, you can take a number and see if it can be written without a whole component, and know if it should be in Z without having any other number in the set first. You can’t do that for a dependent set.

But even if you’re right, it doesn’t impact the argument, as all you’d do is show that even for those sets there must exist an element in the set whose existence is not determined by any other element in the set, which would then justify the cosmological arguments.

]]>Well, how would you do that? By looking at the numerical representation and seeing that it doesn’t have a decimal point? That’s just an artefact of notation. Mathematically, we check if x belong to Z by seeing if we can find two positive integers i and j such that x=i-j. So in order to check if x belong to Z, we need access to the set of all positive integers, named N.

And how do we define N? Well, exactly in the same way as we defined Y above, except that n(0)=1.

(Point of nitpicking: we actually start with saying that 1 belongs to N, then say that every element of N has a successor, and then define “add one” as moving from an element n(m-1) to its successor n(m). Same thing.)

So, mathemathically, the set of all integers has a privileged first member. The number 1. (It is also possible to define the integers starting from n(0)=0, but at any rate you need a first member.)

Your move ðŸ™‚

]]>As I pointed out, when you compare that causal chain to infinite DEPENDENT sets, the math doesn’t work out as well.

]]>Thanks!

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