# Conceptual questions about the Principle of Infinite Descent

I was recently encouraged to develop a deeper understanding of the Principle of Infinite Descent so that I could become more comfortable with its usage in proofs. I have a few questions that I wanted to run by the community in order to confirm that I have sufficiently absorbed the concept.

The below excerpt, taken from http://elib.mi.sanu.ac.rs/files/journals/tm/31/tm1622.pdf, provides this statement: Given what we know about the Well-Ordering Principle for Natural Numbers, this statement is logically sound.

Equipped with the above logical statement, the aforementioned math website goes on to prove the following proposition:

There is no infinite strictly decreasing sequence of natural numbers

This proposition is proven using the following argument: Firstly, I wanted to confirm that the reason "we know" $$1 \notin A$$ (where $$1$$ here is functioning as the $$0$$ that I am more familiar with) is that by Peano's Axioms and the Well-Ordering Principle, there is no element smaller than $$1$$.

Therefore, if $$1$$ WAS a member of set $$A$$, there could be no element less than $$1$$ to continue the infinite sequence...which would necessarily make $$A$$ finite (contradiction). So, to avoid contradiction, the author moves on with the claim $$1 \notin A$$.

Secondly, why is the initial assumption that "$$\exists$$ an infinite set $$A$$" an acceptable assumption? Is it purely because we know that there are an infinite number of natural numbers and therefore, by defining members of $$A$$ as "$$a_n \in \mathbb N$$ and $$n \in \mathbb N$$" we recognize that the natural number indexing of the elements permits an infinite list?

Finally, I recognize the contradiction. Specifically, we arrive at the simultaneous conclusion that the statement $$p(n)$$ is true for $$\forall n \in \mathbb N$$ (i.e. no elements of $$n$$ are in $$A$$) while claiming the existence of an infinite set that contains members of $$\mathbb N$$ (and therefore provide infinitely many instances of $$n$$'s where $$p(n)$$ is false). My question is thus, "What is the negation that is performed on the initial assumption"? Is it simply that The set $$A$$ does not exist? Consequently, given the properties attached to $$A$$, the conclusion is:

No infinitely sized set can be constructed from strictly decreasing elements selected from the $$\mathbb N$$.

Is this all correct? Cheers~

Regarding your first question, yes that is precisely the reason we know $$1 ∉ A$$.
Regarding your second question, the existence of an infinite set is typically an axiom in whatever system you are working in. Since I see that the $$∈$$ relation is being used in the text that you cite, I am assuming that the system is something at least adjacent to ZFC, for which you can find a list of its axioms (including the axiom of infinity). What you state is more or less correct. $$ℕ$$ (but actually $$ω$$) is constructed with the axiom of infinity as the intersection of all inductive sets, and from there we can define functions with the whole of (or only a part of) $$ℕ$$ (but really $$ω$$) as the domain. So when we write $$\{a_{1},a_{2},\dots \}$$ we are in some sense saying that there is a function with $$\{1,2,\dots \}$$ as the domain and $$\{a_{1},a_{2},\dots \}$$ as the range.
Your last comment is correct in that the negation of the initial assumption is that the set $$A$$ does not exist. Your observation of the consequence is also correct in the system I assume we are working in.