# On the definition of Inductive set. Equivalences

As far as I know, the definition of Inductive set is given by the Axiom of infinity:

A set $S$ is called inductive if and only if $\emptyset \in S$ and for ecah $x\in S$, $x\cup \{x\}\in S$.

With this definition, I can understand more or less what it is.

Now, let me introduce the definition of an inductive set given by my book. I need a previous definition (J.L. Kelley 141 p. 273).

Nest. A partial ordered class $N$ is called a nest if and only if for all $a,b\in N$, then $aRb$ or $bRa$.

Remark. Kelley introduces nests with the partial order $\subseteq$, so I suppose I must add the condition or $x=y$ to my previous definition. Also note that, a priori, this definition is the same of trichotomy of the partial order $R$.

Now, the definition of inductive set (not stated in Kelley's book):

Inductive class. A class $C$ is inductive if and only if and only if every nest $N\subseteq C$ has an upper bound.

I have two comments about that definition, each of them with one question:

1.- First, I think with this definition $\mathbb N$ is not an inductive set. Because if I set $N=\mathbb N$, then there isn't an upper bound. I'm right? This fact also conditions my second comment.

2.- Even when I were right in 1.- and $\mathbb N$ is not inductive, there is an equivalence between both definitions? If I'm right in 1. the definitions can't be equivalent. But you know, the fact $\mathbb N$ is not an inductive set may be a little problem. You can consider $\mathbb N\cup\{\infty\}$. Has the second definition more important (and unsolvable) problems? If not, can you show the equivalence?

Thanks.

• These are two different concepts. Feb 17, 2018 at 22:20
• OK. So, with the usual definition, it isn't true that every inductive set has a maximal element (Zorn's lemma), is it? Feb 17, 2018 at 22:47
• $\mathbb N$ is a counterexample. Feb 17, 2018 at 22:49
• Perfect. Thanks a lot. Finally: is there another name for what he calls inductive sets? Feb 17, 2018 at 22:57
• I am not sure there is a standard name. Sometimes people call them c.p.o.s, or complete partial orders, for instance in work on semantics of programming languages. But, as the link indicates, there are many variants. Feb 17, 2018 at 23:16

The two concepts are different. For example, $\omega$, the first infinite ordinal, is the standard example of an inductive set according to the first definition, but is not inductive in the second sense. In fact, no set can be inductive in both senses (any such putative set would contain all ordinals).