Hi all,
I have a question that came about from my Foundations of Mathematics class. It's not a homework question, but trying to understand something said in the lecture. I tried asking my professor about it, but his answer actually confused me more. I'll try and explain the best I can... We started with what the textbook calls the "axiom of infinity" to define a new set:
\(\displaystyle \vdash \exists \mathbb{I}[(\emptyset \in \mathbb{I}) \wedge (\forall C\left\{ (C \in \mathbb{I}) \implies [(C \cup {C}) \in \mathbb{I}]\right\})]\)
I understand that this says that the empty set is an member of this set; and for every set inside \(\displaystyle \mathbb{I}\), the new set produced by the union of that set with a singleton containing that element is also a member of \(\displaystyle \mathbb{I}\). We then continued to define the natural numbers:
We also defined the entire set of natural numbers such that \(\displaystyle \mathbb{N} := \bigcap \mathbb{I}\). I follow up to this point as well. But here's where I get lost. It was then established that \(\displaystyle \mathbb{N} \in \mathbb{I}\), meaning that \(\displaystyle \bigcap \mathbb{I} \in \mathbb{I}\). I don't understand how that can be possible. My thought process for why it's impossible is as follows. To know what's in the intersection of \(\displaystyle \mathbb{I}\), I first unravel the definition:
\(\displaystyle \bigcap S := \left\{ X \in \bigcup S \: : \: \forall Y [(Y \in S) \implies (X \in Y)] \right\}\)
So I now need to know what's inside the union. Unraveling the definition of union leads to:
\(\displaystyle \forall X[(X \in \bigcup S) \iff \left\{ \exists Y [(Y \in S) \wedge (X \in Y)] \right\}]\)
At this point, I think I'm stuck in an infinite loop. Let's say that I have some set comprised of other sets:
\(\displaystyle S := \left\{ \left\{2, 3, 5, 7 \right\}, \left\{4, 6, 8, 9 \right\}, \bigcap S \right \}\)
Accordingly the union would be:
\(\displaystyle \bigcup S = \left\{ 2, 3, 4, 5, 6, 7, 8, 9, ??? \right \}\)
It's here that I get stuck in an infinite loop. To know what's in the union of S I need to extract any elements of the intersection of S, but that requires me to first know what's in the union...
When I asked about this, my professor said something about how it was okay because we'd stop when got to the intersection of S because it's the smallest set that's a member of S. But I don't know what the heck that means. Any help would be greatly appreciated. I'm just so very confused right now, and the textbook and my professor and even Google have failed me.
I have a question that came about from my Foundations of Mathematics class. It's not a homework question, but trying to understand something said in the lecture. I tried asking my professor about it, but his answer actually confused me more. I'll try and explain the best I can... We started with what the textbook calls the "axiom of infinity" to define a new set:
\(\displaystyle \vdash \exists \mathbb{I}[(\emptyset \in \mathbb{I}) \wedge (\forall C\left\{ (C \in \mathbb{I}) \implies [(C \cup {C}) \in \mathbb{I}]\right\})]\)
I understand that this says that the empty set is an member of this set; and for every set inside \(\displaystyle \mathbb{I}\), the new set produced by the union of that set with a singleton containing that element is also a member of \(\displaystyle \mathbb{I}\). We then continued to define the natural numbers:
- \(\displaystyle 0 := \emptyset\)
- \(\displaystyle 1 := 0 \cup \left\{ 0 \right\}\)
- \(\displaystyle 2 := 1 \cup \left\{ 1 \right\}\)
- ...
- \(\displaystyle n + 1 := n \cup \left\{ n \right\}\)
We also defined the entire set of natural numbers such that \(\displaystyle \mathbb{N} := \bigcap \mathbb{I}\). I follow up to this point as well. But here's where I get lost. It was then established that \(\displaystyle \mathbb{N} \in \mathbb{I}\), meaning that \(\displaystyle \bigcap \mathbb{I} \in \mathbb{I}\). I don't understand how that can be possible. My thought process for why it's impossible is as follows. To know what's in the intersection of \(\displaystyle \mathbb{I}\), I first unravel the definition:
\(\displaystyle \bigcap S := \left\{ X \in \bigcup S \: : \: \forall Y [(Y \in S) \implies (X \in Y)] \right\}\)
So I now need to know what's inside the union. Unraveling the definition of union leads to:
\(\displaystyle \forall X[(X \in \bigcup S) \iff \left\{ \exists Y [(Y \in S) \wedge (X \in Y)] \right\}]\)
At this point, I think I'm stuck in an infinite loop. Let's say that I have some set comprised of other sets:
\(\displaystyle S := \left\{ \left\{2, 3, 5, 7 \right\}, \left\{4, 6, 8, 9 \right\}, \bigcap S \right \}\)
Accordingly the union would be:
\(\displaystyle \bigcup S = \left\{ 2, 3, 4, 5, 6, 7, 8, 9, ??? \right \}\)
It's here that I get stuck in an infinite loop. To know what's in the union of S I need to extract any elements of the intersection of S, but that requires me to first know what's in the union...
When I asked about this, my professor said something about how it was okay because we'd stop when got to the intersection of S because it's the smallest set that's a member of S. But I don't know what the heck that means. Any help would be greatly appreciated. I'm just so very confused right now, and the textbook and my professor and even Google have failed me.