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Diffstat (limited to 'mindmap/natural number.org')
| -rw-r--r-- | mindmap/natural number.org | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/mindmap/natural number.org b/mindmap/natural number.org index 9861df6..8b3f31a 100644 --- a/mindmap/natural number.org +++ b/mindmap/natural number.org @@ -13,8 +13,8 @@ I will start with the Peano arithmetic formulation. First, we define an immediat successor function $S:\mathbb{N}\rightarrow\mathbb{N}$ which effectively "adds one" (although we haven't defined addition yet), and a number we call $0 \in \mathbb{N}$. We then define an axiom: \begin{align*} -\forall n \in \mathbb{N} \; \nexists S(n) \; s.t. S(n) = 0; \\ -\forall n \in \mathbb{N} \; S(n) \in \mathbb{N}. +\forall n \in \mathbb{N}, \nexists S(n), s.t. S(n) = 0; \\ +\forall n \in \mathbb{N}, S(n) \in \mathbb{N}. \end{align*} which is equivalent to saying: adding one to any natural number makes it not equal to zero, and any natural number's successor is a natural number. Because zero is a natural number, we can define @@ -42,7 +42,7 @@ however, they don't allow for the ability for us to extrapolate properties of na ** [[id:16b06b82-99cc-4343-b171-fb2166c46a30][Induction]] Let's introduce our last axiom: \begin{align*} -\forall n \in \mathbb{N} \: \forall P(n) \; P(0) \land (P(n) \rightarrow P(S(n))) \rightarrow P(n) \; \forall n \in \mathbb{N} +\forall n \in \mathbb{N}, \forall P(x), P(0) \land (P(x) \rightarrow P(S(x))) \rightarrow P(n) \end{align*} now, this is the principle of [[id:16b06b82-99cc-4343-b171-fb2166c46a30][induction]] specific to natural numbers. What it is saying is that a property $P(n)$ is true for all $n$ if there is a "base case" $P(0)$ which is true, and you can show that $P(1)$ is |