Formatting help

Question

I am having a problem with parts being italicized that aren't supposed to be, and am struggling to make the list both line up as in start from the same point on the left. Trying to figure out how to make it a list of (a) and (b). Also, I would like to stop tex from labeling the theorem so it would be labelled as Theorem 3.1.2.

\newtheorem{theorem}[]{Theorem 3.1.2}
    \begin{theorem}
        Let $P(n)$ be a statement that is either true or false for each n \in \mathbb{N}. Then $P(n)$ is true for all $n$ \in \mathbb{N}, provided that
            \begin{enumerate}
                \item $P(n)$ is true, and
                \item for each $k$\in\mathbb{N}, if $P(k)$ is true, then $P(k+1)$ is true.
            \end{enumerate}
    \end{theorem}   

Answer 1

There are several issues here. The dollar sign is a toggle between math mode and text mode, so keep entire math expressions between them.

$n$ \in \mathbb{N}

should be

$n \in \mathbb{N}$

etc. You can also use \( to enter math mode and \) to leave math mode, so the code

\(n \in \mathbb{N}\)

will produce the same as above. Since you use \mathbb I am assuming you are loading the amsfonts package. In the future you should include this in your post as part of a "Minimal Working Example" (MWE).

The theorem is mostly typeset in italics because that is the default for amsthm, known as \theoremstyle{plain}. If you want your theorem to appear in roman (upright) style, use \theoremstyle{definition}.

A simple way to get your custom theorem number is to use the starred version of \newtheorem, as in \newtheorem*{thm*}{Theorem} (which will suppress the theorem number), and then add your own number. For a more robust solution, look at this post on custom theorem numbering.

Lastly, there are several ways to enumerate with letters instead of numbers. One is suggested in the comment by @marmot, below. Another is to use the enumitem package and add \setenumerate[0]{label=(\alph*)} to the preamble.

Here is sample code to get you started.

\documentclass{article}
\usepackage{amsmath,amsthm,amsfonts}
\usepackage{enumitem}

\setenumerate[0]{label=(\alph*)}

\theoremstyle{definition}
\newtheorem*{thm*}{Theorem}

\begin{document}

\begin{thm*}[\textbf{3.1.2}]
    Let $P(n)$ be a statement that is either true or false for each $n \in \mathbb{N}$. Then $P(n)$ is true for all $n \in \mathbb{N}$, provided that
        \begin{enumerate}
            \item $P(n)$ is true, and
            \item for each $k\in\mathbb{N}$, if $P(k)$ is true, then $P(k+1)$ is true.
        \end{enumerate}
\end{thm*} 

\end{document}