I cannot access vim or vi under the cygwin shell. The only available selection in the setup program under Editors is 'vim-minimal' which is allegedly installed (says 'Keep' beside it in the installer's tree view).
$ vim
/cygdrive/c/Program Files (x86)/git/bin/vim: line 3: /share/vim/vim74/vim: No such file or directory
/cygdrive/c/Program Files (x86)/git/bin/vim: line 3: exec: /share/vim/vim74/vim: cannot execute: No such file or directory
I found this nice bit of magic to try: http://simonmcc.blogspot.in/2005/06/cygwin-vim-rxvt-arrow-key-problems.html ...but got blocked because I cannot get to the dir where the vimrc file resides...
I can list the vim directory from my home dir:
$ pwd
/cygdrive/c/Users/Michael
$ ls -a /usr/share/vim
.
..
autoload
filetype.vim
ftoff.vim
menu.vim
optwin.vim
plugin
scripts.vim
syntax
tutor
vim.exe
...but when I try to go there...
$cd /usr/share/vim
bash: cd: /usr/share/vim: No such file or directory
...and trying to sneak up on it, I find...
$ cd /usr/share
$ ls -a
.
..
awk
bash-completion
doc
file
groff
info
locale
magic
man
misc
p11-kit
pki
tabset
terminfo
zoneinfo
zoneinfo-leaps
zoneinfo-posix
No 'vim' directory! Gak!
Package vim-miminal provides vi, package vim provides vim. They are both there in my setup program. Don't just look under 'Editors', try searching for 'vim' in the search box at the top of the setup program window.
Your setup program says you have vi, what happens when you type vi or vi.exe in the console?
I don't know what's going on with your /usr/share/vim directory, or why there's a vim.exe in your /usr/share directory, that's all really odd. vim.exe should be in /usr/bin. What happens when you do which vi or which vim?
The cygcheck program checks the status of Cygwin packages. cygcheck -c vim should return with 'OK' if it's installed correctly, nothing if not at all, or 'Incomplete' if something is wrong.
Try grabbing a new setup-x86_64.exe and running setup again, and/or choose another mirror. There should be both a vim and a vim-common package (as well as gvim and vim-latex).