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Why is there an h in "ghost", when there is none in "go" or "god"?
You might be tempted, looking at the word "ghoul", which we also see a lot at this time of year, to leap to the etymological conclusion that the words are connected. But you would be wrong. "Ghoul" comes from the Arabic ghūl , from a verbal root meaning ‘to seize’, and entered English only in the late 1700s.
"Ghost", on the other hand, dates back to the Anglo-Saxons, and was spelled without an H by them and for centuries afterward. For them, the word meant "soul", a usage which now survives only in the phrase "give up the ghost". It was also used for the Spirit of God, a usage that survives, but only barely, in some Christian denominations who still refer to the Holy Ghost. Most, however, have switched to "Holy Spirit", probably because generations of Sunday School students have been perplexed by what Casper is doing as part of the Holy Trinity. The now predominant "spectre" sense dates from the late 1300s.
We owe this particular orthographic annoyance to the printer Caxton, who set up the first printing press in England in 1476. He had discovered printing in Flanders, and was probably influenced by the Flemish spelling of the word, gheest.
Despite the influence of the printers in standardizing spellings, though, it is quite surprising that this quirky spelling established itself. Its ultimate success can perhaps be ascribed to the trend in the 1500s to insert silent letters into words to reflect their etymology. Not knowing any better, people perhaps thought that the new "h" in "ghost" was like the new "b" in "debt" or the new "p" in "receipt", which evoked the Latin origins of those words.
Caxton also tried to impose this (to quote the Oxford English Dictionary) "capricious substitute for g" in ghoos , ghoot , gherle, which you will probably not recognize as "goose", "goat", and "girl". Obviously (and fortunately!) he did not succeed with these.
For the perennial question whether to spell Halloween with an apostrophe or not, see this post and for the pronunciation, see this post. For the etymology of "pumpkin", see this post.