Why did Dumbledore say the sword was the only known relic of Gryffindor?

Question

In the Half-Blood Prince, Dumbledore explains his theories to Harry. He shows him memories about Voldemort's youth, his interest in Horcruxes, his interest toward Hogwarts's founders and how he got some of their old items.

He also explains that Voldemort would probably have seeked then something belonging to Ravenclaw and something to Gryffindor.

"The remaining two, assuming again that he created a total of six, are more of a problem, but I will hazard a guess that, having secured objects from Hufflepuff and Slytherin, he set out to track down objects owned by Gryffindor or Ravenclaw. Four objects from the four founders would, I am sure, have exerted a powerful pull over Voldemort's imagination. I cannot answer for whether he ever managed to find anything of Ravenclaw's. I am confident, however, that the only known relic of Gryffindor remains safe."

Dumbledore pointed his blackened fingers to the wall behind him, where a ruby-encrusted sword reposed within a glass case.

(Half Blood Prince, Chapter 23, Horcruxes) - emphasis mine

He's obviously telling Harry that the only know relic of Gryffindor is his sword, and that it's safe in his office.

But we do know about another item that belonged to Gryffindor himself: The Sorting hat told once in his (its?) song that he used to be Gryffindor's.

While still alive they did divide
Their favorites from the throng,
Yet how to pick the worthy ones
When they were dead and gone?
Twas Gryffindor who found the way,
He whipped me off his head
The founders put some brains in me
So I could choose instead!
(Goblet of Fire, Chapter 12, The Triwizard tournament) - emphasis mine

Surely Dumbledore was perfectly aware of this: he was present when the hat sang this, and I'm sure he even knew it before (he had lived in Hogwarts for... almost ever, and more specifically, had shared a room (his office) with the Hat for decades.

Did he forget? (Dumbledore??) Did he purposely omit to count the Hat? (but then why?)

(Of course, this would not have changed a lot of things, the Hat was just as safe as the sword, in the same office, but he could have said "I'm confident the only two known relics of Gryffindor remain safe" then pointed at each one of them, and I don't understand why he didn't...)

Answer 1

Hard to tell, but there are some possibilities

The Hat is a person, not a relic

This might be nitpicking, but the Hat possesses a considerable degree of sentience. It is possible that Dumbledore did not consider the Hat a relic for this reason.

The Hat isn't grand enough to be a relic

Perhaps Voldemort would not have wanted to put his soul into a dirty old hat. Or perhaps he would have disdained the hat as simply a tool for selecting children. He does tend to underestimate a lot of magic. It is even possible that Dumbledore does not consider it a relic in this sense.

The Hat belongs to all the Founders

Indeed, perhaps Voldemort would have considered it unsuitable for completing the set for that reason alone.

(Credit to Himarm)

Answer 2

The Sorting Hat was part of Hogwarts, not solely Gryffindor.

In the Sorting Hat’s song in Harry’s first year, the Hat refers to himself as “the Hogwarts Sorting Hat”, so he considers himself part of Hogwarts as a whole.

“You can keep your bowlers black,
Your top hats sleek and tall,
For I’m the Hogwarts Sorting Hat
And I can cap them all.”
_{^{- Harry Potter and the Philosopher's Stone, Chapter 7 (The Sorting Hat)}}

So, despite originally being Godric Gryffindor’s hat, the Sorting Hat considers himself of all of Hogwarts - and since he clearly had intelligence, a mind of his own, and knew all four founders and their original intentions, I think we can trust him to decide “whose” he is.

In addition, Tom Riddle himself considered the Sorting Hat to belong to Hogwarts rather than Godric Gryffindor specifically, which is significant since he’s the one who would grow up intending to collect an object from each of the four founders to use as his Horcruxes.

“And that –’ said Riddle, now eyeing the ragged thing that Fawkes had dropped, ‘that’s the old school Sorting Hat.’

So it was. Patched, frayed and dirty, the Hat lay motionless at Harry’s feet.”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)}}

He didn’t seem to consider it a “Gryffindor relic” either, so wouldn’t use it to complete the collection.

Answer 3

A relic is something that is imbued with the power of the creator and only the creator. Also, it is sometthing that is explicitly connected to the individual. For example the use of the term in the Catholic church as some personal memorial of a saint, martyr, or other sacred person, preserved as worthy of veneration.

The Sorting Hat is not a "relic" in that sense, it is a tool created by the four founders to be part of the school. As such, it would not be a memorial of Griffindor as is the sword, or the other objects of each of the founders. Also, each of the relics symbolized the basic character of the founder that it was attached to and had powers based on the essential character of that founder.

Answer 4

Dumbledore left out the Sorting Hat because it wasn't relevant, and he wanted Harry to remember the Sword

The Sorting Hat could be considered a Relic of Gryffindor. It could definitely be considered a relic of Hogwarts, which would be desirable to Voldemort. But its location has been known and tracked for centuries, and Voldemort never had the opportunity for to make it a Horcrux.

This means that it isn't necessary for Dumbledore to mention it, except for the purposes of completion and accuracy (neither of which Dumbledore has shown much adherence to in the past), and this conversation has a secondary purpose that bringing up the Hat would distract from.

Dumbledore knows that the Sword will be a useful tool for destroying Horcruxes. For whatever reason he doesn't want to actually mention this fact to Harry, but drawing Harry's attention to the Sword here means that Harry is more likely to remember the Sword when the time comes to use it. Mentioning the Hat would distract from this purpose.